Abstract group 384.5269: $(C_2\times D_8):C_{12}$

• Label: 384.5269
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $23$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$(C_2\times D_8):C_{12}$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2\times C_2.C_2^6.C_2^5$, of order \(8192\)\(\medspace = 2^{13} \)
Composition factors:	$C_2$ x 7, $C_3$
Nilpotency class:	$3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	39	2	24	78	64	48	128	384
Conjugacy classes	1	9	2	8	18	14	16	28	96
Divisions	1	9	1	6	9	8	6	8	48
Autjugacy classes	1	5	1	5	5	4	5	4	30

Dimension	1	2	4	8	16
Irr. complex chars.	48	36	12	0	0	96
Irr. rational chars.	8	20	14	4	2	48

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$2184$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	6	12

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{6}=d^{8}=[a,d]=[b,c]=[c,d]=1, a^{4}=d^{4}, b^{a}=bc^{3}, c^{a}=cd^{4}, d^{b}=d^{7} \rangle$
Permutation group:	Degree $23$ $\langle(1,2,5,11,6,9,14,10)(3,7,8,15,12,16,13,4)(17,18,20,19), (1,3,5,13,6,12,14,8) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(\OD_{16}.D_4)$
Semidirect product:	$(C_6\times D_8)$ $\,\rtimes\,$ $C_4$ (4)	$(C_2\times D_8)$ $\,\rtimes\,$ $C_{12}$ (4)	$(C_2^3.Q_8)$ $\,\rtimes\,$ $C_6$	$(C_2^3.D_4)$ $\,\rtimes\,$ $C_6$ (2)	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_8.D_4)$ (2)	$(C_4:C_{12})$ . $D_4$ (2)	$(C_2\times C_{24})$ . $D_4$ (4)	$C_{12}$ . $(C_4\times D_4)$	all 50

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

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 824 subgroups in 316 conjugacy classes, 112 normal (52 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_4:D_4$
Commutator:	$G' \simeq$ $C_2\times C_4$	$G/G' \simeq$ $C_2^2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2\times D_8):C_{12}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2\times D_8):C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_4:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 384: $(C_2\times D_8):C_{12}$

Order 192: $(C_2\times D_4):C_{12}$ x 2, $C_2^4.C_{12}$ x 2, $C_2\times C_8.C_{12}$, $\OD_{16}:C_{12}$, $C_{24}:C_2^3$

Order 128: $\OD_{16}.D_4$

Order 96: $C_6\times D_8$ x 8, $D_4:C_{12}$ x 4, $\OD_{16}:C_6$ x 4, $C_6\times \OD_{16}$ x 3, $C_8.C_{12}$ x 2, $C_{12}:C_2^3$ x 2, $C_8:C_{12}$, $C_4\times C_{24}$, $C_2^2\times C_{24}$, $C_4^2:C_6$

Order 64: $C_2^3.D_4$ x 2, $\OD_{16}:C_2^2$ x 2, $\OD_{16}:C_4$, $C_2^2\times D_8$, $C_2^3.Q_8$

Order 48: $C_6\times D_4$ x 14, $C_2\times C_{24}$ x 9, $C_3\times D_8$ x 8, $C_3\times \OD_{16}$ x 7, $C_2^3\times C_6$ x 2, $C_4:C_{12}$ x 2, $C_2^2\times C_{12}$, $C_2^2:C_{12}$, $C_4\times C_{12}$

Order 32: $C_2\times D_8$ x 8, $D_4:C_4$ x 4, $\OD_{16}:C_2$ x 4, $C_2\times \OD_{16}$ x 3, $C_2^2\times D_4$ x 2, $C_8.C_4$ x 2, $C_8:C_4$, $C_2^2\times C_8$, $C_4\times C_8$, $C_4^2:C_2$

Order 24: $C_2^2\times C_6$ x 13, $C_3\times D_4$ x 12, $C_2\times C_{12}$ x 8, $C_{24}$ x 8

Order 16: $C_2\times D_4$ x 14, $C_2\times C_8$ x 9, $D_8$ x 8, $\OD_{16}$ x 7, $C_4:C_4$ x 2, $C_2^4$ x 2, $C_2^2:C_4$, $C_4^2$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 21, $C_{12}$ x 6

