Abstract group 944784.sc: $C_9^4.(C_6\times S_4)$

• Label: 944784.sc
• Order: \( 2^{4} \cdot 3^{10} \)
• Exponent: \( 2^{2} \cdot 3^{3} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_9^4.(C_6\times S_4)$
Order:	\(944784\)\(\medspace = 2^{4} \cdot 3^{10} \)
Exponent:	\(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
Automorphism group:	$C_9^3.(C_9\times A_4).C_6^2.C_2$, of order \(5668704\)\(\medspace = 2^{5} \cdot 3^{11} \)
Composition factors:	$C_2$ x 4, $C_3$ x 10
Derived length:	$4$

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	27	36
Elements	1	11475	6074	8748	222534	66096	69984	297432	104976	157464	944784
Conjugacy classes	1	5	14	2	28	114	6	107	15	6	298
Divisions	1	5	12	2	22	102	4	92	12	4	256
Autjugacy classes	1	5	13	2	26	61	6	61	7	6	188

Dimension	1	2	3	4	6	8	12	16	18	24	32	36	48	72	96	144
Irr. complex chars.	12	12	12	3	20	18	28	9	6	28	0	25	20	86	0	19	298
Irr. rational chars.	4	8	4	5	12	7	16	9	6	33	3	25	16	86	3	19	256

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:	${\langle a, b, c, d, e, f \mid b^{6}=c^{18}=d^{18}=e^{9}=f^{9}=[c,f]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,22,32)(2,24,31,3,23,33)(4,5)(7,27,35,21,13,11)(8,25,34,19,15,10)(9,26,36,20,14,12) \!\cdots\! \rangle$
Transitive group:	36T36473				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:	$(C_9\wr S_4)$ . $C_6$	$(C_9^4:S_4)$ . $C_6$	$(C_9^4.C_6)$ . $S_4$	$C_9^4$ . $(C_6\times S_4)$	all 36

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

Homology

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

Subgroups

There are 38 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_9\wr A_4$
Frattini:	a subgroup isomorphic to $C_3^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^4.C_3^2$

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 162 (order at least 5832) are shown, as well as all normal subgroups of any index.

Order 944784: $C_9^4.(C_6\times S_4)$

Order 472392: $C_9^4.(C_6\times A_4)$, $C_9^4.(C_3\times S_4)$, $C_9^4:(C_3\times S_4)$

Order 314928: $C_9^4.(C_2\times S_4)$, $C_9^4.(C_6\times D_4)$, $C_3^5.S_3\wr S_3$

Order 236196: $(C_3^2\times C_9^2).C_3^4.C_2^2$, $C_9^4:(C_3\times A_4)$

Order 157464: $C_9^4:(C_3\times D_4)$ x 2, $C_9^3.C_6.C_6^2$ x 2, $C_9^4.C_6.C_2^2$, $C_9^4:(C_2\times A_4)$, $C_9^4:(C_2\times A_4)$, $C_9^4:S_4$, $C_9^3.A_4.C_{18}$, $C_9\wr S_4$, $C_9^3.(C_3\times A_4).C_6$, $C_9^3.(C_3^2.S_4)$, $C_3^5.S_3\wr C_3$, $C_9^3:(C_3^2\times S_4)$, $C_9^4.(C_2\times C_{12})$, $C_9^4.(C_2^2\times C_6)$

Order 118098: $C_9^4.C_3.C_6$, $C_9^2.C_9^2.C_3.C_6$, $C_9^2.C_3^3.C_3^3.C_2$

Order 104976: $C_3:D_9^3:C_6$ x 2, $D_9^2.D_9^2$, $C_9^3:(S_3\times S_4)$, $C_9^3:(C_6\times S_4)$

Order 78732: $C_9^4.C_6.C_2$ x 3, $C_9^4.C_{12}$ x 2, $C_9^4:(C_2\times C_6)$ x 2, $C_9^3.C_3^3.C_2^2$, $C_9^2.C_3^3.(C_6\times S_3)$, $C_9^2.(C_3\times C_9).C_6^2$, $C_3^3.C_3^3.C_3^3.C_2^2$, $(C_3\times C_9^3).C_6^2$, $(C_3\times C_9^2).C_3^4.C_2^2$, $(C_3\times C_9^2).C_3^3.D_6$, $C_3^4.C_3\wr A_4$, $C_9^4:A_4$, $C_9^3:(C_9\times A_4)$, $C_9\wr A_4$, $C_9^3:(C_3^2\times A_4)$, $C_3^4.C_3\wr A_4$

