Abstract group 104976.pd: $C_9^2\wr C_2.D_4$

• Label: 104976.pd
• Order: \( 2^{4} \cdot 3^{8} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{3} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_9^2\wr C_2.D_4$
Order:	\(104976\)\(\medspace = 2^{4} \cdot 3^{8} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_9^4.C_6.C_6^2.C_2^4$, of order \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \)
Composition factors:	$C_2$ x 4, $C_3$ x 8
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	8199	80	29160	4680	6480	5832	33048	17496	104976
Conjugacy classes	1	5	10	4	14	504	2	228	6	774
Divisions	1	5	10	2	14	168	1	76	1	278
Autjugacy classes	1	5	5	2	7	15	1	11	1	48

Dimension	1	2	4	8	12	16	24	48
Irr. complex chars.	8	2	112	332	0	320	0	0	774
Irr. rational chars.	4	4	14	9	30	2	109	106	278

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	8	not computed	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{4}=b^{18}=c^{18}=d^{9}=e^{9}=[b,d]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,12,14,34)(2,10,13,36)(3,11,15,35)(4,31,17,8)(5,32,16,7)(6,33,18,9)(19,28,21,29) \!\cdots\! \rangle$
Transitive group:	36T20371				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:	$(C_9^4.C_2)$ . $D_4$	$(C_9^2\wr C_2)$ . $D_4$	$D_9^2$ . $(C_9^2:C_4)$	$C_9^2$ . $(D_9^2:C_4)$	all 21

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2} \times C_{18}$
Commutator length:	$1$

Subgroups

There are 4246951 subgroups in 10425 conjugacy classes, 25 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_9^2\wr C_2.D_4$
Commutator:	$G' \simeq$ $C_9^2\wr C_2$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_3^4$	$G/\Phi \simeq$ $(C_3^2\times S_3^2):C_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_9^4$	$G/\operatorname{Fit} \simeq$ $C_2^2:C_4$
Radical:	$R \simeq$ $C_9^2\wr C_2.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $(C_3^2\times S_3^2):C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^4$

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

Order 104976: $C_9^2\wr C_2.D_4$

Order 52488: $C_9^4.C_2^3$, $C_9^4:(C_2\times C_4)$, $C_9^4.(C_2\times C_4)$

Order 26244: $C_9^4.C_4$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9\wr C_4$

Order 17496: $C_9^3.D_6.C_2$ x 2, $(C_3\times C_9^3).C_2^3$, $C_9^3:(C_4\times S_3)$

Order 13122: $C_9^2\wr C_2$ x 2, $C_9\times C_9^3:C_2$, $C_9^4.C_2$, $C_9^4.C_2$

Order 11664: $C_3^2.C_{18}^2:C_4$, $C_9^2:(C_6^2:C_4)$

Order 8748: $C_9^3.D_6$ x 9, $C_9\times D_9\times C_3:D_9$ x 4, $(C_3\times C_9^3).C_2^2$ x 3, $C_3\times C_9^3.C_2^2$ x 2, $C_9^3.D_6$ x 2, $(C_3\times C_9).C_{18}^2$, $C_9:D_9\times C_3:D_9$, $C_9\times S_3\times C_9:D_9$, $C_9^3.D_6$, $C_9^3.C_6.C_2$, $C_9^3.D_6$, $(C_3\times C_9^2):C_{36}$, $C_9^3:C_{12}$

Order 6561: $C_9^4$

Order 5832: $D_9^3$ x 6, $C_9^2.S_3^2.C_2$ x 4, $(C_3^2\times C_9^2):(C_2\times C_4)$ x 2, $(C_3^3\times C_9).D_6.C_2$, $C_2\times C_9^3.C_2^2$, $(C_3^3\times C_9).D_6.C_2$, $C_9^2:C_4\times D_9$, $C_9:D_9.S_3^2$, $C_3^2\wr C_2.S_3^2$

