Abstract group 384.11577: $C_2\times D_{12}:C_8$

• Label: 384.11577
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \)
• Perm deg.: $17$
• Trans deg.: $192$
• Rank: $4$

Group information

Description:	$C_2\times D_{12}:C_8$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3:(C_2^9.C_2^5)$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \)
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	24
Elements	1	55	2	72	14	128	48	64	384
Conjugacy classes	1	15	1	24	7	40	16	16	120
Divisions	1	15	1	14	7	10	10	4	62
Autjugacy classes	1	5	1	6	4	3	5	1	26

Dimension	1	2	4	8
Irr. complex chars.	64	48	8	0	120
Irr. rational chars.	16	20	18	8	62

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$192$
Rank:	$4$
Inequivalent generating quadruples:	$131040$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{8}=d^{12}=[a,b]=[a,c]=[a,d]=[b,c]=1, d^{b}=d^{11}, d^{c}=d^{5} \rangle$
Permutation group:	Degree $17$ $\langle(1,2)(3,4)(8,9,11,13,10,12,14,15)(16,17), (2,4)(6,7)(8,10)(9,12)(11,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 41 & 24 \\ 24 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 37 & 39 \\ 24 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 10 \\ 0 & 5 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/48\Z)$
Direct product:	$C_2$ $\, \times\, $ $(D_{12}:C_8)$
Semidirect product:	$(C_6:C_8)$ $\,\rtimes\,$ $D_4$	$(C_4:C_8)$ $\,\rtimes\,$ $D_6$	$C_6$ $\,\rtimes\,$ $(C_8\times D_4)$	$C_4$ $\,\rtimes\,$ $(C_8\times D_6)$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $(Q_8:D_6)$	$C_4$ . $(D_4\times D_6)$	$(C_4\times D_{12})$ . $C_4$	$C_{12}$ . $(C_4\times D_4)$	all 63

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

Homology

Abelianization:	$C_{2}^{3} \times C_{8} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1518 subgroups in 616 conjugacy classes, 253 normal (37 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 384: $C_2\times D_{12}:C_8$

Order 192: $D_{12}:C_8$ x 8, $C_2\times D_6:C_8$ x 2, $C_{24}:C_2^3$ x 2, $C_4^2.C_{12}$, $C_{12}.C_4^2$, $C_4^2:D_6$

Order 128: $C_4^2.C_2^3$

Order 96: $C_8\times D_6$ x 16, $C_4\times D_{12}$ x 8, $D_6:C_8$ x 8, $C_{12}:C_8$ x 4, $C_4:C_{24}$ x 4, $C_{12}:C_2^3$ x 2, $C_2^2\times C_{24}$ x 2, $C_2^2.D_{12}$ x 2, $C_{12}.C_2^3$ x 2, $C_2^2\times D_{12}$, $C_2\times C_4\times C_{12}$, $C_2^2.D_{12}$

Order 64: $C_8\times D_4$ x 8, $C_2^3:C_8$ x 2, $C_2^3\times C_8$ x 2, $C_2\times C_4\times C_8$, $C_4^2:C_2^2$, $C_4^2.C_4$

Order 48: $S_3\times C_8$ x 16, $C_4\times D_6$ x 16, $C_2\times D_{12}$ x 12, $C_6:C_8$ x 10, $C_2\times C_{24}$ x 8, $D_6:C_4$ x 8, $C_4\times C_{12}$ x 4, $C_{12}:C_4$ x 4, $C_2^2\times C_{12}$ x 3, $C_2^2\times D_6$ x 2, $C_6.C_2^3$ x 2

Order 32: $C_2^2\times C_8$ x 20, $C_2^2:C_8$ x 8, $C_4\times D_4$ x 8, $C_4\times C_8$ x 4, $C_4:C_8$ x 4, $C_2^3\times C_4$ x 2, $C_2^3:C_4$ x 2, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 24: $C_2\times D_6$ x 20, $C_2\times C_{12}$ x 16, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_6:C_4$ x 8, $C_3:C_8$ x 6, $C_{24}$ x 4, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 34, $C_2^2\times C_4$ x 21, $C_2\times D_4$ x 12, $C_2^2:C_4$ x 8, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 12: $D_6$ x 32, $C_{12}$ x 10, $C_2\times C_6$ x 7, $C_3:C_4$ x 4

