Abstract group 864.2523: $C_9\times \SL(2,3):C_2^2$

• Label: 864.2523
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{3} \)
• Perm deg.: $33$
• Trans deg.: $216$
• Rank: $2$

Group information

Description:	$C_9\times \SL(2,3):C_2^2$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3\times S_3\times S_4^2$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	19	26	12	62	54	168	162	360	864
Conjugacy classes	1	4	8	4	14	18	26	36	60	171
Divisions	1	4	4	4	7	3	13	6	10	52
Autjugacy classes	1	2	2	2	3	2	3	3	3	21

Dimension	1	2	3	4	6	8	18	24
Irr. complex chars.	108	0	36	27	0	0	0	0	171
Irr. rational chars.	4	16	4	1	16	4	4	3	52

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$216$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid b^{18}=c^{4}=d^{2}=[b,c]=[c,d]=1, a^{6}=c^{2}, b^{a}=b^{10}c^{2}d, c^{a}=c^{3}, d^{a}=b^{9}c^{3}d, d^{b}=c^{2}d \rangle$
Permutation group:	Degree $33$ $\langle(1,2,7,6)(3,10,5,8)(4,9,15,16)(11,13,14,17)(12,18,22,20)(19,21,23,24), (1,3,7,5) \!\cdots\! \rangle$
Direct product:	$C_9$ $\, \times\, $ $(\SL(2,3):C_2^2)$
Semidirect product:	$(C_{36}.C_2^3)$ $\,\rtimes\,$ $C_3$	$(\SL(2,3):C_{18})$ $\,\rtimes\,$ $C_2$	$(\SL(2,3):C_2)$ $\,\rtimes\,$ $C_{18}$	$\SL(2,3)$ $\,\rtimes\,$ $(C_2\times C_{18})$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(Q_8\times C_9)$ . $A_4$	$Q_8$ . $(C_9\times A_4)$	$(C_{12}.A_4)$ . $C_6$	$C_{36}$ . $(C_2\times A_4)$	all 22

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

Homology

Abelianization:	$C_{6} \times C_{18} \simeq C_{2}^{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 533 subgroups in 182 conjugacy classes, 68 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{18}$	$G/Z \simeq$ $C_2^2\times A_4$
Commutator:	$G' \simeq$ $Q_8$	$G/G' \simeq$ $C_6\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_2^2:C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{36}.C_2^3$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_9\times \SL(2,3):C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2^2:C_6^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

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 864: $C_9\times \SL(2,3):C_2^2$

Order 432: $\SL(2,3):C_{18}$ x 3

Order 288: $C_{12}.(C_2\times A_4)$ x 2, $Q_8.C_6^2$, $C_{36}.C_2^3$

Order 216: $C_{12}.C_{18}$, $C_9\times \SL(2,3)$

Order 144: $C_{12}.A_4$ x 6, $D_4:C_{18}$ x 4, $D_4\times C_{18}$ x 3, $\SL(2,3):C_6$ x 3

Order 108: $C_3\times C_{36}$ x 3

Order 96: $\SL(2,3):C_2^2$ x 3, $C_6.C_2^4$

Order 72: $D_4\times C_9$ x 6, $Q_8\times C_9$ x 4, $C_2\times C_{36}$ x 3, $Q_8:C_9$ x 2, $C_2^2\times C_{18}$ x 2, $Q_8\times C_3^2$, $C_3\times \SL(2,3)$

Order 54: $C_3\times C_{18}$

Order 48: $\SL(2,3):C_2$ x 9, $D_4:C_6$ x 4, $C_6\times D_4$ x 3

Order 36: $C_{36}$ x 10, $C_2\times C_{18}$ x 5, $C_3\times C_{12}$ x 3

Order 32: $D_4:C_2^2$

Order 27: $C_3\times C_9$

Order 24: $C_3\times D_4$ x 6, $C_3\times Q_8$ x 5, $C_2\times C_{12}$ x 3, $\SL(2,3)$ x 3, $C_2^2\times C_6$ x 2

Order 18: $C_{18}$ x 6, $C_3\times C_6$

Order 16: $D_4:C_2$ x 4, $C_2\times D_4$ x 3

Order 12: $C_{12}$ x 13, $C_2\times C_6$ x 5

Order 9: $C_9$ x 3, $C_3^2$

Order 8: $D_4$ x 6, $C_2\times C_4$ x 3, $C_2^3$ x 2, $Q_8$ x 2

Order 6: $C_6$ x 7

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $C_9\times \SL(2,3):C_2^2$

Order 432: $\SL(2,3):C_{18}$

Order 288: $Q_8.C_6^2$, $C_{36}.C_2^3$, $C_{12}.(C_2\times A_4)$

Order 216: $C_{12}.C_{18}$, $C_9\times \SL(2,3)$

Order 144: $D_4:C_{18}$ x 2, $D_4\times C_{18}$, $C_{12}.A_4$, $\SL(2,3):C_6$

Order 108: $C_3\times C_{36}$

Order 96: $C_6.C_2^4$, $\SL(2,3):C_2^2$

Order 72: $Q_8\times C_9$ x 3, $D_4\times C_9$ x 2, $C_2\times C_{36}$, $Q_8\times C_3^2$, $Q_8:C_9$, $C_3\times \SL(2,3)$, $C_2^2\times C_{18}$

