LMFDB - Abstract group 2016.b: $C

Show commands: Gap / Magma

Group information

Description:	$C_6\times \PGL(2,7)$
Order:	$2016$ $\medspace = 2^{5} \cdot 3^{2} \cdot 7$
Exponent:	$168$ $\medspace = 2^{3} \cdot 3 \cdot 7$
Automorphism group:	$C_2^2\times \PGL(2,7)$ , of order $1344$ $\medspace = 2^{6} \cdot 3 \cdot 7$ (generators)
Outer automorphisms:	$C_2^2$ , of order $4$ $\medspace = 2^{2}$
Composition factors:	$C_2$ x 2, $C_3$ , $\GL(3,2)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	14	21	24	42
Elements	1	99	170	84	702	48	168	168	48	96	336	96	2016
Conjugacy classes	1	5	5	2	19	1	4	4	1	2	8	2	54
Divisions	1	5	3	2	11	1	2	2	1	1	2	1	32
Autjugacy classes	1	4	3	2	8	1	2	2	1	1	2	1	28

Dimension	1	2	6	7	8	12	14	16	24
Irr. complex chars.	12	0	18	12	12	0	0	0	0	54
Irr. rational chars.	4	4	2	4	4	4	4	4	2	32

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$828$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	6	8	8

Constructions

Show commands: Gap / Magma

Groups of Lie type:	$\CO(3,7)$
Permutation group:	Degree $13$ $\langle(9,10,11)(12,13), (1,2,4,5,3,6), (2,4,5)(6,7,8), (1,3)(2,5)(7,8)(12,13)\rangle$ (9,10,11)(12,13), (1,2,4,5,3,6), (2,4,5)(6,7,8), (1,3)(2,5)(7,8)(12,13)
comment:Define the group as a permutation group magma:G := PermutationGroup< 13 \| (9,10,11)(12,13), (1,2,4,5,3,6), (2,4,5)(6,7,8), (1,3)(2,5)(7,8)(12,13) >; gap:G := Group( (9,10,11)(12,13), (1,2,4,5,3,6), (2,4,5)(6,7,8), (1,3)(2,5)(7,8)(12,13) );
Direct product:	$C_2$ $\, \times\,$ $C_3$ $\, \times\,$ $\PGL(2,7)$
Semidirect product:	$(C_6\times \GL(3,2))$ $\,\rtimes\,$ $C_2$	$(C_2\times \GL(3,2))$ $\,\rtimes\,$ $C_6$	$\GL(3,2)$ $\,\rtimes\,$ $(C_2\times C_6)$	$(C_3\times \GL(3,2))$ $\,\rtimes\,$ $C_2^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_7\times \GL(3,2))$	$\Aut(C_9\times \GL(3,2))$	$\Aut(C_7\times \SL(2,7))$	$\Aut(C_7\times \PGL(2,7))$	all 7

Elements of the group are displayed as matrices in $\CO(3,7)$ .

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 3362 subgroups in 157 conjugacy classes, 14 normal (10 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $\PGL(2,7)$
Commutator:	$G' \simeq$ $\GL(3,2)$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_6\times \PGL(2,7)$
Fitting:	$\operatorname{Fit} \simeq$ $C_6$	$G/\operatorname{Fit} \simeq$ $\PGL(2,7)$
Radical:	$R \simeq$ $C_6$	$G/R \simeq$ $\PGL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_6\times \GL(3,2)$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Subgroup information

Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 2016:

C_6\times \PGL(2,7)

Order 1008:

C_3\times \PGL(2,7)

x 2,

C_6\times \GL(3,2)

Order 672:

C_2\times \PGL(2,7)

Order 504:

C_3\times \GL(3,2)

Order 336:

\PGL(2,7)

x 2,

C_2\times \GL(3,2)

Order 252:

C_6\times F_7

Order 168:

\GL(3,2)

Order 144:

C_6\times S_4

Order 126:

C_3\times F_7

x 2,

C_{21}:C_6

Order 96:

C_6\times D_8

Order 84:

C_2\times F_7

x 3,

C_3\times D_{14}

Order 72:

C_3\times S_4

x 2,

C_6\times D_6

C_6\times A_4

Order 63:

C_{21}:C_3

Order 48:

C_3\times D_8

x 4,

C_6\times D_4

x 2,

C_2\times S_4

C_2\times C_{24}

Order 42:

F_7

x 6,

C_7:C_6

x 3,

C_3\times D_7

x 2,

C_{42}

Order 36:

C_6\times S_3

x 6,

C_6^2

C_3\times A_4

Order 32:

C_2\times D_8

Order 28:

D_{14}

Order 24:

C_3\times D_4

x 6,

C_{24}

x 2,

C_2^2\times C_6

x 2,

C_2\times A_4

x 2,

S_4

x 2,

C_2\times C_{12}

C_2\times D_6

Order 21:

C_7:C_3

x 3,

C_{21}

Order 18:

C_3\times S_3

x 4,

C_3\times C_6

x 3

Order 16:

D_8

x 4,

C_2\times D_4

x 2,

C_2\times C_8

Order 14:

D_7

x 2,

C_{14}

Order 12:

C_2\times C_6

x 9,

D_6

x 6,

A_4

x 2,

C_{12}

x 2

Order 9:

C_3^2

Order 8:

D_4

x 6,

C_2^3

x 2,

C_8

x 2,

C_2\times C_4

Order 7:

C_7

Order 6:

C_6

x 11,

S_3

x 4

Order 4:

C_2^2

x 7,

C_4

x 2

Order 3:

C_3

x 3

Order 2:

C_2

x 5

Order 1:

C_1

Classes of subgroups up to automorphism

Order 2016:

C_6\times \PGL(2,7)

Order 1008:

C_6\times \GL(3,2)

C_3\times \PGL(2,7)

Order 672:

C_2\times \PGL(2,7)

Order 504:

C_3\times \GL(3,2)

Order 336:

C_2\times \GL(3,2)

\PGL(2,7)

Order 252:

C_6\times F_7

Order 168:

\GL(3,2)

Order 144:

C_6\times S_4

Order 126:

C_3\times F_7

C_{21}:C_6

Order 96:

C_6\times D_8

Order 84:

C_2\times F_7

x 2,

C_3\times D_{14}

Order 72:

C_3\times S_4

x 2,

C_6\times D_6

C_6\times A_4

Order 63:

C_{21}:C_3

Order 48:

C_6\times D_4

x 2,

C_3\times D_8

x 2,

C_2\times S_4

C_2\times C_{24}

Order 42:

C_7:C_6

x 2,

F_7

x 2,

C_{42}

C_3\times D_7

Order 36:

C_6\times S_3

x 4,

C_6^2

C_3\times A_4

Order 32:

C_2\times D_8

Order 28:

D_{14}

Order 24:

C_3\times D_4

x 5,

C_2^2\times C_6

x 2,

C_2\times A_4

x 2,

S_4

x 2,

C_2\times C_{12}

C_{24}

C_2\times D_6

Order 21:

C_7:C_3

x 2,

C_{21}

Order 18:

C_3\times S_3

x 3,

C_3\times C_6

x 2

Order 16:

D_8

x 2,

C_2\times D_4

x 2,

C_2\times C_8

Order 14:

C_{14}

D_7

Order 12:

C_2\times C_6

x 8,

D_6

x 4,

A_4

x 2,

C_{12}

x 2

Order 9:

C_3^2

Order 8:

D_4

x 5,

C_2^3

x 2,

C_2\times C_4

C_8

Order 7:

C_7

Order 6:

C_6

x 8,

S_3

x 3

Order 4:

C_2^2

x 6,

C_4

x 2

Order 3:

C_3

x 3

Order 2:

C_2

x 4

Order 1:

C_1

Normal subgroups (quotient in parentheses)

Order 2016:

C_6\times \PGL(2,7)

(

C_1

)

Order 1008:

C_3\times \PGL(2,7)

(

C_2

) x 2,

C_6\times \GL(3,2)

(

C_2

)

Order 672:

C_2\times \PGL(2,7)

(

C_3

)

Order 504:

C_3\times \GL(3,2)

(

C_2^2

)

Order 336:

\PGL(2,7)

(

C_6

) x 2,

C_2\times \GL(3,2)

(

C_6

)

Order 168:

\GL(3,2)

(

C_2\times C_6

)

Order 6:

C_6

(

\PGL(2,7)

)

Order 3:

C_3

(

C_2\times \PGL(2,7)

)

Order 2:

C_2

(

C_3\times \PGL(2,7)

)

Order 1:

C_1

(

C_6\times \PGL(2,7)

)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2016:

C_6\times \PGL(2,7)

(

C_1

)

Order 1008:

C_6\times \GL(3,2)

(

C_2

C_3\times \PGL(2,7)

(

C_2

)

Order 672:

C_2\times \PGL(2,7)

(

C_3

)

Order 504:

C_3\times \GL(3,2)

(

C_2^2

)

Order 336:

C_2\times \GL(3,2)

(

C_6

\PGL(2,7)

(

C_6

)

Order 168:

\GL(3,2)

(

C_2\times C_6

)

Order 6:

C_6

(

\PGL(2,7)

)

Order 3:

C_3

(

C_2\times \PGL(2,7)

)

Order 2:

C_2

(

C_3\times \PGL(2,7)

)

Order 1:

C_1

(

C_6\times \PGL(2,7)

)

Series

Derived series	$C_6\times \PGL(2,7)$	$\rhd$	$\GL(3,2)$
Chief series	$C_6\times \PGL(2,7)$	$\rhd$	$C_6\times \GL(3,2)$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_6\times \PGL(2,7)$	$\rhd$	$\GL(3,2)$
Upper central series	$C_1$	$\lhd$	$C_6$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $54 \times 54$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi
Siegel

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Modular curves
Higher genus families
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Belyi maps

Galois groups
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Abstract groups
Lattices

Label	2016.b
Order	$2^{5} \cdot 3^{2} \cdot 7$
Exponent	$2^{3} \cdot 3 \cdot 7$

Nilpotent	no
Solvable	no
$\card{G^{\mathrm{ab}}}$	$2^{2} \cdot 3$
$\card{Z(G)}$	$2 \cdot 3$
$\card{\mathrm{Aut}(G)}$	$2^{6} \cdot 3 \cdot 7$
$\card{\mathrm{Out}(G)}$	$2^{2}$
Perm deg.	$13$
Trans deg.	$48$
Rank	$2$

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Group information

Group statistics

Minimal presentations

Minimal degrees of faithful linear representations

Constructions

Homology

Subgroups

Special subgroups

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

Subgroup information

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups (quotient in parentheses)

Normal subgroups up to automorphism (quotient in parentheses)

Series

Supergroups

Character theory

Complex character table

Rational character table

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.