Order 4374: $C_9^3.S_3$ x 10, $C_9\times (C_3\times C_9^2):C_2$ x 9, $C_9^3.C_6$ x 5, $C_3\times C_9^3:C_2$ x 4, $C_9^2\times C_3:D_9$ x 2, $C_3\times C_9\times C_9:D_9$ x 2, $C_9^3:S_3$ x 2, $C_9^3:C_6$ x 2, $C_9\times C_9^2\times S_3$, $C_3\times C_9^3:C_2$

Order 2916: $C_9^3.C_2^2$ x 74, $C_9\times D_9\times D_9$ x 12, $(C_3\times C_9^2).D_6$ x 9, $C_9^2.C_3:S_3.C_2$ x 6, $C_9:D_9^2$ x 6, $C_9\times D_9^2$ x 6, $(C_3\times C_9^2).D_6$ x 5, $C_9^2.S_3^2$ x 4, $C_9\times S_3\times C_3:D_9$ x 4, $(C_3\times C_9^2).D_6$ x 4, $C_3\times (C_3\times C_9^2):C_2^2$ x 4, $C_9^2.S_3^2$ x 4, $C_3\times D_9\times C_3:D_9$ x 4, $(C_3:D_9)^2$ x 2, $C_9:D_9\times D_9$ x 2, $C_3\times (C_3\times C_9^2):C_2^2$, $(C_3^3\times C_9).D_6$, $(C_3\times C_9^2).D_6$, $C_9\times D_9\times C_3:S_3$, $C_2\times C_9^3:C_2$, $C_9^2.(C_6\times S_3)$, $D_9\times C_9:D_9$, $(C_3\times C_9^2).D_6$, $C_3^4.C_6^2$, $(C_3\times C_9^2).D_6$, $C_9^2:C_{36}$, $C_9^3:C_4$, $(C_3^2\times C_9):C_{36}$, $(C_3\times C_9^2):C_{12}$, $C_9^2:D_{18}$, $D_{18}\times C_9^2$

Order 2187: $C_3\times C_9^3$ x 10

Order 1944: $S_3\times D_9^2$ x 14, $C_6:D_9^2$ x 4, $C_9:S_3^3$ x 2, $C_3^2.S_3^3$, $C_3^3:(C_4\times D_9)$, $S_3\times C_9:D_{18}$, $C_9^2:C_4\times S_3$

Order 1458: $C_9^3:C_2$ x 169, $D_9\times C_9^2$ x 75, $C_9^2:C_{18}$ x 75, $(C_3\times C_9^2):S_3$ x 28, $(C_3\times C_9)\wr C_2$ x 12, $C_3\times C_9^2:S_3$ x 9, $C_9\times C_3^2:D_9$ x 9, $C_9\times C_3^2\times D_9$ x 9, $C_3\times S_3\times C_9^2$ x 9, $C_3\times C_9^2:C_6$ x 2, $C_9\times C_3^2:C_{18}$ x 2, $C_9^2\times C_{18}$

Order 1296: $C_{18}^2:C_4$, $D_9^2:C_4$, $(C_3^2\times S_3^2):C_4$

Order 972: $C_3:D_9^2$ x 128, $C_9^2:D_6$ x 71, $C_9^2:D_6$ x 28, $C_3:D_9^2$ x 14, $C_3\times D_9^2$ x 14, $C_3^3:D_{18}$ x 9, $C_9^2:D_6$ x 9, $C_3^3:D_{18}$ x 4, $C_3^3:D_{18}$ x 4, $C_9^2:D_6$ x 4, $C_3\times C_9\times D_{18}$ x 4, $C_3^3:D_{18}$ x 2, $C_3^3.C_6^2$ x 2, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $S_3\times C_3^2:C_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:C_{36}$, $(C_3^2\times C_9):C_{12}$, $C_3\times C_9:D_{18}$, $D_6\times C_9^2$, $(C_3\times C_9^2):C_4$, $C_9^2:C_{12}$