Order 8: $C_2^3$ x 13, $D_4$ x 12, $C_2\times C_4$ x 8, $C_8$ x 8

Order 6: $C_6$ x 9

Order 4: $C_2^2$ x 21, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $(C_2\times D_8):C_{12}$

Order 192: $C_2\times C_8.C_{12}$, $(C_2\times D_4):C_{12}$, $C_2^4.C_{12}$, $\OD_{16}:C_{12}$, $C_{24}:C_2^3$

Order 128: $\OD_{16}.D_4$

Order 96: $C_6\times D_8$ x 4, $C_6\times \OD_{16}$ x 2, $C_8.C_{12}$, $D_4:C_{12}$, $\OD_{16}:C_6$, $C_8:C_{12}$, $C_4\times C_{24}$, $C_{12}:C_2^3$, $C_2^2\times C_{24}$, $C_4^2:C_6$

Order 64: $C_2^3.D_4$, $\OD_{16}:C_2^2$, $\OD_{16}:C_4$, $C_2^2\times D_8$, $C_2^3.Q_8$

Order 48: $C_6\times D_4$ x 6, $C_2\times C_{24}$ x 6, $C_3\times \OD_{16}$ x 4, $C_3\times D_8$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_4:C_{12}$, $C_2^2:C_{12}$, $C_4\times C_{12}$

Order 32: $C_2\times D_8$ x 4, $C_2\times \OD_{16}$ x 2, $D_4:C_4$, $\OD_{16}:C_2$, $C_2^2\times D_4$, $C_8:C_4$, $C_2^2\times C_8$, $C_4\times C_8$, $C_4^2:C_2$, $C_8.C_4$

Order 24: $C_2\times C_{12}$ x 7, $C_2^2\times C_6$ x 5, $C_{24}$ x 4, $C_3\times D_4$ x 4

Order 16: $C_2\times C_8$ x 6, $C_2\times D_4$ x 6, $\OD_{16}$ x 4, $D_8$ x 2, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2^4$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 8, $C_{12}$ x 5

Order 8: $C_2\times C_4$ x 7, $C_2^3$ x 5, $D_4$ x 4, $C_8$ x 4

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 8, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $(C_2\times D_8):C_{12}$ ($C_1$)

Order 192: $(C_2\times D_4):C_{12}$ ($C_2$) x 2, $C_2^4.C_{12}$ ($C_2$) x 2, $C_2\times C_8.C_{12}$ ($C_2$), $\OD_{16}:C_{12}$ ($C_2$), $C_{24}:C_2^3$ ($C_2$)

Order 128: $\OD_{16}.D_4$ ($C_3$)

Order 96: $C_6\times D_8$ ($C_4$) x 4, $C_6\times \OD_{16}$ ($C_2^2$) x 3, $C_{12}:C_2^3$ ($C_2^2$) x 2, $C_2^2\times C_{24}$ ($C_2^2$), $C_4^2:C_6$ ($C_2^2$)

Order 64: $C_2^3.D_4$ ($C_6$) x 2, $\OD_{16}:C_2^2$ ($C_6$) x 2, $\OD_{16}:C_4$ ($C_6$), $C_2^2\times D_8$ ($C_6$), $C_2^3.Q_8$ ($C_6$)

Order 48: $C_6\times D_4$ ($C_2\times C_4$) x 4, $C_2\times C_{24}$ ($D_4$) x 4, $C_3\times \OD_{16}$ ($D_4$) x 2, $C_2\times C_{24}$ ($C_2\times C_4$) x 2, $C_4:C_{12}$ ($D_4$) x 2, $C_2^2\times C_{12}$ ($C_2^3$)

Order 32: $C_2\times D_8$ ($C_{12}$) x 4, $C_2\times \OD_{16}$ ($C_2\times C_6$) x 3, $C_2^2\times D_4$ ($C_2\times C_6$) x 2, $C_2^2\times C_8$ ($C_2\times C_6$), $C_4^2:C_2$ ($C_2\times C_6$)

