Abstract group 576.8416: $C_2\times C_6^2.D_4$

• Label: 576.8416
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $24$
• Trans deg.: $96$
• Rank: $4$

Group information

Description:	$C_2\times C_6^2.D_4$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_6^2.C_2^6.C_2^3$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	31	8	144	104	144	144	576
Conjugacy classes	1	7	2	10	14	4	10	48
Divisions	1	7	2	10	14	4	6	44
Autjugacy classes	1	4	2	5	7	1	2	22

Dimension	1	2	4	8
Irr. complex chars.	16	4	26	2	48
Irr. rational chars.	16	4	18	6	44

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$96$
Rank:	$4$
Inequivalent generating quadruples:	$1834560$

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:	${\langle a, b, c, d, e \mid a^{2}=c^{8}=d^{3}=e^{6}=[a,b]=[a,d]=[a,e]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(4,6)(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21)(23,24), (1,2) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_6^2.D_4)$
Semidirect product:	$(C_6^2:Q_8)$ $\,\rtimes\,$ $C_2$	$(C_6^2.D_4)$ $\,\rtimes\,$ $C_2$	$(C_6^2.D_4)$ $\,\rtimes\,$ $C_2$	$(C_3^2:Q_{16})$ $\,\rtimes\,$ $C_2^2$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6.D_6)$ . $D_4$ (2)	$(D_6:S_3)$ . $C_2^3$	$(D_6:D_6)$ . $C_2^2$	$(D_6.D_6)$ . $C_2^2$	all 19

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 2146 subgroups in 418 conjugacy classes, 89 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3^2:C_2^2$
Commutator:	$G' \simeq$ $C_3^2:C_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6^2:D_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $C_2\times C_6^2.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 576: $C_2\times C_6^2.D_4$

Order 288: $C_6^2.D_4$ x 8, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2:Q_8$, $C_2.D_6^2$, $C_2\times C_3^2:\OD_{16}$

Order 144: $C_2^2.S_3^2$ x 10, $C_3^2:Q_{16}$ x 8, $C_3^2:\SD_{16}$ x 8, $D_6.D_6$ x 6, $C_3^2:\OD_{16}$ x 4, $C_6^2.C_4$ x 2, $C_6^2:C_4$, $C_6^2:C_2^2$, $C_6^2:C_4$, $D_6:D_6$, $D_6.D_6$

Order 96: $D_4:D_6$, $C_6.C_2^4$

Order 72: $C_3^2:Q_8$ x 13, $C_6:C_{12}$ x 7, $C_6.D_6$ x 6, $C_6\wr C_2$ x 4, $C_6.D_6$ x 4, $C_3^2:C_8$ x 4, $D_6:S_3$ x 3, $C_2\times C_6^2$, $C_6\times D_6$

Order 64: $Q_{16}:C_2^2$

Order 48: $C_6:Q_8$ x 13, $D_4:S_3$ x 8, $C_6.C_2^3$ x 4, $C_6:D_4$ x 2, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_4\times D_6$

Order 36: $C_3:C_{12}$ x 6, $C_6^2$ x 5, $C_3^2:C_4$ x 4, $C_6\times S_3$ x 4

Order 32: $Q_{16}:C_2$ x 8, $C_2\times Q_{16}$ x 2, $C_2\times \SD_{16}$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_2\times \OD_{16}$

Order 24: $C_6:C_4$ x 23, $C_3:Q_8$ x 20, $C_3:D_4$ x 8, $C_2\times C_{12}$ x 7, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 3, $C_2\times D_6$

Order 18: $C_3\times C_6$ x 5, $C_3\times S_3$ x 2

Order 16: $C_2\times Q_8$ x 10, $Q_{16}$ x 8, $\SD_{16}$ x 8, $D_4:C_2$ x 6, $\OD_{16}$ x 4, $C_2^2\times C_4$ x 3, $C_2\times C_8$ x 2, $C_2\times D_4$ x 2

Order 12: $C_2\times C_6$ x 16, $C_3:C_4$ x 14, $C_{12}$ x 6, $D_6$ x 4

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 17, $Q_8$ x 13, $D_4$ x 7, $C_8$ x 4, $C_2^3$ x 2

Order 6: $C_6$ x 14, $S_3$ x 2

Order 4: $C_4$ x 10, $C_2^2$ x 9

Order 3: $C_3$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_2\times C_6^2.D_4$

Order 288: $C_6^2:Q_8$, $C_2.D_6^2$, $C_2\times C_3^2:\OD_{16}$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$

Order 144: $C_2^2.S_3^2$ x 5, $D_6.D_6$ x 2, $C_6^2:C_4$, $C_6^2:C_2^2$, $C_6^2:C_4$, $D_6:D_6$, $D_6.D_6$, $C_3^2:\OD_{16}$, $C_6^2.C_4$, $C_3^2:Q_{16}$, $C_3^2:\SD_{16}$

Order 96: $D_4:D_6$, $C_6.C_2^4$

Order 72: $C_3^2:Q_8$ x 5, $C_6.D_6$ x 4, $C_6:C_{12}$ x 3, $D_6:S_3$ x 2, $C_2\times C_6^2$, $C_6\times D_6$, $C_6\wr C_2$, $C_6.D_6$, $C_3^2:C_8$