Order 729: $C_9^3$ x 168, $C_3^2\times C_9^2$ x 28

Order 648: $D_9\times D_{18}$ x 13, $D_9\times S_3^2$ x 10, $C_{18}:S_3^2$ x 4, $C_2\times C_9^2:C_4$ x 3, $C_3:S_3^3$, $C_3^4:(C_2\times C_4)$, $C_3^3:(C_4\times S_3)$, $C_3:S_3\times D_{18}$, $C_3^2:C_4\times D_9$, $C_{18}.D_{18}$, $C_{18}:D_{18}$

Order 486: $C_9^2:S_3$ x 679, $C_9^2:C_6$ x 120, $C_9^2:S_3$ x 120, $S_3\times C_9^2$ x 69, $C_9^2:C_6$ x 69, $C_3^3:D_9$ x 10, $C_3^3:D_9$ x 9, $C_3^3:C_{18}$ x 9, $C_3\times C_9\times C_{18}$ x 9, $C_3^3:C_{18}$ x 4, $C_3^3:D_9$ x 4, $C_3^3:C_{18}$, $D_9\times C_3^3$

Order 324: $D_9^2$ x 133, $C_9:S_3^2$ x 108, $C_9:D_{18}$ x 69, $C_9^2:C_4$ x 47, $C_9\times D_{18}$ x 25, $C_3^2:D_{18}$ x 21, $C_3^2:D_{18}$ x 20, $C_9:S_3^2$ x 10, $C_9\times S_3^2$ x 10, $C_3^2:D_{18}$ x 9, $C_3^2:D_{18}$ x 4, $C_3^2.C_6^2$ x 4, $C_{18}^2$, $C_9:C_{36}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^4:C_4$, $C_3\wr C_4$, $C_3:S_3\times C_{18}$, $C_3^2\times D_{18}$, $(C_3^2\times C_9):C_4$, $C_3^2:C_{36}$

Order 243: $C_3\times C_9^2$ x 670, $C_3^3\times C_9$ x 10

Order 216: $S_3\times D_{18}$ x 17, $S_3^3$ x 2, $C_6:D_{18}$ x 2, $C_6.D_{18}$, $C_6:S_3^2$, $C_3^2:C_4\times S_3$

Order 162: $C_9:D_9$ x 1483, $C_3^2:D_9$ x 257, $C_9\times D_9$ x 214, $C_3^2:D_9$ x 100, $C_3^2:C_{18}$ x 100, $C_9\times C_{18}$ x 67, $C_3^2:C_{18}$ x 19, $C_3^2\times D_9$ x 19, $C_3^2\times C_{18}$ x 9, $C_3^2\wr C_2$ x 2, $C_3^3:S_3$, $C_3^3:C_6$, $S_3\times C_3^3$

Order 144: $C_6^2:C_4$, $S_3^2:C_4$

Order 108: $S_3\times D_9$ x 140, $C_3:D_{18}$ x 98, $S_3\times C_{18}$ x 17, $C_3\times D_{18}$ x 17, $C_3:S_3^2$ x 15, $C_3\times S_3^2$ x 6, $C_3:S_3^2$ x 2, $C_6\times C_{18}$ x 2, $C_3:C_{36}$, $C_9:C_{12}$, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 81: $C_9^2$ x 1416, $C_3^2\times C_9$ x 248, $C_3^4$

Order 72: $C_2\times D_{18}$ x 7, $S_3\times D_6$ x 5, $C_2\times C_3^2:C_4$ x 3, $C_4\times D_9$, $C_6:D_6$, $C_6.D_6$

Order 54: $C_3:D_9$ x 762, $S_3\times C_9$ x 106, $C_3\times D_9$ x 106, $C_3\times C_{18}$ x 92, $C_3^2:C_6$ x 13, $S_3\times C_3^2$ x 13, $C_3^2:S_3$ x 11, $C_3^2\times C_6$