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

Order 16: $C_2\times C_8$ ($C_3\times D_4$) x 4, $C_2\times D_4$ ($C_2\times C_{12}$) x 4, $\OD_{16}$ ($C_3\times D_4$) x 2, $C_2\times C_8$ ($C_2\times C_{12}$) x 2, $C_4:C_4$ ($C_3\times D_4$) x 2, $C_2^2\times C_4$ ($C_2^2\times C_6$)

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

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

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

Order 4: $C_2^2$ ($C_4^2:C_6$), $C_2^2$ ($C_{12}:D_4$), $C_2^2$ ($D_4\times C_{12}$), $C_4$ ($C_{12}:D_4$), $C_4$ ($C_{12}:D_4$), $C_4$ ($D_4\times C_{12}$), $C_4$ ($C_2^3:C_{12}$)

Order 3: $C_3$ ($\OD_{16}.D_4$)

Order 2: $C_2$ ($C_{24}.D_4$) x 2, $C_2$ ($(C_2\times C_{12}):D_4$)

Order 1: $C_1$ ($(C_2\times D_8):C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $(C_2\times D_8):C_{12}$ ($C_1$)

Order 192: $C_2\times C_8.C_{12}$ ($C_2$), $(C_2\times D_4):C_{12}$ ($C_2$), $C_2^4.C_{12}$ ($C_2$), $\OD_{16}:C_{12}$ ($C_2$), $C_{24}:C_2^3$ ($C_2$)

Order 128: $\OD_{16}.D_4$ ($C_3$)

Order 96: $C_6\times \OD_{16}$ ($C_2^2$) x 2, $C_{12}:C_2^3$ ($C_2^2$), $C_6\times D_8$ ($C_4$), $C_2^2\times C_{24}$ ($C_2^2$), $C_4^2:C_6$ ($C_2^2$)

Order 64: $C_2^3.D_4$ ($C_6$), $\OD_{16}:C_2^2$ ($C_6$), $\OD_{16}:C_4$ ($C_6$), $C_2^2\times D_8$ ($C_6$), $C_2^3.Q_8$ ($C_6$)

Order 48: $C_2\times C_{24}$ ($D_4$) x 3, $C_6\times D_4$ ($C_2\times C_4$), $C_2^2\times C_{12}$ ($C_2^3$), $C_3\times \OD_{16}$ ($D_4$), $C_2\times C_{24}$ ($C_2\times C_4$), $C_4:C_{12}$ ($D_4$)

Order 32: $C_2\times \OD_{16}$ ($C_2\times C_6$) x 2, $C_2^2\times D_4$ ($C_2\times C_6$), $C_2\times D_8$ ($C_{12}$), $C_2^2\times C_8$ ($C_2\times C_6$), $C_4^2:C_2$ ($C_2\times C_6$)

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

Order 16: $C_2\times C_8$ ($C_3\times D_4$) x 3, $\OD_{16}$ ($C_3\times D_4$), $C_2\times C_8$ ($C_2\times C_{12}$), $C_4:C_4$ ($C_3\times D_4$), $C_2\times D_4$ ($C_2\times C_{12}$), $C_2^2\times C_4$ ($C_2^2\times C_6$)

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

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

Order 6: $C_6$ ($C_2^3.C_2^3$), $C_6$ ($C_8.D_4$)

Order 4: $C_2^2$ ($C_4^2:C_6$), $C_2^2$ ($C_{12}:D_4$), $C_2^2$ ($D_4\times C_{12}$), $C_4$ ($C_{12}:D_4$), $C_4$ ($C_{12}:D_4$), $C_4$ ($D_4\times C_{12}$), $C_4$ ($C_2^3:C_{12}$)

Order 3: $C_3$ ($\OD_{16}.D_4$)

Order 2: $C_2$ ($C_{24}.D_4$), $C_2$ ($(C_2\times C_{12}):D_4$)

Order 1: $C_1$ ($(C_2\times D_8):C_{12}$)

Series

Derived series

$(C_2\times D_8):C_{12}$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$(C_2\times D_8):C_{12}$

$\rhd$

$\OD_{16}:C_{12}$

$\rhd$

$C_2^2\times C_{24}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2\times D_8):C_{12}$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2^2\times C_{12}$

$\lhd$

$(C_2\times D_8):C_{12}$

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.