Abstract group 2304.cr: $C_2^8:C_3^2$

• Label: 2304.cr
• Order: \( 2^{8} \cdot 3^{2} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3^{6} \cdot 5^{2} \cdot 7 \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 17 \)
• Perm deg.: $19$
• Trans deg.: $144$
• Rank: $5$

Group information

Description:	$C_2^8:C_3^2$
Order:	\(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	Group of order \(9096501657600\)\(\medspace = 2^{22} \cdot 3^{6} \cdot 5^{2} \cdot 7 \cdot 17 \)
Composition factors:	$C_2$ x 8, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	3	6
Elements	1	255	1538	510	2304
Conjugacy classes	1	85	8	170	264
Divisions	1	85	4	85	175
Autjugacy classes	1	1	2	1	5

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$144$
Rank:	$5$
Inequivalent generating 5-tuples:	$4840$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid a^{3}=b^{6}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $19$ $\langle(1,2)(3,4)(5,6)(7,8)(17,18,19), (2,3,4)(5,7,8)(9,10,11)(13,14,16), (5,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{153} & \alpha^{46} \\ \alpha^{158} & \alpha^{153} \\ \end{array}\right), \left(\begin{array}{ll}1 & 0 \\ \alpha^{226} & \alpha^{85} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{218} & \alpha^{24} \\ \alpha^{136} & \alpha^{218} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{38} & \alpha^{128} \\ \alpha^{240} & \alpha^{38} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{73} & \alpha^{144} \\ \alpha & \alpha^{73} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{203} & \alpha^{129} \\ \alpha^{241} & \alpha^{203} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{99} & \alpha^{212} \\ \alpha^{69} & \alpha^{99} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{94} & \alpha^{146} \\ \alpha^{3} & \alpha^{94} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{107} & \alpha^{2} \\ \alpha^{114} & \alpha^{107} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{85} & 0 \\ 0 & \alpha^{85} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{256}) = \GL_{2}(\F_{2}[\alpha]/(\alpha^{8} + \alpha^{4} + \alpha^{3} + \alpha^{2} + 1))$
Direct product:	$C_3$ $\, \times\, $ $(C_2^6:A_4)$
Semidirect product:	$C_2^8$ $\,\rtimes\,$ $C_3^2$	$(C_2^5\times C_6)$ $\,\rtimes\,$ $A_4$	$C_2^6$ $\,\rtimes\,$ $(C_3\times A_4)$	$(C_2^7\times C_6)$ $\,\rtimes\,$ $C_3$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\GL_{2}(\F_{256})$.

Homology

Abelianization:	$C_{3}^{2} $
Schur multiplier:	$C_{2}^{15} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 881394 subgroups in 280954 conjugacy classes, 1062 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_2^6:A_4$
Commutator:	$G' \simeq$ $C_2^8$	$G/G' \simeq$ $C_3^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^8:C_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^7\times C_6$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_2^8:C_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^7\times C_6$	$G/\operatorname{soc} \simeq$ $C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 2304: $C_2^8:C_3^2$

Order 768: $C_2^6:A_4$ x 3, $C_2^7\times C_6$

Order 576: $C_2^6:C_3^2$ x 85

Order 384: $C_2^6\times C_6$ x 85

Order 256: $C_2^8$

Order 192: $C_2^5\times C_6$ x 3655, $C_2^4:A_4$ x 255

Order 144: $C_2^4:C_3^2$ x 357

Order 128: $C_2^7$ x 85

Order 96: $C_2^4\times C_6$ x 32385

Order 64: $C_2^6$ x 3655

Order 48: $C_2^3\times C_6$ x 67167, $C_2^2:A_4$ x 1071

Order 36: $C_3\times A_4$ x 85

Order 32: $C_2^5$ x 32385

Order 24: $C_2^2\times C_6$ x 32385

Order 16: $C_2^4$ x 67167

Order 12: $C_2\times C_6$ x 3655, $A_4$ x 255

Order 9: $C_3^2$

Order 8: $C_2^3$ x 32385

Order 6: $C_6$ x 85

Order 4: $C_2^2$ x 3655

Order 3: $C_3$ x 4

Order 2: $C_2$ x 85

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2304: $C_2^8:C_3^2$

Order 768: $C_2^7\times C_6$, $C_2^6:A_4$

Order 576: $C_2^6:C_3^2$

Order 384: $C_2^6\times C_6$

Order 256: $C_2^8$

Order 192: $C_2^5\times C_6$ x 2, $C_2^4:A_4$

Order 144: $C_2^4:C_3^2$

Order 128: $C_2^7$

Order 96: $C_2^4\times C_6$ x 2

Order 64: $C_2^6$ x 2

Order 48: $C_2^3\times C_6$ x 3, $C_2^2:A_4$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$ x 2

Order 24: $C_2^2\times C_6$ x 2

Order 16: $C_2^4$ x 3

Order 12: $C_2\times C_6$ x 2, $A_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 2

Order 6: $C_6$

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2304: $C_2^8:C_3^2$ ($C_1$)

Order 768: $C_2^6:A_4$ ($C_3$) x 3, $C_2^7\times C_6$ ($C_3$)

Order 256: $C_2^8$ ($C_3^2$)

Order 192: $C_2^5\times C_6$ ($A_4$) x 85

Order 64: $C_2^6$ ($C_3\times A_4$) x 85

Order 48: $C_2^3\times C_6$ ($C_2^2:A_4$) x 357

Order 16: $C_2^4$ ($C_2^4:C_3^2$) x 357

Order 12: $C_2\times C_6$ ($C_2^4:A_4$) x 85

Order 4: $C_2^2$ ($C_2^6:C_3^2$) x 85

Order 3: $C_3$ ($C_2^6:A_4$)

Order 1: $C_1$ ($C_2^8:C_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2304: $C_2^8:C_3^2$ ($C_1$)

Order 768: $C_2^7\times C_6$ ($C_3$), $C_2^6:A_4$ ($C_3$)

Order 256: $C_2^8$ ($C_3^2$)

Order 192: $C_2^5\times C_6$ ($A_4$)

Order 64: $C_2^6$ ($C_3\times A_4$)

Order 48: $C_2^3\times C_6$ ($C_2^2:A_4$)

Order 16: $C_2^4$ ($C_2^4:C_3^2$)

Order 12: $C_2\times C_6$ ($C_2^4:A_4$)

Order 4: $C_2^2$ ($C_2^6:C_3^2$)

Order 3: $C_3$ ($C_2^6:A_4$)

Order 1: $C_1$ ($C_2^8:C_3^2$)

Series

Derived series

$C_2^8:C_3^2$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^8:C_3^2$

$\rhd$

$C_2^6:A_4$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^8:C_3^2$

$\rhd$

$C_2^8$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $264 \times 264$ character table is not available for this group.

Rational character table

The $175 \times 175$ rational character table is not available for this group.