Order 36: $D_{18}$ x 97, $S_3^2$ x 25, $C_6:S_3$ x 13, $C_6\times S_3$ x 9, $C_3^2:C_4$ x 7, $C_2\times C_{18}$ x 7, $C_3:C_{12}$, $C_{36}$, $C_6^2$, $C_9:C_4$

Order 27: $C_3\times C_9$ x 670, $C_3^3$ x 10

Order 24: $C_2\times D_6$ x 3, $C_4\times S_3$

Order 18: $D_9$ x 244, $C_{18}$ x 76, $C_3:S_3$ x 39, $C_3\times S_3$ x 30, $C_3\times C_6$ x 11

Order 16: $C_2^2:C_4$

Order 12: $D_6$ x 23, $C_2\times C_6$ x 3, $C_{12}$, $C_3:C_4$

Order 9: $C_9$ x 168, $C_3^2$ x 28

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

Order 6: $S_3$ x 24, $C_6$ x 14

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

Order 3: $C_3$ x 10

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 104976: $C_9^2\wr C_2.D_4$

Order 52488: $C_9^4.C_2^3$, $C_9^4:(C_2\times C_4)$, $C_9^4.(C_2\times C_4)$

Order 26244: $C_9^4.C_4$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9^4.C_2^2$, $C_9\wr C_4$

Order 17496: $C_9^3.D_6.C_2$, $(C_3\times C_9^3).C_2^3$, $C_9^3:(C_4\times S_3)$

Order 13122: $C_9^2\wr C_2$ x 2, $C_9\times C_9^3:C_2$, $C_9^4.C_2$, $C_9^4.C_2$

Order 11664: $C_3^2.C_{18}^2:C_4$, $C_9^2:(C_6^2:C_4)$

Order 8748: $C_9^3.D_6$ x 3, $(C_3\times C_9^3).C_2^2$ x 2, $(C_3\times C_9).C_{18}^2$, $C_9:D_9\times C_3:D_9$, $C_9\times S_3\times C_9:D_9$, $C_9^3.D_6$, $C_3\times C_9^3.C_2^2$, $C_9\times D_9\times C_3:D_9$, $C_9^3.C_6.C_2$, $C_9^3.D_6$, $C_9^3.D_6$, $(C_3\times C_9^2):C_{36}$, $C_9^3:C_{12}$

Order 6561: $C_9^4$

Order 5832: $(C_3^2\times C_9^2):(C_2\times C_4)$ x 2, $(C_3^3\times C_9).D_6.C_2$, $C_2\times C_9^3.C_2^2$, $(C_3^3\times C_9).D_6.C_2$, $C_9^2.S_3^2.C_2$, $C_9^2:C_4\times D_9$, $D_9^3$, $C_9:D_9.S_3^2$, $C_3^2\wr C_2.S_3^2$

Order 4374: $C_9^3.S_3$ x 5, $C_9\times (C_3\times C_9^2):C_2$ x 3, $C_9^3.C_6$ x 2, $C_3\times C_9\times C_9:D_9$ x 2, $C_9^3:S_3$ x 2, $C_3\times C_9^3:C_2$, $C_9^2\times C_3:D_9$, $C_9\times C_9^2\times S_3$, $C_3\times C_9^3:C_2$, $C_9^3:C_6$

