Abstract group 472392.sq: $C_3^6.C_3^3:(C_4\times S_3)$

• Label: 472392.sq
• Order: \( 2^{3} \cdot 3^{10} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{14} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{4} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^6.C_3^3:(C_4\times S_3)$
Order:	\(472392\)\(\medspace = 2^{3} \cdot 3^{10} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(153055008\)\(\medspace = 2^{5} \cdot 3^{14} \)
Composition factors:	$C_2$ x 3, $C_3$ x 10
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	8019	18224	26244	64152	40824	52488	104976	157464	472392
Conjugacy classes	1	3	70	4	7	108	4	39	12	248
Divisions	1	3	66	2	7	36	2	13	2	132
Autjugacy classes	1	3	17	1	6	12	1	5	1	47

Dimension	1	2	4	8	12	18	24	36	48	54	72	108	144	216
Irr. complex chars.	8	8	6	8	38	24	49	32	0	0	75	0	0	0	248
Irr. rational chars.	4	6	8	8	2	0	29	14	1	4	24	12	3	17	132

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid c^{9}=d^{3}=e^{9}=f^{3}=g^{3}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,12,20,29,15,36,31,18,26,22,8,4,2,10,21,28,13,34,32,17,27,23,9,6,3,11,19,30,14,35,33,16,25,24,7,5) \!\cdots\! \rangle$
Transitive group:	36T30298				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^6.C_3.S_3)$ . $S_3^2$	$(C_3^6.C_3^3:C_4)$ . $S_3$	$(C_3^6.C_3\wr C_4)$ . $C_2$	$(C_3^6.C_3^3.D_6)$ . $C_2$	all 17

Elements of the group are displayed as permutations of degree 36.

Homology

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

Subgroups

There are 29 normal subgroups (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^6.C_3^3:(C_4\times S_3)$
Commutator:	$G' \simeq$ $C_3^6.C_3^4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_3^6$	$G/\Phi \simeq$ $C_3^3:(C_4\times S_3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6.C_3^4$	$G/\operatorname{Fit} \simeq$ $C_2\times C_4$
Radical:	$R \simeq$ $C_3^6.C_3^3:(C_4\times S_3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $\He_3^2:(C_2\times C_4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6.C_3^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 9 (order at least 52488) are shown, as well as all normal subgroups of any index.

Order 472392: $C_3^6.C_3^3:(C_4\times S_3)$

Order 236196: $C_3^6.C_3\wr C_4$ x 2, $C_3^6.C_3^3.D_6$

Order 157464: $C_3^4.C_3^4:(C_4\times S_3)$ x 2

Order 118098: $C_3^6.C_3^4.C_2$, $C_3^5.C_3^4.C_6$, $C_3^6.C_3^4.C_2$

Order 78732: $C_3^5.C_3^3.D_6$ x 2, $C_3^5.\He_3.D_6$ x 2, $C_3^4.C_3^4:C_{12}$ x 2, $C_3^6.C_3^3:C_4$ x 2, $C_3^6.C_3^2.D_6$

Order 59049: $C_3^6.C_3^4$

Order 52488: $C_3^6.C_3:S_3.C_2^2$, $C_3^4:C_{18}.D_{18}$

Order 39366: $C_3^6.C_3^3.C_2$ x 2

Order 19683: $C_3^6.C_3^3$ x 2

Order 13122: $C_3^6.C_3.S_3$ x 2

Order 6561: $C_3^6.C_3^2$ x 2

Order 2187: $C_3\times C_3^5.C_3$ x 2

Order 729: $C_3^6$

Order 81: $C_3^4$

Order 27: $C_3^3$ x 2

Order 9: $C_3^2$ x 2

Order 8: $C_2\times C_4$

Order 3: $C_3$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 9 (order at least 52488) are shown, as well as all normal subgroups of any index.

Order 472392: $C_3^6.C_3^3:(C_4\times S_3)$

Order 236196: $C_3^6.C_3^3.D_6$, $C_3^6.C_3\wr C_4$

Order 157464: $C_3^4.C_3^4:(C_4\times S_3)$

Order 118098: $C_3^6.C_3^4.C_2$, $C_3^5.C_3^4.C_6$, $C_3^6.C_3^4.C_2$

Order 78732: $C_3^5.C_3^3.D_6$ x 2, $C_3^6.C_3^2.D_6$, $C_3^5.\He_3.D_6$, $C_3^4.C_3^4:C_{12}$, $C_3^6.C_3^3:C_4$

