Abstract group 288.1017: $C_6^2:D_4$

• Label: 288.1017
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3^{2} \)
• Perm deg.: $14$
• Trans deg.: $144$
• Rank: $4$

Group information

Description:	$C_6^2:D_4$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_2^4\times C_3:S_3).C_2^6.S_3^2$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6
Elements	1	87	8	72	120	288
Conjugacy classes	1	15	4	4	60	84
Divisions	1	15	4	4	44	68
Autjugacy classes	1	4	1	1	3	10

Dimension	1	2	4
Irr. complex chars.	16	68	0	84
Irr. rational chars.	16	36	16	68

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$144$
Rank:	$4$
Inequivalent generating quadruples:	$2730$

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:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{6}=d^{6}=[a,c]=[a,d]=[c,d]=1, b^{a}=b^{3}, c^{b}=c^{5}, d^{b}=d^{5} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,4)(5,6)(7,8)(10,11)(13,14), (5,7)(6,8), (5,8)(6,7), (2,4)(5,6)(7,8), (1,3)(2,4), (12,13,14), (9,10,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 17 & 12 \\ 12 & 5 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 9 & 22 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 14 & 7 \\ 3 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_6^2:C_2)$
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $D_4$	$C_6^2$ $\,\rtimes\,$ $C_2^3$	$C_6$ $\,\rtimes\,$ $(C_6:D_4)$	$(C_2^3\times C_6)$ $\,\rtimes\,$ $S_3$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_2^3$	$C_2^3$ . $(C_6:S_3)$	$(C_2^2\times C_6)$ . $D_6$	$C_6$ . $(C_2^2\times D_6)$	all 10

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 2404 subgroups in 708 conjugacy classes, 213 normal (11 characteristic).

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

Special subgroups

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 288: $C_6^2:D_4$

Order 144: $C_6^2:C_2^2$ x 12, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_6^2:C_4$

Order 96: $C_2^3:D_6$ x 4

Order 72: $C_6^2:C_2$ x 16, $C_2\times C_6^2$ x 11, $C_6:D_6$ x 10, $C_6.D_6$ x 6

Order 48: $C_6:D_4$ x 48, $C_2^3\times C_6$ x 4, $C_2^2\times D_6$ x 4, $C_6.C_2^3$ x 4

Order 36: $C_6^2$ x 23, $C_6:S_3$ x 16, $C_3^2:C_4$ x 4

Order 32: $C_2^2\times D_4$

Order 24: $C_3:D_4$ x 64, $C_2^2\times C_6$ x 44, $C_2\times D_6$ x 40, $C_6:C_4$ x 24

Order 18: $C_3\times C_6$ x 11, $C_3:S_3$ x 4

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 92, $D_6$ x 64, $C_3:C_4$ x 16

Order 9: $C_3^2$

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $C_6$ x 44, $S_3$ x 16

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_6^2:D_4$

Order 144: $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6^2:C_4$

Order 96: $C_2^3:D_6$

Order 72: $C_2\times C_6^2$ x 3, $C_6:D_6$ x 2, $C_6^2:C_2$, $C_6.D_6$

Order 48: $C_2^3\times C_6$, $C_2^2\times D_6$, $C_6:D_4$, $C_6.C_2^3$

Order 36: $C_6^2$ x 4, $C_6:S_3$ x 2, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$

Order 24: $C_2^2\times C_6$ x 3, $C_2\times D_6$ x 2, $C_3:D_4$, $C_6:C_4$

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

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

Order 12: $C_2\times C_6$ x 4, $D_6$ x 2, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 3, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_6^2:D_4$ ($C_1$)

Order 144: $C_6^2:C_2^2$ ($C_2$) x 12, $C_2^2\times C_6^2$ ($C_2$), $C_6^2:C_2^2$ ($C_2$), $C_6^2:C_4$ ($C_2$)

Order 72: $C_6^2:C_2$ ($C_2^2$) x 16, $C_2\times C_6^2$ ($C_2^2$) x 7, $C_6:D_6$ ($C_2^2$) x 6, $C_6.D_6$ ($C_2^2$) x 6

Order 48: $C_2^3\times C_6$ ($S_3$) x 4

Order 36: $C_6^2$ ($C_2^3$) x 7, $C_3^2:C_4$ ($C_2^3$) x 4, $C_6^2$ ($D_4$) x 4, $C_6:S_3$ ($C_2^3$) x 4

Order 24: $C_2^2\times C_6$ ($D_6$) x 28

Order 18: $C_3\times C_6$ ($C_2\times D_4$) x 6, $C_3\times C_6$ ($C_2^4$)

Order 16: $C_2^4$ ($C_3:S_3$)

Order 12: $C_2\times C_6$ ($C_2\times D_6$) x 28, $C_2\times C_6$ ($C_3:D_4$) x 16

Order 9: $C_3^2$ ($C_2^2\times D_4$)

Order 8: $C_2^3$ ($C_6:S_3$) x 7

Order 6: $C_6$ ($C_6:D_4$) x 24, $C_6$ ($C_2^2\times D_6$) x 4

Order 4: $C_2^2$ ($C_6:D_6$) x 7, $C_2^2$ ($C_6^2:C_2$) x 4

Order 3: $C_3$ ($C_2^3:D_6$) x 4

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

Order 1: $C_1$ ($C_6^2:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_6^2:D_4$ ($C_1$)

Order 144: $C_2^2\times C_6^2$ ($C_2$), $C_6^2:C_2^2$ ($C_2$), $C_6^2:C_2^2$ ($C_2$), $C_6^2:C_4$ ($C_2$)

Order 72: $C_2\times C_6^2$ ($C_2^2$) x 2, $C_6:D_6$ ($C_2^2$), $C_6^2:C_2$ ($C_2^2$), $C_6.D_6$ ($C_2^2$)

Order 48: $C_2^3\times C_6$ ($S_3$)

Order 36: $C_6^2$ ($C_2^3$) x 2, $C_3^2:C_4$ ($C_2^3$), $C_6^2$ ($D_4$), $C_6:S_3$ ($C_2^3$)

Order 24: $C_2^2\times C_6$ ($D_6$) x 2

Order 18: $C_3\times C_6$ ($C_2^4$), $C_3\times C_6$ ($C_2\times D_4$)

Order 16: $C_2^4$ ($C_3:S_3$)

Order 12: $C_2\times C_6$ ($C_2\times D_6$) x 2, $C_2\times C_6$ ($C_3:D_4$)

Order 9: $C_3^2$ ($C_2^2\times D_4$)

Order 8: $C_2^3$ ($C_6:S_3$) x 2

Order 6: $C_6$ ($C_2^2\times D_6$), $C_6$ ($C_6:D_4$)

Order 4: $C_2^2$ ($C_6:D_6$) x 2, $C_2^2$ ($C_6^2:C_2$)

Order 3: $C_3$ ($C_2^3:D_6$)

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

Order 1: $C_1$ ($C_6^2:D_4$)

Series

Derived series

$C_6^2:D_4$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^2:D_4$

$\rhd$

$C_6^2:C_2^2$

$\rhd$

$C_2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2:D_4$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^4$

Supergroups

This group is a maximal subgroup of 81 larger groups in the database.

This group is a maximal quotient of 62 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.