Order 2916: $C_9^3.C_2^2$ x 9, $(C_3\times C_9^2).D_6$ x 3, $C_9^2.C_3:S_3.C_2$ x 3, $(C_3\times C_9^2).D_6$ x 2, $C_9:D_9\times D_9$ x 2, $C_9\times D_9\times D_9$, $C_3\times (C_3\times C_9^2):C_2^2$, $(C_3^3\times C_9).D_6$, $(C_3\times C_9^2).D_6$, $C_9\times D_9\times C_3:S_3$, $C_2\times C_9^3:C_2$, $C_9^2.S_3^2$, $C_9\times S_3\times C_3:D_9$, $(C_3\times C_9^2).D_6$, $(C_3:D_9)^2$, $C_9^2.(C_6\times S_3)$, $C_3\times (C_3\times C_9^2):C_2^2$, $D_9\times C_9:D_9$, $(C_3\times C_9^2).D_6$, $C_3^4.C_6^2$, $(C_3\times C_9^2).D_6$, $C_9^2.S_3^2$, $C_3\times D_9\times C_3:D_9$, $C_9^2:C_{36}$, $C_9^3:C_4$, $C_9:D_9^2$, $C_9\times D_9^2$, $(C_3^2\times C_9):C_{36}$, $(C_3\times C_9^2):C_{12}$, $C_9^2:D_{18}$, $D_{18}\times C_9^2$

Order 2187: $C_3\times C_9^3$ x 5

Order 1944: $S_3\times D_9^2$ x 2, $C_9:S_3^3$, $C_3^2.S_3^3$, $C_3^3:(C_4\times D_9)$, $S_3\times C_9:D_{18}$, $C_6:D_9^2$, $C_9^2:C_4\times S_3$

Order 1458: $C_9^3:C_2$ x 16, $D_9\times C_9^2$ x 10, $C_9^2:C_{18}$ x 10, $(C_3\times C_9^2):S_3$ x 10, $(C_3\times C_9)\wr C_2$ x 4, $C_3\times C_9^2:S_3$ x 3, $C_9\times C_3^2:D_9$ x 3, $C_9\times C_3^2\times D_9$ x 3, $C_3\times S_3\times C_9^2$ x 3, $C_3\times C_9^2:C_6$ x 2, $C_9\times C_3^2:C_{18}$ x 2, $C_9^2\times C_{18}$

Order 1296: $C_{18}^2:C_4$, $D_9^2:C_4$, $(C_3^2\times S_3^2):C_4$

Order 972: $C_3:D_9^2$ x 17, $C_9^2:D_6$ x 10, $C_3^3:D_{18}$ x 3, $C_3^3:D_{18}$ x 3, $C_9^2:D_6$ x 3, $C_9^2:D_6$ x 3, $C_3:D_9^2$ x 2, $C_3\times D_9^2$ x 2, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $S_3\times C_3^2:C_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3.C_6^2$, $C_3^3:D_{18}$, $C_3^3:C_{36}$, $(C_3^2\times C_9):C_{12}$, $C_3\times C_9:D_{18}$, $C_9^2:D_6$, $D_6\times C_9^2$, $C_3\times C_9\times D_{18}$, $(C_3\times C_9^2):C_4$, $C_9^2:C_{12}$

Order 729: $C_9^3$ x 15, $C_3^2\times C_9^2$ x 10

Order 648: $C_2\times C_9^2:C_4$ x 3, $D_9\times S_3^2$ x 2, $D_9\times D_{18}$ x 2, $C_3:S_3^3$, $C_3^4:(C_2\times C_4)$, $C_3^3:(C_4\times S_3)$, $C_{18}:S_3^2$, $C_3:S_3\times D_{18}$, $C_3^2:C_4\times D_9$, $C_{18}.D_{18}$, $C_{18}:D_{18}$

Order 486: $C_9^2:S_3$ x 54, $C_9^2:C_6$ x 15, $C_9^2:S_3$ x 15, $S_3\times C_9^2$ x 8, $C_9^2:C_6$ x 8, $C_3^3:D_9$ x 5, $C_3^3:D_9$ x 3, $C_3^3:C_{18}$ x 3, $C_3^3:D_9$ x 3, $C_3^3:C_{18}$ x 3, $C_3\times C_9\times C_{18}$ x 3, $C_3^3:C_{18}$, $D_9\times C_3^3$