Order 59049: $C_9^4.C_3^2$

Order 52488: $C_9^3.C_3^2.C_2^3$ x 3, $C_9^3:(S_3\times A_4)$ x 2, $C_9^4:D_4$ x 2, $C_9^3:(D_4\times C_9)$ x 2, $C_9\wr D_4$ x 2, $C_9^3:C_6\wr C_2$ x 2, $C_9^3:(C_3\times D_{12})$ x 2, $C_9^3:(S_3\times C_{12})$ x 2, $C_9^3.C_6^2.C_2$ x 2, $C_3^5.S_3^3$ x 2, $C_9^3.C_2.C_6^2$ x 2, $C_9^3:(C_3\times S_4)$ x 2, $C_9^4.C_2^3$, $C_3^3.C_3^5.C_2^3$, $(C_3\times C_9^3):S_4$, $(C_3\times C_9^3).S_4$, $C_3\times C_9^3:S_4$, $C_9^4.(C_2\times C_4)$, $C_9:D_9^3$, $D_9^3:C_3^2$, $C_9^3:(C_3\times S_4)$

Order 39366: $C_9^2.C_3^3.C_3.C_6$ x 5, $C_3^3.C_3^5.C_6$ x 4, $C_9^4.C_6$ x 3, $C_9^2.C_3^2.C_3^3.C_2$ x 2, $(C_3\times C_9^2).C_3^4.C_2$ x 2, $\He_3:C_3:C_3.C_9.C_3.C_6$, $C_9^3.C_3^2.C_6$, $C_9^2.C_9^2.C_6$, $C_9^2.C_9.C_3^3.C_2$, $C_9^2.(C_9^2\times S_3)$, $C_3^4.C_3^4.C_6$, $C_3\times C_9^2.C_3^3.C_6$, $(C_3^3\times C_9).\He_3:S_3$, $(C_3^3\times C_9).C_3^4.C_2$, $(C_3^2\times C_9^2).C_3^3.C_2$, $(C_3\times C_9^2).\He_3.C_6$, $(C_3\times C_9^2).C_9.C_3.C_6$, $(C_3\times C_9^2).C_3^3.C_6$

Order 34992: $(C_3\times D_9^2):D_{18}$ x 2, $D_9^3:C_6$ x 2, $C_3^5.(S_3\times S_4)$, $C_9^2.(C_6\times S_3\wr C_2)$, $D_9^2:(C_3\times S_3^2)$, $D_9\wr S_3$

Order 26244: $C_9^3.(C_6\times S_3)$ x 13, $C_9^3:(C_6\times S_3)$ x 9, $C_3^3.C_3^5.C_2^2$ x 7, $C_9^4.C_2^2$ x 4, $C_9^2.C_9:C_9.C_2^2$ x 3, $C_9^3:(C_6\times S_3)$ x 3, $C_9^2.C_3^3.C_6.C_2$ x 2, $C_3^3.\He_3.(C_6\times S_3)$ x 2, $C_3^2.C_3^5.C_6.C_2$ x 2, $C_3\times C_9^2.C_9.C_6.C_2$ x 2, $C_3\times C_9^3:A_4$ x 2, $(C_3\times C_9^3).A_4$ x 2, $C_9^3:C_{36}$ x 2, $C_9\wr C_4$ x 2, $C_9^3:(C_2\times C_{18})$ x 2, $C_9\wr C_2^2$ x 2, $(C_3\times C_9^3):C_{12}$ x 2, $C_3\times C_9^3:C_{12}$ x 2, $C_9^3:C_6^2$ x 2, $C_9^3:(C_3\times A_4)$ x 2, $C_9^3.S_3^2$, $C_9^3.C_{18}.C_2$, $C_9^3.C_6^2$, $C_9^2.C_3^4.C_2^2$, $C_9^2.C_3^3.D_6$, $C_9\times C_9^3.C_2^2$, $C_3^4.C_3^4.C_2^2$, $C_3^3.C_3^4.C_6.C_2$, $C_3^3.C_3^3.S_3^2$, $C_3^2.\He_3:C_9.D_6$, $C_3\times (C_3\times C_9^2).C_6^2$, $(C_3^3\times C_9).C_3.C_6^2$

Order 19683: $(C_3\times C_9^3).C_3^2$ x 15, $C_9^4.C_3$ x 2, $C_3\times C_9^3.C_3^2$, $C_9^4.C_3$