Order 8: $C_2\times C_4$ x 40, $C_2^3$ x 21, $D_4$ x 16, $C_8$ x 10

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$ x 39, $C_4$ x 14

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2\times D_{12}:C_8$

Order 192: $C_4^2.C_{12}$, $C_2\times D_6:C_8$, $C_{12}.C_4^2$, $D_{12}:C_8$, $C_{24}:C_2^3$, $C_4^2:D_6$

Order 128: $C_4^2.C_2^3$

Order 96: $C_4\times D_{12}$ x 2, $C_{12}.C_2^3$ x 2, $C_8\times D_6$ x 2, $C_{12}:C_8$, $C_4:C_{24}$, $D_6:C_8$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^2\times C_{24}$, $C_2\times C_4\times C_{12}$, $C_2^2.D_{12}$, $C_2^2.D_{12}$

Order 64: $C_2^3:C_8$, $C_2\times C_4\times C_8$, $C_2^3\times C_8$, $C_4^2:C_2^2$, $C_8\times D_4$, $C_4^2.C_4$

Order 48: $C_6:C_8$ x 4, $C_4\times D_6$ x 4, $C_2^2\times C_{12}$ x 3, $C_2\times D_{12}$ x 2, $C_2\times C_{24}$ x 2, $C_4\times C_{12}$ x 2, $C_2^2\times D_6$, $C_6.C_2^3$, $S_3\times C_8$, $D_6:C_4$, $C_{12}:C_4$

Order 32: $C_2^2\times C_8$ x 5, $C_4\times D_4$ x 2, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_4\times C_8$, $C_2^2.D_4$, $C_2^3:C_4$, $C_2\times C_4^2$, $C_4:C_8$

Order 24: $C_2\times C_{12}$ x 8, $C_2\times D_6$ x 4, $C_4\times S_3$ x 3, $C_6:C_4$ x 2, $C_3:C_8$ x 2, $D_{12}$, $C_{24}$, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 8, $C_2\times C_8$ x 7, $C_4^2$ x 2, $C_2\times D_4$ x 2, $C_4:C_4$, $C_2^2:C_4$, $C_2^4$

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 4, $D_6$ x 4, $C_3:C_4$

Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 5, $C_8$ x 3, $D_4$

Order 6: $C_6$ x 4, $S_3$

Order 4: $C_2^2$ x 8, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2\times D_{12}:C_8$ ($C_1$)

Order 192: $D_{12}:C_8$ ($C_2$) x 8, $C_2\times D_6:C_8$ ($C_2$) x 2, $C_{24}:C_2^3$ ($C_2$) x 2, $C_4^2.C_{12}$ ($C_2$), $C_{12}.C_4^2$ ($C_2$), $C_4^2:D_6$ ($C_2$)

Order 96: $D_6:C_8$ ($C_2^2$) x 8, $C_8\times D_6$ ($C_2^2$) x 8, $C_{12}:C_8$ ($C_2^2$) x 4, $C_4\times D_{12}$ ($C_2^2$) x 4, $C_4\times D_{12}$ ($C_4$) x 4, $C_4:C_{24}$ ($C_2^2$) x 4, $C_{12}:C_2^3$ ($C_2^2$) x 2, $C_2^2\times C_{24}$ ($C_2^2$) x 2, $C_2^2.D_{12}$ ($C_4$) x 2, $C_{12}.C_2^3$ ($C_2^2$) x 2, $C_2^2\times D_{12}$ ($C_4$), $C_2\times C_4\times C_{12}$ ($C_2^2$), $C_2^2.D_{12}$ ($C_4$)

Order 64: $C_4^2.C_4$ ($S_3$)

Order 48: $C_2\times D_{12}$ ($C_8$) x 8, $D_6:C_4$ ($C_2\times C_4$) x 8, $C_6:C_8$ ($C_2^3$) x 4, $C_6:C_8$ ($D_4$) x 4, $C_2\times D_{12}$ ($C_2\times C_4$) x 4, $C_4\times D_6$ ($C_2^3$) x 4, $C_4\times D_6$ ($C_2\times C_4$) x 4, $C_2\times C_{24}$ ($C_2^3$) x 4, $C_{12}:C_4$ ($C_2\times C_4$) x 4, $C_2^2\times D_6$ ($C_2\times C_4$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_4$) x 2, $C_6.C_2^3$ ($C_2\times C_4$) x 2, $C_4\times C_{12}$ ($C_2^3$) x 2, $C_4\times C_{12}$ ($C_2\times C_4$) x 2, $C_2^2\times C_{12}$ ($C_2^3$)

