Abstract group 1417176.ss: $C_3^5.\He_3^2:(C_2\times C_4)$

• Label: 1417176.ss
• Order: \( 2^{3} \cdot 3^{11} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{15} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{4} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^5.\He_3^2:(C_2\times C_4)$
Order:	\(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(918330048\)\(\medspace = 2^{6} \cdot 3^{15} \)
Composition factors:	$C_2$ x 3, $C_3$ x 11
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
Elements	1	8019	54674	236196	208494	122472	472392	314928	1417176
Conjugacy classes	1	3	163	4	14	81	4	20	290
Divisions	1	3	114	2	13	54	2	12	201
Autjugacy classes	1	2	22	1	8	15	1	5	55

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	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid d^{3}=e^{9}=f^{9}=g^{3}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,7,26,20,3,8,27,19,2,9,25,21)(4,23,6,24)(5,22)(10,30,35,16,12,29,36,17,11,28,34,18) \!\cdots\! \rangle$
Transitive group:	36T40527				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^6.C_3^3.D_6)$ . $S_3$	$(C_3^6.C_3^4.C_2)$ . $D_6$	$(C_3^7.C_3^4.C_4)$ . $C_2$ (3)	$(C_3^6.C_3^5.C_2)$ . $C_4$ (2)	all 18

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 32 normal subgroups (12 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_3^6$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^7.C_3^4$

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 81 (order at least 17496) are shown, as well as all normal subgroups of any index.

Order 1417176: $C_3^5.\He_3^2:(C_2\times C_4)$

Order 708588: $C_3^7.C_3^4.C_4$ x 2, $C_3^6.C_3^5.C_2^2$

Order 472392: $C_3^6.C_3^4:(C_2\times C_4)$

Order 354294: $C_3^6.C_3^5.C_2$ x 3

Order 236196: $C_3^6.C_3^4.C_4$ x 2, $C_3^6.C_3^4.C_2^2$ x 2, $C_3^5.C_3^5.C_2^2$ x 2, $C_3^6.C_3^3.D_6$

Order 177147: $C_3^7.C_3^4$

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

Order 118098: $C_3^6.C_3^4.C_2$ x 21, $C_3^5.C_3^4.C_6$ x 3, $C_3^5.C_3^5.C_2$ x 2, $C_3^3.C_3^5.C_3.C_6$

Order 78732: $C_3^7.C_3^2.C_4$ x 6, $C_3^4.C_3^5.C_2^2$ x 6, $C_3^6.C_3^3.C_4$ x 4, $C_3^5.C_3^4.C_2^2$ x 4, $C_3^6.C_3^3.C_2^2$ x 2, $C_3^5.\He_3.D_6$ x 2, $C_3^5.C_3^4.C_4$ x 2, $C_3^6.C_3^2.D_6$, $C_3^5.C_3^3.D_6$

Order 59049: $C_3^6.C_3^4$ x 14, $C_3^7.C_3^3$ x 11

Order 52488: $(C_3\times C_9^2):S_3.S_3^2$ x 4, $C_3^6.C_3:S_3.C_2^2$ x 2

Order 39366: $C_3^5.C_3^4.C_2$ x 37, $C_3^4.C_3^5.C_2$ x 29, $C_3^6.C_3^3.C_2$ x 23, $C_3^5.\He_3.C_6$ x 12, $C_3^5.\He_3:S_3$ x 8, $C_3^5.C_3^3.C_6$ x 8, $C_3^3.C_3^5.C_6$ x 8, $C_3^3.C_3^4.C_3.C_6$ x 5, $C_3^4.C_3^4.C_6$ x 4, $C_3^4.C_3^3.C_3.C_6$ x 4, $C_3.C_3^5.C_3^3.C_2$ x 3, $C_3^7.C_3.C_6$ x 2, $C_3^5.\He_3.S_3$

Order 26244: $C_3^5.C_3^3.C_4$ x 24, $C_3^6.C_3^2.C_4$ x 22, $C_3^4.C_3^4.C_2^2$ x 20, $C_3^5.C_3^3.C_2^2$ x 6, $C_3^5.C_3^2.D_6$ x 6, $C_3^3.C_3^5.C_2^2$ x 6, $C_3^4.C_3^3.D_6$ x 3, $C_3^7.C_{12}$ x 2, $C_3^6.S_3^2$ x 2, $C_3^6.C_3.D_6$, $C_3^6.(C_6\times S_3)$, $C_3^4.C_3^4.C_4$, $C_3^3.C_3^5.C_4$

Order 19683: $C_3^6.C_3^3$ x 193, $C_3^5.C_3^4$ x 54, $C_3^7.C_3^2$ x 33, $C_3^4.C_3^5$

Order 17496: $(C_3^2\times C_9^2):(C_4\times S_3)$ x 4, $C_9^2:(C_2\times C_3^3:C_4)$ x 2, $C_3^7:(C_2\times C_4)$

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

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

Order 2187: $C_3^7$

Order 729: $C_3^6$ x 3

Order 243: $C_3^5$ x 3

Order 81: $C_3^4$

Order 9: $C_3^2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1417176: $C_3^5.\He_3^2:(C_2\times C_4)$ ($C_1$)

Order 708588: $C_3^7.C_3^4.C_4$ ($C_2$) x 2, $C_3^6.C_3^5.C_2^2$ ($C_2$)

Order 354294: $C_3^6.C_3^5.C_2$ ($C_4$) x 2, $C_3^6.C_3^5.C_2$ ($C_2^2$)

Order 236196: $C_3^6.C_3^3.D_6$ ($S_3$)

Order 177147: $C_3^7.C_3^4$ ($C_2\times C_4$)

Order 118098: $C_3^6.C_3^4.C_2$ ($D_6$), $C_3^6.C_3^4.C_2$ ($C_3:C_4$), $C_3^5.C_3^4.C_6$ ($C_3:C_4$)

Order 59049: $C_3^6.C_3^4$ ($C_6:C_4$)

Order 39366: $C_3^6.C_3^3.C_2$ ($C_3^2:C_4$) x 2

Order 19683: $C_3^7.C_3^2$ ($C_2\times C_3^2:C_4$) x 2

Order 13122: $C_3^6.C_3.S_3$ ($C_3^3:C_4$) x 2

Order 6561: $C_3^6.C_3^2$ ($C_2\times C_3^3:C_4$) x 2

Order 2187: $C_3^7$ ($C_3^4:(C_2\times C_4)$)

Order 729: $C_3^6$ ($C_3^3:S_3^2.C_2$) x 2, $C_3^6$ ($C_3^5:(C_2\times C_4)$)

Order 243: $C_3^5$ ($C_3.C_3^5.C_4.C_2$) x 2, $C_3^5$ ($\He_3^2:(C_2\times C_4)$)

Order 81: $C_3^4$ ($\He_3^2:(C_6:C_4)$)

Order 9: $C_3^2$ ($C_3^5.C_3^4.C_2^2.C_2$), $C_3^2$ ($C_3^4.C_3^5.C_4.C_2$)

Order 1: $C_1$ ($C_3^5.\He_3^2:(C_2\times C_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

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

Character theory

Complex character table

The $290 \times 290$ character table is not available for this group.

Rational character table

The $201 \times 201$ rational character table is not available for this group.