Abstract group $C_{12}.D_{16}$

• Label: 384.45
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $27$
• Trans deg.: $384$
• Rank: $2$

Group information

Description:	$C_{12}.D_{16}$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	Group of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) (generators)
Outer automorphisms:	$C_2^4\times C_4$, of order \(64\)\(\medspace = 2^{6} \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	16	24
Elements	1	3	2	12	6	144	24	96	96	384
Conjugacy classes	1	3	1	8	3	16	8	16	16	72
Divisions	1	3	1	5	3	5	5	3	4	30
Autjugacy classes	1	3	1	5	3	5	5	2	4	29

Dimension	1	2	4	8	16
Irr. complex chars.	16	44	12	0	0	72
Irr. rational chars.	4	6	10	8	2	30

Minimal Presentations

Permutation degree:	$27$
Transitive degree:	$384$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{8}=b^{16}=c^{3}=[a,c]=1, b^{a}=b^{15}, c^{b}=c^{2} \rangle$
Permutation group:	Degree $27$ $\langle(1,2,3,5,4,6,7,8)(12,13)(14,15)(16,26)(17,20)(18,21)(19,24)(22,27)(23,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 34 \\ 68 & 18 \end{array}\right), \left(\begin{array}{rr} 0 & 72 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 32 & 0 \\ 0 & 77 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 81 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 34 & 44 \\ 3 & 51 \end{array}\right), \left(\begin{array}{rr} 38 & 0 \\ 0 & 38 \end{array}\right), \left(\begin{array}{rr} 69 & 0 \\ 0 & 69 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/85\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3:C_{16})$ $\,\rtimes\,$ $C_8$ (2)	$C_3$ $\,\rtimes\,$ $(C_{16}:C_8)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $Q_{32}$	$C_{12}$ . $D_{16}$	$C_{24}$ . $\OD_{16}$	$(C_8:C_8)$ . $S_3$	all 39

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 166 subgroups in 70 conjugacy classes, 45 normal (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_3:D_8$
Commutator:	$G' \simeq$ $C_{24}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_4\times C_8$	$G/\Phi \simeq$ $D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_8:C_{24}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{12}.D_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_8:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}:C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 384: $C_{12}.D_{16}$

Order 192: $C_{12}:C_{16}$, $C_{24}:C_8$, $C_8:C_{24}$

Order 128: $C_{16}:C_8$

Order 96: $C_6:C_{16}$ x 2, $C_4:C_{24}$, $C_4\times C_{24}$, $C_{12}:C_8$

Order 64: $C_8:C_8$ x 2, $C_4\times C_{16}$

Order 48: $C_2\times C_{24}$ x 3, $C_3:C_{16}$ x 3, $C_6:C_8$, $C_4\times C_{12}$

Order 32: $C_2\times C_{16}$ x 2, $C_4:C_8$ x 2, $C_4\times C_8$

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 3, $C_3:C_8$

Order 16: $C_2\times C_8$ x 4, $C_{16}$ x 3, $C_4^2$

Order 12: $C_{12}$ x 5, $C_2\times C_6$

Order 8: $C_8$ x 5, $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

Order 4: $C_4$ x 5, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_{12}.D_{16}$

Order 192: $C_{12}:C_{16}$, $C_{24}:C_8$, $C_8:C_{24}$

Order 128: $C_{16}:C_8$

Order 96: $C_6:C_{16}$ x 2, $C_4:C_{24}$, $C_4\times C_{24}$, $C_{12}:C_8$

Order 64: $C_8:C_8$ x 2, $C_4\times C_{16}$

Order 48: $C_2\times C_{24}$ x 3, $C_3:C_{16}$ x 2, $C_6:C_8$, $C_4\times C_{12}$

Order 32: $C_2\times C_{16}$ x 2, $C_4:C_8$ x 2, $C_4\times C_8$

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 3, $C_3:C_8$

Order 16: $C_2\times C_8$ x 4, $C_{16}$ x 2, $C_4^2$

Order 12: $C_{12}$ x 5, $C_2\times C_6$

Order 8: $C_8$ x 5, $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

Order 4: $C_4$ x 5, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_{12}.D_{16}$ ($C_1$)

Order 192: $C_{12}:C_{16}$ ($C_2$), $C_{24}:C_8$ ($C_2$), $C_8:C_{24}$ ($C_2$)

Order 96: $C_6:C_{16}$ ($C_4$) x 2, $C_4\times C_{24}$ ($C_2^2$)

Order 64: $C_8:C_8$ ($S_3$)