Order 32: $C_4:C_8$ ($D_6$) x 4, $C_2^2\times C_8$ ($D_6$) x 2, $C_2\times C_4^2$ ($D_6$)

Order 24: $D_{12}$ ($C_2\times C_8$) x 16, $C_2\times D_6$ ($C_2\times C_8$) x 8, $C_2\times C_{12}$ ($C_2^2\times C_4$) x 5, $C_2\times C_{12}$ ($C_2\times C_8$) x 4, $C_6:C_4$ ($C_2^2\times C_4$) x 4, $C_2\times D_6$ ($C_2^2\times C_4$) x 4, $C_3:C_8$ ($C_2\times D_4$) x 4, $C_2\times C_{12}$ ($D_4:C_2$) x 2, $C_2\times C_{12}$ ($C_2\times D_4$) x 2, $C_2\times C_{12}$ ($C_2^4$), $C_2^2\times C_6$ ($C_2^2\times C_4$)

Order 16: $C_2\times C_8$ ($C_2\times D_6$) x 4, $C_4^2$ ($C_4\times S_3$) x 2, $C_4^2$ ($C_2\times D_6$) x 2, $C_2^2\times C_4$ ($C_4\times S_3$) x 2, $C_2^2\times C_4$ ($C_2\times D_6$)

Order 12: $D_6$ ($C_2^2\times C_8$) x 8, $C_{12}$ ($C_2^2\times C_8$) x 4, $C_2\times C_6$ ($\OD_{16}:C_2$) x 2, $C_2\times C_6$ ($C_2^2\times C_8$) x 2, $C_2\times C_6$ ($C_4\times D_4$) x 2, $C_{12}$ ($C_4\times D_4$) x 2, $C_2\times C_6$ ($C_2^3\times C_4$), $C_{12}$ ($D_4:C_2^2$), $C_{12}$ ($C_2^2\times D_4$)

Order 8: $C_2\times C_4$ ($C_4\times D_6$) x 5, $C_2\times C_4$ ($S_3\times C_8$) x 4, $C_2\times C_4$ ($D_{12}:C_2$) x 2, $C_2\times C_4$ ($S_3\times D_4$) x 2, $C_2^3$ ($C_4\times D_6$), $C_2\times C_4$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_8\times D_4$) x 4, $C_6$ ($C_8.C_2^3$), $C_6$ ($C_2^3\times C_8$), $C_6$ ($C_4^2:C_2^2$)

Order 4: $C_4$ ($C_8\times D_6$) x 4, $C_2^2$ ($C_8.D_6$) x 2, $C_2^2$ ($C_8\times D_6$) x 2, $C_2^2$ ($D_{12}:C_4$) x 2, $C_4$ ($D_{12}:C_4$) x 2, $C_2^2$ ($C_{12}:C_2^3$), $C_4$ ($Q_8:D_6$), $C_4$ ($D_4\times D_6$)

Order 3: $C_3$ ($C_4^2.C_2^3$)

Order 2: $C_2$ ($D_{12}:C_8$) x 4, $C_2$ ($\OD_{16}:D_6$), $C_2$ ($C_{24}:C_2^3$), $C_2$ ($C_2\times D_{12}:C_4$)

Order 1: $C_1$ ($C_2\times D_{12}:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2\times D_{12}:C_8$ ($C_1$)

Order 192: $C_4^2.C_{12}$ ($C_2$), $C_2\times D_6:C_8$ ($C_2$), $C_{12}.C_4^2$ ($C_2$), $D_{12}:C_8$ ($C_2$), $C_{24}:C_2^3$ ($C_2$), $C_4^2:D_6$ ($C_2$)

Order 96: $C_{12}.C_2^3$ ($C_2^2$) x 2, $C_{12}:C_8$ ($C_2^2$), $C_4\times D_{12}$ ($C_2^2$), $C_4\times D_{12}$ ($C_4$), $C_4:C_{24}$ ($C_2^2$), $D_6:C_8$ ($C_2^2$), $C_2^2\times D_{12}$ ($C_4$), $C_{12}:C_2^3$ ($C_2^2$), $C_2^2\times C_{24}$ ($C_2^2$), $C_2\times C_4\times C_{12}$ ($C_2^2$), $C_2^2.D_{12}$ ($C_4$), $C_2^2.D_{12}$ ($C_4$), $C_8\times D_6$ ($C_2^2$)

