Abstract group 384.6844: $(C_2\times C_4^2).D_6$

• Label: 384.6844
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $27$
• Trans deg.: $192$
• Rank: $4$

Group information

Description:	$(C_2\times C_4^2).D_6$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	Group of order \(24576\)\(\medspace = 2^{13} \cdot 3 \)
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	12
Elements	1	31	2	224	14	112	384
Conjugacy classes	1	9	1	28	7	20	66
Divisions	1	9	1	18	7	8	44
Autjugacy classes	1	8	1	14	7	7	38

Dimension	1	2	4	8
Irr. complex chars.	16	36	14	0	66
Irr. rational chars.	16	8	12	8	44

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$192$
Rank:	$4$
Inequivalent generating quadruples:	$262080$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	12	14

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{4}=d^{12}=[a,c]=[b,c]=1, b^{a}=b^{3}d^{6}, d^{a}=b^{2}c^{2}d^{11}, d^{b}=c^{2}d^{7}, d^{c}=d^{11} \rangle$
Permutation group:	Degree $27$ $\langle(1,2,5,7)(3,4,6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,21)(22,24) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^2:C_4)$ $\,\rtimes\,$ $S_3$	$(C_4^2:C_{12})$ $\,\rtimes\,$ $C_2$	$(C_4\times C_6.D_4)$ $\,\rtimes\,$ $C_2$	$C_3$ $\,\rtimes\,$ $(C_2^3.C_2^4)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^2$ . $(D_4:D_6)$	$C_2^2$ . $(D_4:D_6)$ (4)	$C_2^2$ . $(D_4.D_6)$	$(D_6:C_4^2)$ . $C_2$	all 46

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1278 subgroups in 412 conjugacy classes, 125 normal (113 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^2\times D_6$
Commutator:	$G' \simeq$ $C_2^2\times C_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2:C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2\times C_4^2).D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^4$
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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Classes of subgroups up to conjugation

Order 384: $(C_2\times C_4^2).D_6$

Order 192: $C_4^2:C_{12}$, $(C_2\times C_4^2):S_3$, $C_4\times C_6.D_4$, $(C_2\times D_6).Q_8$, $(C_2\times D_6).D_4$, $(C_2\times C_4).D_{12}$, $(C_2\times C_4).D_{12}$, $D_6:C_4^2$, $(C_4\times D_6):C_4$, $C_2^2.(S_3\times D_4)$, $(C_2\times C_{12}).Q_8$, $C_6.(C_4\times D_4)$, $C_6.(C_4\times D_4)$, $(C_2\times C_4).D_{12}$, $C_2^2.(C_4\times D_6)$

Order 128: $C_2^3.C_2^4$

Order 96: $C_6.C_4^2$ x 10, $C_6.C_4^2$ x 6, $C_2^2.D_{12}$ x 6, $C_2^3.D_6$ x 6, $C_6:C_4^2$ x 3, $C_2^2.D_{12}$ x 2, $C_{12}:C_2^3$, $C_2\times C_4\times C_{12}$

Order 64: $C_2^3.C_2^3$ x 3, $C_2^3.C_2^3$ x 2, $C_2^3.C_2^3$, $C_2^3.C_2^3$, $C_2^3.D_4$, $C_2^3.Q_8$, $C_2^3.D_4$, $C_4^2:C_4$, $C_4^2:C_4$, $C_2^3.D_4$, $C_4:C_4^2$, $C_2^2:C_4^2$

Order 48: $D_6:C_4$ x 11, $C_6.D_4$ x 9, $C_2^2\times C_{12}$ x 7, $C_6.C_2^3$ x 7, $C_4\times D_6$ x 5, $C_{12}:C_4$ x 5, $C_{12}:C_4$ x 4, $C_4\times C_{12}$ x 2, $C_2^2\times D_6$

Order 32: $C_2.C_4^2$ x 16, $C_2^2.D_4$ x 8, $C_2^3:C_4$ x 6, $C_2\times C_4^2$ x 4, $C_2^3\times C_4$

Order 24: $C_6:C_4$ x 24, $C_2\times C_{12}$ x 22, $C_2\times D_6$ x 8, $C_4\times S_3$ x 4, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 19, $C_4:C_4$ x 13, $C_2^2:C_4$ x 11, $C_4^2$ x 7, $C_2^4$

Order 12: $D_6$ x 10, $C_3:C_4$ x 10, $C_{12}$ x 8, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 50, $C_2^3$ x 9