Order 324: $C_9:S_3^2$ x 15, $D_9^2$ x 13, $C_3^2:D_{18}$ x 10, $C_9:D_{18}$ x 9, $C_9^2:C_4$ x 6, $C_9\times D_{18}$ x 3, $C_3^2:D_{18}$ x 3, $C_3^2:D_{18}$ x 3, $C_9:S_3^2$ x 2, $C_9\times S_3^2$ x 2, $C_{18}^2$, $C_9:C_{36}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^4:C_4$, $C_3\wr C_4$, $C_3:S_3\times C_{18}$, $C_3^2:D_{18}$, $C_3^2.C_6^2$, $C_3^2\times D_{18}$, $(C_3^2\times C_9):C_4$, $C_3^2:C_{36}$

Order 243: $C_3\times C_9^2$ x 51, $C_3^3\times C_9$ x 5

Order 216: $S_3\times D_{18}$ x 3, $C_6.D_{18}$, $C_6:S_3^2$, $S_3^3$, $C_3^2:C_4\times S_3$, $C_6:D_{18}$

Order 162: $C_9:D_9$ x 59, $C_3^2:D_9$ x 38, $C_9\times D_9$ x 18, $C_3^2:D_9$ x 13, $C_3^2:C_{18}$ x 13, $C_3^2:C_{18}$ x 8, $C_3^2\times D_9$ x 8, $C_9\times C_{18}$ x 7, $C_3^2\times C_{18}$ x 3, $C_3^2\wr C_2$ x 2, $C_3^3:S_3$, $C_3^3:C_6$, $S_3\times C_3^3$

Order 144: $C_6^2:C_4$, $S_3^2:C_4$

Order 108: $S_3\times D_9$ x 20, $C_3:D_{18}$ x 14, $C_3:S_3^2$ x 8, $S_3\times C_{18}$ x 3, $C_3\times D_{18}$ x 3, $C_3\times S_3^2$ x 2, $C_3:C_{36}$, $C_9:C_{12}$, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_6\times C_{18}$

Order 81: $C_9^2$ x 52, $C_3^2\times C_9$ x 35, $C_3^4$

Order 72: $C_2\times C_3^2:C_4$ x 3, $S_3\times D_6$ x 2, $C_2\times D_{18}$ x 2, $C_4\times D_9$, $C_6:D_6$, $C_6.D_6$

Order 54: $C_3:D_9$ x 63, $S_3\times C_9$ x 14, $C_3\times D_9$ x 14, $C_3\times C_{18}$ x 12, $C_3^2:S_3$ x 6, $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2\times C_6$

Order 36: $D_{18}$ x 16, $S_3^2$ x 9, $C_6:S_3$ x 7, $C_3^2:C_4$ x 4, $C_6\times S_3$ x 3, $C_2\times C_{18}$ x 2, $C_3:C_{12}$, $C_{36}$, $C_6^2$, $C_9:C_4$

Order 27: $C_3\times C_9$ x 51, $C_3^3$ x 5

Order 24: $C_2\times D_6$ x 2, $C_4\times S_3$

Order 18: $D_9$ x 26, $C_3:S_3$ x 15, $C_{18}$ x 11, $C_3\times S_3$ x 10, $C_3\times C_6$ x 5

Order 16: $C_2^2:C_4$

Order 12: $D_6$ x 12, $C_2\times C_6$ x 2, $C_{12}$, $C_3:C_4$

Order 9: $C_9$ x 15, $C_3^2$ x 10

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

Order 6: $S_3$ x 12, $C_6$ x 7

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 104976: $C_9^2\wr C_2.D_4$ ($C_1$)

Order 52488: $C_9^4.C_2^3$ ($C_2$), $C_9^4:(C_2\times C_4)$ ($C_2$), $C_9^4.(C_2\times C_4)$ ($C_2$)

Order 26244: $C_9^4.C_2^2$ ($C_4$), $C_9^4.C_2^2$ ($C_2^2$), $C_9^4.C_2^2$ ($C_4$)