Order 59049: $C_3^6.C_3^4$

Order 52488: $C_3^6.C_3:S_3.C_2^2$, $C_3^4:C_{18}.D_{18}$

Order 39366: $C_3^6.C_3^3.C_2$

Order 19683: $C_3^6.C_3^3$

Order 13122: $C_3^6.C_3.S_3$ x 2

Order 6561: $C_3^6.C_3^2$ x 2

Order 2187: $C_3\times C_3^5.C_3$

Order 729: $C_3^6$

Order 81: $C_3^4$

Order 27: $C_3^3$

Order 9: $C_3^2$ x 2

Order 8: $C_2\times C_4$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 472392: $C_3^6.C_3^3:(C_4\times S_3)$ ($C_1$)

Order 236196: $C_3^6.C_3\wr C_4$ ($C_2$) x 2, $C_3^6.C_3^3.D_6$ ($C_2$)

Order 118098: $C_3^6.C_3^4.C_2$ ($C_2^2$), $C_3^5.C_3^4.C_6$ ($C_4$), $C_3^6.C_3^4.C_2$ ($C_4$)

Order 78732: $C_3^6.C_3^3:C_4$ ($S_3$) x 2

Order 59049: $C_3^6.C_3^4$ ($C_2\times C_4$)

Order 39366: $C_3^6.C_3^3.C_2$ ($D_6$) x 2

Order 19683: $C_3^6.C_3^3$ ($C_4\times S_3$) x 2

Order 13122: $C_3^6.C_3.S_3$ ($C_3^2:C_4$), $C_3^6.C_3.S_3$ ($S_3^2$)

Order 6561: $C_3^6.C_3^2$ ($C_2\times C_3^2:C_4$), $C_3^6.C_3^2$ ($C_6.D_6$)

Order 2187: $C_3\times C_3^5.C_3$ ($C_3^2:C_4\times S_3$) x 2

Order 729: $C_3^6$ ($C_3^3:(C_4\times S_3)$)

Order 81: $C_3^4$ ($\He_3^2:(C_2\times C_4)$)

Order 27: $C_3^3$ ($C_3^4.C_3^3.C_4.C_2$) x 2

Order 9: $C_3^2$ ($C_3^4.C_3^4.C_2^2.C_2$), $C_3^2$ ($C_3^4.C_3^4.C_2^2.C_2$)

Order 3: $C_3$ ($C_3^5.C_3^4.C_2^2.C_2$) x 2

Order 1: $C_1$ ($C_3^6.C_3^3:(C_4\times S_3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 472392: $C_3^6.C_3^3:(C_4\times S_3)$ ($C_1$)

Order 236196: $C_3^6.C_3^3.D_6$ ($C_2$), $C_3^6.C_3\wr C_4$ ($C_2$)

Order 118098: $C_3^6.C_3^4.C_2$ ($C_2^2$), $C_3^5.C_3^4.C_6$ ($C_4$), $C_3^6.C_3^4.C_2$ ($C_4$)

Order 78732: $C_3^6.C_3^3:C_4$ ($S_3$)

Order 59049: $C_3^6.C_3^4$ ($C_2\times C_4$)

Order 39366: $C_3^6.C_3^3.C_2$ ($D_6$)

Order 19683: $C_3^6.C_3^3$ ($C_4\times S_3$)

Order 13122: $C_3^6.C_3.S_3$ ($C_3^2:C_4$), $C_3^6.C_3.S_3$ ($S_3^2$)

Order 6561: $C_3^6.C_3^2$ ($C_2\times C_3^2:C_4$), $C_3^6.C_3^2$ ($C_6.D_6$)

Order 2187: $C_3\times C_3^5.C_3$ ($C_3^2:C_4\times S_3$)

Order 729: $C_3^6$ ($C_3^3:(C_4\times S_3)$)

Order 81: $C_3^4$ ($\He_3^2:(C_2\times C_4)$)

Order 27: $C_3^3$ ($C_3^4.C_3^3.C_4.C_2$)

Order 9: $C_3^2$ ($C_3^4.C_3^4.C_2^2.C_2$), $C_3^2$ ($C_3^4.C_3^4.C_2^2.C_2$)

Order 3: $C_3$ ($C_3^5.C_3^4.C_2^2.C_2$)

Order 1: $C_1$ ($C_3^6.C_3^3:(C_4\times S_3)$)

Series

Derived series

$C_3^6.C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^6.C_3^4$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^6.C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^6.C_3^3.D_6$

$\rhd$

$C_3^6.C_3^4.C_2$

$\rhd$

$C_3^6.C_3^4$

$\rhd$

$C_3^6.C_3^2$

$\rhd$

$C_3\times C_3^5.C_3$

$\rhd$

$C_3^6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^6.C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^6.C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $248 \times 248$ 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 $132 \times 132$ rational character table (warning: may be slow to load).