Order 48: $C_3:C_{16}$ ($C_8$) x 2, $C_2\times C_{24}$ ($Q_8$), $C_2\times C_{24}$ ($C_2\times C_4$), $C_4\times C_{12}$ ($D_4$)

Order 32: $C_4\times C_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($Q_{16}$), $C_2\times C_{12}$ ($D_8$), $C_2\times C_{12}$ ($C_4:C_4$), $C_{24}$ ($\OD_{16}$), $C_{24}$ ($C_2\times C_8$)

Order 16: $C_2\times C_8$ ($C_4\times S_3$), $C_2\times C_8$ ($C_3:Q_8$), $C_4^2$ ($C_3:D_4$)

Order 12: $C_2\times C_6$ ($C_8:C_4$), $C_{12}$ ($Q_{32}$), $C_{12}$ ($D_{16}$), $C_{12}$ ($C_8.C_4$), $C_{12}$ ($C_4:C_8$)

Order 8: $C_2\times C_4$ ($C_3:Q_{16}$), $C_2\times C_4$ ($C_3:D_8$), $C_2\times C_4$ ($C_6.D_4$), $C_8$ ($C_{24}:C_2$), $C_8$ ($S_3\times C_8$)

Order 6: $C_6$ ($C_{16}.C_4$), $C_6$ ($C_{16}:C_4$), $C_6$ ($C_8:C_8$)

Order 4: $C_2^2$ ($C_6.D_8$), $C_4$ ($C_3:Q_{32}$), $C_4$ ($C_3:D_{16}$), $C_4$ ($C_{12}.D_4$), $C_4$ ($C_{12}.D_4$)

Order 3: $C_3$ ($C_{16}:C_8$)

Order 2: $C_2$ ($C_{12}.D_8$), $C_2$ ($C_6.D_{16}$), $C_2$ ($C_{12}.D_8$)

Order 1: $C_1$ ($C_{12}.D_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_{12}.D_{16}$ ($C_1$)

Order 192: $C_{12}:C_{16}$ ($C_2$), $C_{24}:C_8$ ($C_2$), $C_8:C_{24}$ ($C_2$)

Order 96: $C_6:C_{16}$ ($C_4$) x 2, $C_4\times C_{24}$ ($C_2^2$)

Order 64: $C_8:C_8$ ($S_3$)

Order 48: $C_2\times C_{24}$ ($Q_8$), $C_2\times C_{24}$ ($C_2\times C_4$), $C_4\times C_{12}$ ($D_4$), $C_3:C_{16}$ ($C_8$)

Order 32: $C_4\times C_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($Q_{16}$), $C_2\times C_{12}$ ($D_8$), $C_2\times C_{12}$ ($C_4:C_4$), $C_{24}$ ($\OD_{16}$), $C_{24}$ ($C_2\times C_8$)

Order 16: $C_2\times C_8$ ($C_4\times S_3$), $C_2\times C_8$ ($C_3:Q_8$), $C_4^2$ ($C_3:D_4$)

Order 12: $C_2\times C_6$ ($C_8:C_4$), $C_{12}$ ($Q_{32}$), $C_{12}$ ($D_{16}$), $C_{12}$ ($C_8.C_4$), $C_{12}$ ($C_4:C_8$)

Order 8: $C_2\times C_4$ ($C_3:Q_{16}$), $C_2\times C_4$ ($C_3:D_8$), $C_2\times C_4$ ($C_6.D_4$), $C_8$ ($C_{24}:C_2$), $C_8$ ($S_3\times C_8$)

Order 6: $C_6$ ($C_{16}.C_4$), $C_6$ ($C_{16}:C_4$), $C_6$ ($C_8:C_8$)

Order 4: $C_2^2$ ($C_6.D_8$), $C_4$ ($C_3:Q_{32}$), $C_4$ ($C_3:D_{16}$), $C_4$ ($C_{12}.D_4$), $C_4$ ($C_{12}.D_4$)

Order 3: $C_3$ ($C_{16}:C_8$)

Order 2: $C_2$ ($C_{12}.D_8$), $C_2$ ($C_6.D_{16}$), $C_2$ ($C_{12}.D_8$)

Order 1: $C_1$ ($C_{12}.D_{16}$)

Series

Derived series

$C_{12}.D_{16}$

$\rhd$

$C_{24}$

$\rhd$

$C_1$

Chief series

$C_{12}.D_{16}$

$\rhd$

$C_8:C_{24}$

$\rhd$

$C_4\times C_{24}$

$\rhd$

$C_4\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}.D_{16}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2$

$\lhd$

$C_4\times C_8$

$\lhd$

$C_8:C_8$

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.