Order 64: $C_4^2.C_4$ ($S_3$)

Order 48: $C_6:C_8$ ($C_2^3$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_4$) x 2, $C_6:C_8$ ($D_4$), $C_2^2\times D_6$ ($C_2\times C_4$), $C_2^2\times C_{12}$ ($C_2^3$), $C_6.C_2^3$ ($C_2\times C_4$), $C_2\times D_{12}$ ($C_2\times C_4$), $C_2\times D_{12}$ ($C_8$), $C_4\times D_6$ ($C_2^3$), $C_4\times D_6$ ($C_2\times C_4$), $C_2\times C_{24}$ ($C_2^3$), $C_4\times C_{12}$ ($C_2^3$), $C_4\times C_{12}$ ($C_2\times C_4$), $D_6:C_4$ ($C_2\times C_4$), $C_{12}:C_4$ ($C_2\times C_4$)

Order 32: $C_2^2\times C_8$ ($D_6$), $C_2\times C_4^2$ ($D_6$), $C_4:C_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($C_2^2\times C_4$) x 3, $C_2\times C_{12}$ ($C_2\times C_8$), $C_2\times C_{12}$ ($C_2^4$), $C_2\times C_{12}$ ($D_4:C_2$), $C_2\times C_{12}$ ($C_2\times D_4$), $C_6:C_4$ ($C_2^2\times C_4$), $D_{12}$ ($C_2\times C_8$), $C_2^2\times C_6$ ($C_2^2\times C_4$), $C_2\times D_6$ ($C_2\times C_8$), $C_2\times D_6$ ($C_2^2\times C_4$), $C_3:C_8$ ($C_2\times D_4$)

Order 16: $C_2^2\times C_4$ ($C_4\times S_3$) x 2, $C_2\times C_8$ ($C_2\times D_6$), $C_4^2$ ($C_4\times S_3$), $C_4^2$ ($C_2\times D_6$), $C_2^2\times C_4$ ($C_2\times D_6$)

Order 12: $C_2\times C_6$ ($C_2^3\times C_4$), $C_2\times C_6$ ($\OD_{16}:C_2$), $C_2\times C_6$ ($C_2^2\times C_8$), $C_2\times C_6$ ($C_4\times D_4$), $D_6$ ($C_2^2\times C_8$), $C_{12}$ ($D_4:C_2^2$), $C_{12}$ ($C_2^2\times D_4$), $C_{12}$ ($C_2^2\times C_8$), $C_{12}$ ($C_4\times D_4$)

Order 8: $C_2\times C_4$ ($C_4\times D_6$) x 3, $C_2^3$ ($C_4\times D_6$), $C_2\times C_4$ ($C_2^2\times D_6$), $C_2\times C_4$ ($D_{12}:C_2$), $C_2\times C_4$ ($S_3\times C_8$), $C_2\times C_4$ ($S_3\times D_4$)

Order 6: $C_6$ ($C_8.C_2^3$), $C_6$ ($C_2^3\times C_8$), $C_6$ ($C_4^2:C_2^2$), $C_6$ ($C_8\times D_4$)

Order 4: $C_2^2$ ($C_{12}:C_2^3$), $C_2^2$ ($C_8.D_6$), $C_2^2$ ($C_8\times D_6$), $C_2^2$ ($D_{12}:C_4$), $C_4$ ($Q_8:D_6$), $C_4$ ($D_4\times D_6$), $C_4$ ($C_8\times D_6$), $C_4$ ($D_{12}:C_4$)

Order 3: $C_3$ ($C_4^2.C_2^3$)

Order 2: $C_2$ ($D_{12}:C_8$), $C_2$ ($\OD_{16}:D_6$), $C_2$ ($C_{24}:C_2^3$), $C_2$ ($C_2\times D_{12}:C_4$)

Order 1: $C_1$ ($C_2\times D_{12}:C_8$)

Series

Derived series

$C_2\times D_{12}:C_8$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_2\times D_{12}:C_8$

$\rhd$

$C_{24}:C_2^3$

$\rhd$

$C_{12}:C_2^3$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times D_{12}:C_8$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_4^2.C_4$

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $62 \times 62$ rational character table.