Order 13122: $C_9^4.C_2$ ($D_4$), $C_9^2\wr C_2$ ($D_4$), $C_9^2\wr C_2$ ($C_2\times C_4$)

Order 6561: $C_9^4$ ($C_2^2:C_4$)

Order 2916: $C_3\times (C_3\times C_9^2):C_2^2$ ($C_3^2:C_4$)

Order 1458: $C_3\times C_9^2:C_6$ ($C_2\times C_3^2:C_4$), $C_3\times C_9^2:C_6$ ($\SOPlus(4,2)$)

Order 729: $C_3^2\times C_9^2$ ($C_6^2:C_4$), $C_3^2\times C_9^2$ ($S_3^2:C_4$)

Order 324: $D_9^2$ ($C_9^2:C_4$)

Order 162: $C_9:D_9$ ($C_2\times C_9^2:C_4$), $C_9:D_9$ ($D_9\wr C_2$)

Order 81: $C_9^2$ ($C_{18}^2:C_4$), $C_9^2$ ($D_9^2:C_4$), $C_3^4$ ($(C_3^2\times S_3^2):C_4$)

Order 9: $C_3^2$ ($C_3^2.C_{18}^2:C_4$), $C_3^2$ ($C_9^2:(C_6^2:C_4)$)

Order 1: $C_1$ ($C_9^2\wr C_2.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 104976: $C_9^2\wr C_2.D_4$ ($C_1$)

Order 52488: $C_9^4.C_2^3$ ($C_2$), $C_9^4:(C_2\times C_4)$ ($C_2$), $C_9^4.(C_2\times C_4)$ ($C_2$)

Order 26244: $C_9^4.C_2^2$ ($C_4$), $C_9^4.C_2^2$ ($C_2^2$), $C_9^4.C_2^2$ ($C_4$)

Order 13122: $C_9^4.C_2$ ($D_4$), $C_9^2\wr C_2$ ($D_4$), $C_9^2\wr C_2$ ($C_2\times C_4$)

Order 6561: $C_9^4$ ($C_2^2:C_4$)

Order 2916: $C_3\times (C_3\times C_9^2):C_2^2$ ($C_3^2:C_4$)

Order 1458: $C_3\times C_9^2:C_6$ ($C_2\times C_3^2:C_4$), $C_3\times C_9^2:C_6$ ($\SOPlus(4,2)$)

Order 729: $C_3^2\times C_9^2$ ($C_6^2:C_4$), $C_3^2\times C_9^2$ ($S_3^2:C_4$)

Order 324: $D_9^2$ ($C_9^2:C_4$)

Order 162: $C_9:D_9$ ($C_2\times C_9^2:C_4$), $C_9:D_9$ ($D_9\wr C_2$)

Order 81: $C_9^2$ ($C_{18}^2:C_4$), $C_9^2$ ($D_9^2:C_4$), $C_3^4$ ($(C_3^2\times S_3^2):C_4$)

Order 9: $C_3^2$ ($C_3^2.C_{18}^2:C_4$), $C_3^2$ ($C_9^2:(C_6^2:C_4)$)

Order 1: $C_1$ ($C_9^2\wr C_2.D_4$)

Series

Derived series

$C_9^2\wr C_2.D_4$

$\rhd$

$C_9^2\wr C_2$

$\rhd$

$C_9^2$

$\rhd$

$C_1$

Chief series

$C_9^2\wr C_2.D_4$

$\rhd$

$C_9^4:(C_2\times C_4)$

$\rhd$

$C_9^4.C_2^2$

$\rhd$

$C_9^4.C_2$

$\rhd$

$C_9^4$

$\rhd$

$C_3^2\times C_9^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_9^2\wr C_2.D_4$

$\rhd$

$C_9^2\wr C_2$

$\rhd$

$C_9^4$

Upper central series

$C_1$

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

The $774 \times 774$ character table is not available for this group.

Rational character table

See the $278 \times 278$ rational character table (warning: may be slow to load).