Order 17496: $D_9^3:C_3$ x 9, $(C_9\times D_9^2):C_6$ x 4, $(C_3\times C_9^3).C_2^3$ x 3, $C_9^2.C_3^3.C_2^3$ x 2, $C_3^4.C_3^3.C_2^3$ x 2, $C_3^3.C_3^4.C_2^3$ x 2, $(C_3\times D_9^2):C_{18}$ x 2, $C_9\times D_9^2:S_3$ x 2, $(C_9\times D_9^2):C_6$ x 2, $C_9^3.D_{12}$ x 2, $C_9^3:(C_4\times S_3)$ x 2, $C_9^3:(C_2\times C_{12})$ x 2, $(C_3\times C_9^2):D_{36}$ x 2, $C_9^3:(C_3\times D_4)$ x 2, $C_3:D_9^3$ x 2, $C_3\times C_9^3:D_4$ x 2, $(C_3^3\times C_9).C_{12}.C_6$ x 2, $C_9^2:S_3^2:C_6$ x 2, $C_3^4.S_3^3$ x 2, $C_3^5.\SOPlus(4,2)$ x 2, $D_9^2:(S_3\times C_3^2)$ x 2, $C_9^3:S_4$ x 2, $D_9\wr C_3$ x 2, $C_9^3.D_6.C_2$, $C_9^3.C_6.C_2^2$, $C_9:C_3\wr S_4$, $C_9:C_3^4:S_4$, $C_3^5.(S_3\times A_4)$, $C_9^2:C_6.S_3^2$, $C_3^4:C_6.S_3^2$, $C_9^3:S_4$

Order 13122: $C_9^3:(C_3\times S_3)$ x 18, $C_3^3.C_3^4.C_6$ x 11, $C_9^3.C_3^2.C_2$ x 10, $C_9.(C_3\times C_9:C_9).C_6$ x 9, $C_9^3:(C_3\times C_6)$ x 9, $C_9^3:(C_3\times S_3)$ x 9, $C_9^3.C_3.C_6$ x 7, $\He_3:C_3:C_3.C_9.C_6$ x 5, $C_9^2.C_9:C_9.C_2$ x 5, $C_9^2.C_3^3.C_6$ x 5, $C_9.C_9:(C_9:C_3).C_6$ x 5, $C_3\times C_9^2.C_3^2.C_6$ x 5, $(C_3\times C_9^3).C_6$ x 5, $C_3^2.\He_3:C_9.C_6$ x 4, $C_3\times (C_3\times C_9).C_3^3.C_6$ x 4, $C_3^2.C_3^5.C_6$ x 3, $(C_3^3\times C_9).C_3^3.C_2$ x 3, $(C_3\times C_9^2).C_9.C_6$ x 3, $C_9^4.C_2$ x 2, $C_3^4.C_3^4.C_2$ x 2, $C_3^3.C_9.C_3^3.C_2$ x 2, $C_3\times C_9^2.C_9.C_6$ x 2, $C_9^2\wr C_2$ x 2, $\He_3.C_9:C_3.C_3.C_6$, $C_9^3.C_{18}$, $C_9^2.C_9:D_9$, $C_9^2.C_3^2:D_9$, $C_9\times C_9^3:C_2$, $C_9.C_3^4.C_3.C_6$, $C_3^3.C_3^5.C_2$, $C_3^3.(C_3^4.S_3)$, $C_3^2.\He_3:C_9.S_3$, $C_3\times \He_3:C_3.C_3^3.C_2$, $C_3.C_3^5.C_3.C_6$, $(C_3^3\times C_9).\He_3.C_2$, $(C_3^2\times C_9).\He_3:S_3$, $(C_3^2\times C_9).\He_3.C_6$, $(C_3^2\times C_9).C_3^4.C_2$, $(C_3\times C_9^2).\He_3.C_2$, $(C_3\times C_9^2).C_9.S_3$, $(C_3\times C_9).C_9:C_9.C_6$

Order 11664: $D_9^2:D_{18}$ x 2, $D_9^2:C_6\times S_3$ x 2, $C_3^5.(S_3\times D_4)$ x 2, $C_3^3:(D_9\times S_4)$, $S_3^2:D_9^2$, $D_9^2:S_3^2$, $C_3^4:(C_6\times S_4)$