Order 6: $C_6$ x 7, $S_3$ x 2

Order 4: $C_4$ x 18, $C_2^2$ x 17

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $(C_2\times C_4^2).D_6$

Order 192: $C_4^2:C_{12}$, $(C_2\times C_4^2):S_3$, $C_4\times C_6.D_4$, $(C_2\times D_6).Q_8$, $(C_2\times D_6).D_4$, $(C_2\times C_4).D_{12}$, $(C_2\times C_4).D_{12}$, $D_6:C_4^2$, $(C_4\times D_6):C_4$, $C_2^2.(S_3\times D_4)$, $(C_2\times C_{12}).Q_8$, $C_6.(C_4\times D_4)$, $C_6.(C_4\times D_4)$, $(C_2\times C_4).D_{12}$, $C_2^2.(C_4\times D_6)$

Order 128: $C_2^3.C_2^4$

Order 96: $C_6.C_4^2$ x 10, $C_6.C_4^2$ x 6, $C_2^2.D_{12}$ x 6, $C_2^3.D_6$ x 6, $C_6:C_4^2$ x 3, $C_2^2.D_{12}$ x 2, $C_{12}:C_2^3$, $C_2\times C_4\times C_{12}$

Order 64: $C_2^3.C_2^3$ x 3, $C_2^3.C_2^3$ x 2, $C_2^3.C_2^3$, $C_2^3.C_2^3$, $C_2^3.D_4$, $C_2^3.Q_8$, $C_2^3.D_4$, $C_4^2:C_4$, $C_4^2:C_4$, $C_2^3.D_4$, $C_4:C_4^2$, $C_2^2:C_4^2$

Order 48: $C_2^2\times C_{12}$ x 7, $C_6.C_2^3$ x 7, $D_6:C_4$ x 6, $C_6.D_4$ x 6, $C_4\times D_6$ x 3, $C_{12}:C_4$ x 3, $C_{12}:C_4$ x 2, $C_2^2\times D_6$, $C_4\times C_{12}$

Order 32: $C_2.C_4^2$ x 16, $C_2^2.D_4$ x 8, $C_2^3:C_4$ x 6, $C_2\times C_4^2$ x 4, $C_2^3\times C_4$

Order 24: $C_2\times C_{12}$ x 21, $C_6:C_4$ x 21, $C_2\times D_6$ x 7, $C_4\times S_3$, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 17, $C_4:C_4$ x 8, $C_2^2:C_4$ x 6, $C_4^2$ x 4, $C_2^4$

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

Order 8: $C_2\times C_4$ x 43, $C_2^3$ x 8

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

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

Order 3: $C_3$

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $(C_2\times C_4^2).D_6$ ($C_1$)

Order 192: $C_4^2:C_{12}$ ($C_2$), $(C_2\times C_4^2):S_3$ ($C_2$), $C_4\times C_6.D_4$ ($C_2$), $(C_2\times D_6).Q_8$ ($C_2$), $(C_2\times D_6).D_4$ ($C_2$), $(C_2\times C_4).D_{12}$ ($C_2$), $(C_2\times C_4).D_{12}$ ($C_2$), $D_6:C_4^2$ ($C_2$), $(C_4\times D_6):C_4$ ($C_2$), $C_2^2.(S_3\times D_4)$ ($C_2$), $(C_2\times C_{12}).Q_8$ ($C_2$), $C_6.(C_4\times D_4)$ ($C_2$), $C_6.(C_4\times D_4)$ ($C_2$), $(C_2\times C_4).D_{12}$ ($C_2$), $C_2^2.(C_4\times D_6)$ ($C_2$)

Order 96: $C_6.C_4^2$ ($C_2^2$) x 10, $C_6.C_4^2$ ($C_2^2$) x 6, $C_2^2.D_{12}$ ($C_2^2$) x 6, $C_2^3.D_6$ ($C_2^2$) x 6, $C_6:C_4^2$ ($C_2^2$) x 3, $C_2^2.D_{12}$ ($C_2^2$) x 2, $C_{12}:C_2^3$ ($C_2^2$), $C_2\times C_4\times C_{12}$ ($C_2^2$)

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

Order 48: $C_2^2\times C_{12}$ ($C_2^3$) x 7, $C_6.C_2^3$ ($C_2^3$) x 7, $C_2^2\times D_6$ ($C_2^3$)

Order 32: $C_2.C_4^2$ ($D_6$) x 6, $C_2\times C_4^2$ ($D_6$)