Order 54: $C_3\times C_{18}$

Order 48: $D_4:C_6$ x 2, $C_6\times D_4$, $\SL(2,3):C_2$

Order 36: $C_{36}$ x 3, $C_2\times C_{18}$ x 2, $C_3\times C_{12}$

Order 32: $D_4:C_2^2$

Order 27: $C_3\times C_9$

Order 24: $C_3\times Q_8$ x 3, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $\SL(2,3)$, $C_2^2\times C_6$

Order 18: $C_{18}$ x 3, $C_3\times C_6$

Order 16: $D_4:C_2$ x 2, $C_2\times D_4$

Order 12: $C_{12}$ x 3, $C_2\times C_6$ x 2

Order 9: $C_9$ x 2, $C_3^2$

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

Order 6: $C_6$ x 3

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_9\times \SL(2,3):C_2^2$ ($C_1$)

Order 432: $\SL(2,3):C_{18}$ ($C_2$) x 3

Order 288: $C_{12}.(C_2\times A_4)$ ($C_3$) x 2, $Q_8.C_6^2$ ($C_3$), $C_{36}.C_2^3$ ($C_3$)

Order 216: $C_9\times \SL(2,3)$ ($C_2^2$)

Order 144: $C_{12}.A_4$ ($C_6$) x 6, $D_4:C_{18}$ ($C_6$) x 3, $\SL(2,3):C_6$ ($C_6$) x 3

Order 96: $\SL(2,3):C_2^2$ ($C_9$) x 3, $C_6.C_2^4$ ($C_3^2$)

Order 72: $Q_8:C_9$ ($C_2\times C_6$) x 2, $C_3\times \SL(2,3)$ ($C_2\times C_6$), $Q_8\times C_9$ ($C_2\times C_6$), $Q_8\times C_9$ ($A_4$)

Order 48: $\SL(2,3):C_2$ ($C_{18}$) x 9, $D_4:C_6$ ($C_3\times C_6$) x 3

Order 36: $C_{36}$ ($C_2\times A_4$) x 3

Order 32: $D_4:C_2^2$ ($C_3\times C_9$)

Order 24: $\SL(2,3)$ ($C_2\times C_{18}$) x 3, $C_3\times Q_8$ ($C_6^2$), $C_3\times Q_8$ ($C_3\times A_4$)

Order 18: $C_{18}$ ($C_2^2\times A_4$)

Order 16: $D_4:C_2$ ($C_3\times C_{18}$) x 3

Order 12: $C_{12}$ ($C_6\times A_4$) x 3

Order 9: $C_9$ ($\SL(2,3):C_2^2$)

Order 8: $Q_8$ ($C_6\times C_{18}$), $Q_8$ ($C_9\times A_4$)

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

Order 4: $C_4$ ($A_4\times C_{18}$) x 3

Order 3: $C_3$ ($Q_8.C_6^2$)

Order 2: $C_2$ ($C_2\times A_4\times C_{18}$)

Order 1: $C_1$ ($C_9\times \SL(2,3):C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_9\times \SL(2,3):C_2^2$ ($C_1$)

Order 432: $\SL(2,3):C_{18}$ ($C_2$)

Order 288: $Q_8.C_6^2$ ($C_3$), $C_{36}.C_2^3$ ($C_3$), $C_{12}.(C_2\times A_4)$ ($C_3$)

Order 216: $C_9\times \SL(2,3)$ ($C_2^2$)

Order 144: $D_4:C_{18}$ ($C_6$), $C_{12}.A_4$ ($C_6$), $\SL(2,3):C_6$ ($C_6$)

Order 96: $C_6.C_2^4$ ($C_3^2$), $\SL(2,3):C_2^2$ ($C_9$)

Order 72: $Q_8:C_9$ ($C_2\times C_6$), $C_3\times \SL(2,3)$ ($C_2\times C_6$), $Q_8\times C_9$ ($C_2\times C_6$), $Q_8\times C_9$ ($A_4$)

Order 48: $D_4:C_6$ ($C_3\times C_6$), $\SL(2,3):C_2$ ($C_{18}$)

Order 36: $C_{36}$ ($C_2\times A_4$)

Order 32: $D_4:C_2^2$ ($C_3\times C_9$)

Order 24: $\SL(2,3)$ ($C_2\times C_{18}$), $C_3\times Q_8$ ($C_6^2$), $C_3\times Q_8$ ($C_3\times A_4$)

Order 18: $C_{18}$ ($C_2^2\times A_4$)

Order 16: $D_4:C_2$ ($C_3\times C_{18}$)

Order 12: $C_{12}$ ($C_6\times A_4$)

Order 9: $C_9$ ($\SL(2,3):C_2^2$)

Order 8: $Q_8$ ($C_6\times C_{18}$), $Q_8$ ($C_9\times A_4$)

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

Order 4: $C_4$ ($A_4\times C_{18}$)

Order 3: $C_3$ ($Q_8.C_6^2$)

Order 2: $C_2$ ($C_2\times A_4\times C_{18}$)

Order 1: $C_1$ ($C_9\times \SL(2,3):C_2^2$)

Series

Derived series

$C_9\times \SL(2,3):C_2^2$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_9\times \SL(2,3):C_2^2$

$\rhd$

$\SL(2,3):C_{18}$

$\rhd$

$C_9\times \SL(2,3)$

$\rhd$

$C_3\times \SL(2,3)$

$\rhd$

$C_3\times Q_8$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_9\times \SL(2,3):C_2^2$

$\rhd$

$Q_8$

Upper central series

$C_1$

$\lhd$

$C_{18}$

$\lhd$

$Q_8\times C_9$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

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

Character theory

Complex character table

See the $171 \times 171$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $52 \times 52$ rational character table.