Order 8748: $C_9^3.C_6.C_2$ x 78, $C_3^3.C_3^4.C_2^2$ x 25, $C_9^3:(C_2\times C_6)$ x 22, $C_9^3.D_6$ x 13, $C_9^2.C_3^3.C_2^2$ x 12, $C_3^2.C_3^5.C_2^2$ x 11, $(C_3\times C_9^3).C_2^2$ x 10, $C_9^3:(C_2\times C_6)$ x 9, $C_9^3:D_6$ x 9, $C_9^2.(C_{18}\times S_3)$ x 4, $C_3\times (C_3^2\times C_9).C_6^2$ x 4, $C_3.C_3^5.C_6.C_2$ x 4, $(C_3^3\times C_9).C_6^2$ x 4, $C_3^3.C_3^3.D_6$ x 3, $C_3\times C_9^2.C_{18}.C_2$ x 3, $C_9^2:(S_3\times C_{18})$ x 3, $C_9^3:D_6$ x 3, $C_9^2:(C_3\times S_3^2)$ x 3, $C_3^5.S_3^2$ x 3, $C_9^2.(C_3\times S_3^2)$ x 3, $C_9^3:A_4$ x 3, $C_3^4.C_3^3.C_2^2$ x 2, $C_3\times C_9^3.C_2^2$ x 2, $(C_3\times C_9^2):C_{36}$ x 2, $(C_3\times C_9^2):C_{36}$ x 2, $C_9^3:C_{12}$ x 2, $C_9^3:C_{12}$ x 2, $C_9^3:C_{12}$ x 2, $C_3\times C_9:D_9^2$ x 2, $(C_3^2\times C_9^2):C_{12}$ x 2, $(C_3^2\times C_9^2):C_{12}$ x 2, $\He_3:C_3.C_3^3.C_2^2$, $\He_3.C_3^4.C_2^2$, $C_9\times (C_3\times C_9^2):C_2^2$, $C_9.C_3^4.C_6.C_2$, $C_3^3.\He_3.D_6$, $C_3^3.(C_9\times S_3^2)$, $C_3^2\times C_9:(C_9:C_3):C_2^2$, $C_3^2.C_3^4.D_6$, $C_3\times S_3\times C_9:(C_9:C_3):C_2$, $C_3\times D_9\times C_9:D_9$, $C_3\times C_9\times D_9^2$, $C_2\times C_9^3.C_6$, $C_2\times C_9^2.C_3^2.C_6$, $C_2\times (C_3\times C_9).C_3^3.C_6$, $(C_3\times C_9^2).C_{18}.C_2$, $C_9:C_3\wr A_4$

Order 6561: $C_9^3:C_3^2$ x 18, $(C_3^3\times C_9).\He_3$ x 9, $C_3\times C_9^3.C_3$ x 8, $C_3^4.C_3^4$ x 8, $C_3\times C_3^2.\He_3:C_9$ x 6, $C_9^3:C_3^2$ x 6, $C_9^3:C_9$ x 6, $C_3^2.C_3^4.C_3^2$ x 3, $C_3^2.C_3^4.C_3^2$ x 3, $C_3^2.C_3^4.C_3^2$ x 3, $C_3^2.C_3^4.C_3^2$ x 3, $C_3\times C_3^3.C_3^3.C_3$ x 3, $(C_3^3\times C_9).C_3^3$ x 3, $C_3^2.C_3^4.C_3^2$, $C_3^2.C_3^4.C_3^2$, $(C_3^3\times C_9).C_3^3$, $(C_3\times C_9^3).C_3$, $C_3^2.C_3^5.C_3$, $C_3^2.C_3^5.C_3$, $C_3^3.C_3^4.C_3$, $C_9^3.C_3^2$, $(C_3\times C_9^2).C_3^3$, $C_9^4$, $(C_3^3\times C_9).C_3^3$, $C_3^2.C_3^4.C_3^2$, $C_3.C_3^5.C_3^2$, $(C_3^3\times C_9).C_3^3$, $(C_3^3\times C_9).C_3^3$, $(C_3\times C_9^3).C_3$, $C_9^3.C_9$, $C_3\times C_3^2.C_3^4.C_3$, $C_3^3.C_3^5$