Order 24: $C_6:C_4$ ($D_4:C_2$) x 6, $C_2\times C_{12}$ ($D_4:C_2$) x 2, $C_2\times D_6$ ($D_4:C_2$) x 2, $C_2^2\times C_6$ ($C_2^4$)

Order 16: $C_2^2\times C_4$ ($C_2\times D_6$) x 7

Order 12: $C_2\times C_6$ ($D_4:C_2^2$) x 5, $C_2\times C_6$ ($C_4.C_2^3$), $C_2\times C_6$ ($D_4:C_2^2$)

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

Order 6: $C_6$ ($C_4^2.C_2^2$) x 2, $C_6$ ($C_4^2.C_2^2$) x 2, $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2:C_2^2$), $C_6$ ($C_4^2.C_2^2$)

Order 4: $C_2^2$ ($D_4:D_6$) x 4, $C_2^2$ ($D_4.D_6$), $C_2^2$ ($D_4:D_6$), $C_2^2$ ($D_{12}:C_2^2$)

Order 3: $C_3$ ($C_2^3.C_2^4$)

Order 2: $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2:D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$)

Order 1: $C_1$ ($(C_2\times C_4^2).D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $(C_2\times C_4^2).D_6$ ($C_1$)

Order 192: $C_4^2:C_{12}$ ($C_2$), $(C_2\times C_4^2):S_3$ ($C_2$), $C_4\times C_6.D_4$ ($C_2$), $(C_2\times D_6).Q_8$ ($C_2$), $(C_2\times D_6).D_4$ ($C_2$), $(C_2\times C_4).D_{12}$ ($C_2$), $(C_2\times C_4).D_{12}$ ($C_2$), $D_6:C_4^2$ ($C_2$), $(C_4\times D_6):C_4$ ($C_2$), $C_2^2.(S_3\times D_4)$ ($C_2$), $(C_2\times C_{12}).Q_8$ ($C_2$), $C_6.(C_4\times D_4)$ ($C_2$), $C_6.(C_4\times D_4)$ ($C_2$), $(C_2\times C_4).D_{12}$ ($C_2$), $C_2^2.(C_4\times D_6)$ ($C_2$)

Order 96: $C_6.C_4^2$ ($C_2^2$) x 10, $C_6.C_4^2$ ($C_2^2$) x 6, $C_2^2.D_{12}$ ($C_2^2$) x 6, $C_2^3.D_6$ ($C_2^2$) x 6, $C_6:C_4^2$ ($C_2^2$) x 3, $C_2^2.D_{12}$ ($C_2^2$) x 2, $C_{12}:C_2^3$ ($C_2^2$), $C_2\times C_4\times C_{12}$ ($C_2^2$)

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

Order 48: $C_2^2\times C_{12}$ ($C_2^3$) x 7, $C_6.C_2^3$ ($C_2^3$) x 7, $C_2^2\times D_6$ ($C_2^3$)

Order 32: $C_2.C_4^2$ ($D_6$) x 6, $C_2\times C_4^2$ ($D_6$)

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

Order 16: $C_2^2\times C_4$ ($C_2\times D_6$) x 7

Order 12: $C_2\times C_6$ ($D_4:C_2^2$) x 5, $C_2\times C_6$ ($C_4.C_2^3$), $C_2\times C_6$ ($D_4:C_2^2$)

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

Order 6: $C_6$ ($C_4^2.C_2^2$) x 2, $C_6$ ($C_4^2.C_2^2$) x 2, $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2:C_2^2$), $C_6$ ($C_4^2.C_2^2$)

Order 4: $C_2^2$ ($D_4:D_6$) x 4, $C_2^2$ ($D_4.D_6$), $C_2^2$ ($D_4:D_6$), $C_2^2$ ($D_{12}:C_2^2$)

Order 3: $C_3$ ($C_2^3.C_2^4$)

Order 2: $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2:D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$)

Order 1: $C_1$ ($(C_2\times C_4^2).D_6$)

Series

Derived series

$(C_2\times C_4^2).D_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_1$

Chief series

$(C_2\times C_4^2).D_6$

$\rhd$

$D_6:C_4^2$

$\rhd$

$C_2^2.D_{12}$

$\rhd$

$C_2^2\times D_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_4^2).D_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_4^2:C_4$

Character theory

Complex character table

See the $66 \times 66$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $44 \times 44$ rational character table.