Order 64: $Q_{16}:C_2^2$

Order 48: $C_6:Q_8$ x 5, $C_6.C_2^3$ x 4, $C_6:D_4$ x 2, $D_4:S_3$ x 2, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_4\times D_6$

Order 36: $C_3^2:C_4$ x 3, $C_6^2$ x 3, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2

Order 32: $D_4:C_2^2$, $C_2^2\times Q_8$, $Q_{16}:C_2$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_2\times \OD_{16}$

Order 24: $C_6:C_4$ x 12, $C_3:Q_8$ x 5, $C_2\times C_{12}$ x 3, $C_3:D_4$ x 3, $C_2^2\times C_6$ x 3, $C_4\times S_3$, $C_2\times D_6$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 3, $C_3\times S_3$

Order 16: $C_2\times Q_8$ x 5, $C_2^2\times C_4$ x 3, $D_4:C_2$ x 2, $C_2\times D_4$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $C_2\times C_8$

Order 12: $C_2\times C_6$ x 8, $C_3:C_4$ x 8, $D_6$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 8, $Q_8$ x 5, $D_4$ x 3, $C_2^3$ x 2, $C_8$

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

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_2\times C_6^2.D_4$ ($C_1$)

Order 288: $C_6^2.D_4$ ($C_2$) x 8, $C_6^2.D_4$ ($C_2$) x 2, $C_6^2.D_4$ ($C_2$) x 2, $C_6^2:Q_8$ ($C_2$), $C_2.D_6^2$ ($C_2$), $C_2\times C_3^2:\OD_{16}$ ($C_2$)

Order 144: $C_3^2:Q_{16}$ ($C_2^2$) x 8, $C_3^2:\SD_{16}$ ($C_2^2$) x 8, $C_2^2.S_3^2$ ($C_2^2$) x 7, $D_6.D_6$ ($C_2^2$) x 4, $C_3^2:\OD_{16}$ ($C_2^2$) x 4, $C_6^2.C_4$ ($C_2^2$) x 2, $C_6^2:C_4$ ($C_2^2$), $D_6:D_6$ ($C_2^2$)

Order 72: $C_3^2:Q_8$ ($C_2^3$) x 6, $C_3^2:C_8$ ($C_2^3$) x 4, $C_6.D_6$ ($C_2^3$) x 3, $C_6.D_6$ ($D_4$) x 3, $D_6:S_3$ ($C_2^3$) x 2, $C_2\times C_6^2$ ($D_4$)

Order 36: $C_3^2:C_4$ ($C_2\times D_4$) x 3, $C_6^2$ ($C_2\times D_4$) x 3, $C_3^2:C_4$ ($C_2^4$)

Order 18: $C_3\times C_6$ ($Q_{16}:C_2$) x 2, $C_3\times C_6$ ($C_2^2\times D_4$)

Order 9: $C_3^2$ ($Q_{16}:C_2^2$)

Order 8: $C_2^3$ ($\SOPlus(4,2)$)

Order 4: $C_2^2$ ($S_3^2:C_2^2$) x 3

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

Order 1: $C_1$ ($C_2\times C_6^2.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_2\times C_6^2.D_4$ ($C_1$)

Order 288: $C_6^2:Q_8$ ($C_2$), $C_2.D_6^2$ ($C_2$), $C_2\times C_3^2:\OD_{16}$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$)

Order 144: $C_2^2.S_3^2$ ($C_2^2$) x 3, $C_6^2:C_4$ ($C_2^2$), $D_6:D_6$ ($C_2^2$), $D_6.D_6$ ($C_2^2$), $C_3^2:\OD_{16}$ ($C_2^2$), $C_6^2.C_4$ ($C_2^2$), $C_3^2:Q_{16}$ ($C_2^2$), $C_3^2:\SD_{16}$ ($C_2^2$)

Order 72: $C_6.D_6$ ($C_2^3$) x 2, $C_6.D_6$ ($D_4$) x 2, $C_3^2:Q_8$ ($C_2^3$) x 2, $C_2\times C_6^2$ ($D_4$), $D_6:S_3$ ($C_2^3$), $C_3^2:C_8$ ($C_2^3$)

Order 36: $C_3^2:C_4$ ($C_2\times D_4$) x 2, $C_6^2$ ($C_2\times D_4$) x 2, $C_3^2:C_4$ ($C_2^4$)

Order 18: $C_3\times C_6$ ($C_2^2\times D_4$), $C_3\times C_6$ ($Q_{16}:C_2$)

Order 9: $C_3^2$ ($Q_{16}:C_2^2$)

Order 8: $C_2^3$ ($\SOPlus(4,2)$)

Order 4: $C_2^2$ ($S_3^2:C_2^2$) x 2

Order 2: $C_2$ ($C_6^2.D_4$), $C_2$ ($C_6^2:D_4$)

Order 1: $C_1$ ($C_2\times C_6^2.D_4$)

Series

Derived series

$C_2\times C_6^2.D_4$

$\rhd$

$C_3^2:C_4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_2\times C_6^2.D_4$

$\rhd$

$C_6^2.D_4$

$\rhd$

$C_2^2.S_3^2$

$\rhd$

$C_6.D_6$

$\rhd$

$C_3^2:C_4$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_2\times C_6^2.D_4$

$\rhd$

$C_3^2:C_4$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

See the $48 \times 48$ 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.