Order 5832: $C_3^3.S_3^3$ x 18, $D_9^3$ x 9, $C_3.C_3^5.C_2^3$ x 4, $C_9^3:D_4$ x 4, $(C_3\times D_9^2):C_6$ x 4, $(C_3\times C_9^2).D_6.C_2$ x 3, $C_2\times C_9^2.(C_6\times S_3)$ x 2, $C_2\times C_3^3.C_3^3.C_2^2$ x 2, $(C_3^3\times C_9).D_6.C_2$ x 2, $D_9^2:C_{18}$ x 2, $D_9^2:C_{18}$ x 2, $C_9^2:C_4\times D_9$ x 2, $C_9^2:D_{36}$ x 2, $C_9\times C_3^3:S_4$ x 2, $(C_9\times S_3^2):C_{18}$ x 2, $(C_9\times S_3)\wr C_2$ x 2, $S_3\times C_9^2:C_{12}$ x 2, $C_9^2:\SOPlus(4,2)$ x 2, $C_3^3.S_3^3$ x 2, $(C_3\times C_9^2):D_{12}$ x 2, $C_9^2:(C_3\times D_{12})$ x 2, $(C_3\times D_9)\wr C_2$ x 2, $(C_3^3\times S_3^2).S_3$ x 2, $C_9:C_3\wr D_4$ x 2, $C_3^2.C_3\wr D_4$ x 2, $C_3^5.D_{12}$ x 2, $C_3^4.(S_3\times C_{12})$ x 2, $C_3^3.S_3^3$ x 2, $C_3^3.S_3^3$, $C_3\times C_9:D_9.C_6.C_2$, $C_2\times C_9^3.C_2^2$, $C_2\times C_3^2.C_3^4.C_2^2$, $C_3^5:S_4$, $(C_3^3\times C_9):S_4$, $C_9:S_3\wr C_3$, $C_3^3:(A_4\times D_9)$, $C_9:D_9.S_3^2$, $C_3^2\wr C_2.S_3^2$, $C_3\times D_9^2:C_6$, $C_3^5:S_4$, $C_3^4:(C_6\times A_4)$, $C_3^5.S_4$

Order 2916: $C_9:D_9^2$

Order 2187: $C_3\times C_9^3$

Order 729: $C_9:C_3^4$, $C_9^3$

Order 243: $C_3^5$, $C_3^3\times C_9$

Order 81: $C_3^4$

Order 27: $C_3^3$

Order 9: $C_9$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 944784: $C_9^4.(C_6\times S_4)$ ($C_1$)

Order 472392: $C_9^4.(C_6\times A_4)$ ($C_2$), $C_9^4.(C_3\times S_4)$ ($C_2$), $C_9^4:(C_3\times S_4)$ ($C_2$)

Order 314928: $C_9^4.(C_2\times S_4)$ ($C_3$)

Order 236196: $C_9^4:(C_3\times A_4)$ ($C_2^2$)

Order 157464: $C_9^4:(C_2\times A_4)$ ($C_6$), $C_9^4:S_4$ ($C_6$), $C_9\wr S_4$ ($C_6$), $C_9^3:(C_3^2\times S_4)$ ($S_3$), $C_9^4.(C_2^2\times C_6)$ ($S_3$)

Order 78732: $C_9\wr A_4$ ($C_2\times C_6$), $C_9^3:(C_3^2\times A_4)$ ($D_6$), $C_9^4:(C_2\times C_6)$ ($D_6$)

Order 52488: $C_3\times C_9^3:S_4$ ($C_3\times S_3$), $C_9:D_9^3$ ($C_3\times S_3$)

Order 39366: $C_9^4.C_6$ ($S_4$)

Order 26244: $C_3\times C_9^3:A_4$ ($C_6\times S_3$), $C_9\wr C_2^2$ ($C_6\times S_3$), $C_9^3:C_6^2$ ($S_3^2$)

Order 19683: $C_9^4.C_3$ ($C_2\times S_4$)

Order 17496: $C_9^3:S_4$ ($C_9:C_6$)

Order 13122: $C_9^4.C_2$ ($C_3\times S_4$)

Order 8748: $C_3\times C_9:D_9^2$ ($C_3\times S_3^2$), $C_9^3:A_4$ ($C_{18}:C_6$)

Order 6561: $C_3\times C_9^3.C_3$ ($S_3\times S_4$), $C_9^4$ ($C_6\times S_4$)

Order 2916: $C_9:D_9^2$ ($C_3^2.S_3^2$)

Order 2187: $C_3\times C_9^3$ ($C_6^2:D_6$)

Order 729: $C_9:C_3^4$ ($S_3\wr S_3$), $C_9^3$ ($C_6^2.S_3^2$)

Order 243: $C_3^5$ ($C_3^3:(S_3\times S_4)$), $C_3^3\times C_9$ ($C_3\times S_3\wr S_3$)

Order 81: $C_3^4$ ($C_3^4:(C_6\times S_4)$)

Order 27: $C_3^3$ ($C_3^5.(S_3\times S_4)$)

Order 9: $C_9$ ($C_9^3:(C_6\times S_4)$)

Order 3: $C_3$ ($C_3^5.S_3\wr S_3$)

Order 1: $C_1$ ($C_9^4.(C_6\times S_4)$)

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 3 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 $298 \times 298$ 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 $256 \times 256$ rational character table (warning: may be slow to load).