Abstract group $C_2^2\times C_{42}$

• Label: 168.57
• Order: \( 2^{3} \cdot 3 \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Abelian: yes
• $\card{\operatorname{Aut}(G)}$: \( 2^{5} \cdot 3^{2} \cdot 7 \)
• Perm deg.: $16$
• Trans deg.: $168$
• Rank: $3$

Group information

Description:	$C_2^2\times C_{42}$
Order:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$C_2\times C_6\times \GL(3,2)$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) (generators)
Outer automorphisms:	$C_2\times C_6\times \GL(3,2)$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_7$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	7	2	14	6	42	12	84	168
Conjugacy classes	1	7	2	14	6	42	12	84	168
Divisions	1	7	1	7	1	7	1	7	32
Autjugacy classes	1	1	1	1	1	1	1	1	8

Dimension	1	2	6	12
Irr. complex chars.	168	0	0	0	168
Irr. rational chars.	8	8	8	8	32

Minimal Presentations

Permutation degree:	$16$
Transitive degree:	$168$
Rank:	$3$
Inequivalent generating triples:	$741$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	4	9

Constructions

Presentation:	Abelian group $\langle a, b, c \mid a^{2}=b^{2}=c^{42}=1 \rangle$
Permutation group:	Degree $16$ $\langle(1,2), (3,4), (5,6), (7,9,8), (10,16,15,14,13,12,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} z_{6}^{5} + z_{6}^{3} + z_{6}^{2} & z_{6}^{5} + z_{6}^{4} + z_{6}^{2} + z_{6} \\ z_{6}^{4} + z_{6}^{3} + z_{6}^{2} + z_{6} + 1 & z_{6}^{5} + z_{6}^{3} + z_{6}^{2} \end{array}\right), \left(\begin{array}{rr} z_{6}^{5} + z_{6}^{2} & z_{6}^{5} + z_{6}^{3} \\ z_{6}^{5} + z_{6}^{3} + z_{6}^{2} + z_{6} & z_{6}^{5} + z_{6}^{2} \end{array}\right), \left(\begin{array}{rr} z_{6}^{5} + z_{6} & 0 \\ 0 & z_{6}^{5} + z_{6} \end{array}\right), \left(\begin{array}{rr} z_{6}^{5} + z_{6}^{3} + z_{6}^{2} + z_{6} & z_{6}^{5} + z_{6}^{3} + z_{6}^{2} + z_{6} + 1 \\ z_{6}^{5} + z_{6}^{2} & z_{6}^{5} + z_{6}^{3} + z_{6}^{2} + z_{6} \end{array}\right), \left(\begin{array}{rr} z_{6}^{3} + z_{6}^{2} + z_{6} & 0 \\ 0 & z_{6}^{3} + z_{6}^{2} + z_{6} \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{64})$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $C_3$ $\, \times\, $ $C_7$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_{344})$	$\Aut(C_{392})$	$\Aut(C_{516})$	$\Aut(C_{588})$

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Primary decomposition:	$C_{2}^{3} \times C_{3} \times C_{7}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$0$

Subgroups

There are 64 subgroups, all normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{42}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2^2\times C_{42}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times C_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^2\times C_{42}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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

Classes of subgroups up to conjugation

Order 168: $C_2^2\times C_{42}$

Order 84: $C_2\times C_{42}$ x 7

Order 56: $C_2^2\times C_{14}$

Order 42: $C_{42}$ x 7

Order 28: $C_2\times C_{14}$ x 7

Order 24: $C_2^2\times C_6$

Order 21: $C_{21}$

Order 14: $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 7

Order 8: $C_2^3$

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 168: $C_2^2\times C_{42}$

Order 84: $C_2\times C_{42}$

Order 56: $C_2^2\times C_{14}$

Order 42: $C_{42}$

Order 28: $C_2\times C_{14}$

Order 24: $C_2^2\times C_6$

Order 21: $C_{21}$

Order 14: $C_{14}$

Order 12: $C_2\times C_6$

Order 8: $C_2^3$

Order 7: $C_7$

Order 6: $C_6$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 168: $C_2^2\times C_{42}$ ($C_1$)

Order 84: $C_2\times C_{42}$ ($C_2$) x 7

Order 56: $C_2^2\times C_{14}$ ($C_3$)

Order 42: $C_{42}$ ($C_2^2$) x 7

Order 28: $C_2\times C_{14}$ ($C_6$) x 7

Order 24: $C_2^2\times C_6$ ($C_7$)

Order 21: $C_{21}$ ($C_2^3$)

Order 14: $C_{14}$ ($C_2\times C_6$) x 7

Order 12: $C_2\times C_6$ ($C_{14}$) x 7

Order 8: $C_2^3$ ($C_{21}$)

Order 7: $C_7$ ($C_2^2\times C_6$)

Order 6: $C_6$ ($C_2\times C_{14}$) x 7

Order 4: $C_2^2$ ($C_{42}$) x 7

Order 3: $C_3$ ($C_2^2\times C_{14}$)

Order 2: $C_2$ ($C_2\times C_{42}$) x 7

Order 1: $C_1$ ($C_2^2\times C_{42}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 168: $C_2^2\times C_{42}$ ($C_1$)

Order 84: $C_2\times C_{42}$ ($C_2$)

Order 56: $C_2^2\times C_{14}$ ($C_3$)

Order 42: $C_{42}$ ($C_2^2$)

Order 28: $C_2\times C_{14}$ ($C_6$)

Order 24: $C_2^2\times C_6$ ($C_7$)

Order 21: $C_{21}$ ($C_2^3$)

Order 14: $C_{14}$ ($C_2\times C_6$)

Order 12: $C_2\times C_6$ ($C_{14}$)

Order 8: $C_2^3$ ($C_{21}$)

Order 7: $C_7$ ($C_2^2\times C_6$)

Order 6: $C_6$ ($C_2\times C_{14}$)

Order 4: $C_2^2$ ($C_{42}$)

Order 3: $C_3$ ($C_2^2\times C_{14}$)

Order 2: $C_2$ ($C_2\times C_{42}$)

Order 1: $C_1$ ($C_2^2\times C_{42}$)

Series

Derived series	$C_2^2\times C_{42}$	$\rhd$	$C_1$
Chief series	$C_2^2\times C_{42}$	$\rhd$	$C_2\times C_{42}$	$\rhd$	$C_{42}$	$\rhd$	$C_{21}$	$\rhd$	$C_7$	$\rhd$	$C_1$
Lower central series	$C_2^2\times C_{42}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2^2\times C_{42}$

Supergroups

This group is a maximal subgroup of 39 larger groups in the database.

This group is a maximal quotient of 19 larger groups in the database.

Character theory

Complex character table

See the $168 \times 168$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

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