Abstract group 104976.pg: $C_9^4:(C_2\times D_4)$

• Label: 104976.pg
• Order: \( 2^{4} \cdot 3^{8} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_9^4:(C_2\times D_4)$
Order:	\(104976\)\(\medspace = 2^{4} \cdot 3^{8} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_9^4.(C_6\times D_4).C_6.C_2^3$, of order \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \)
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	18
Elements	1	7371	80	26244	6480	6480	58320	104976
Conjugacy classes	1	7	10	2	16	450	108	594
Divisions	1	7	10	2	16	150	36	222
Autjugacy classes	1	3	3	1	3	12	5	28

Dimension	1	2	4	8	12	16	24	48
Irr. complex chars.	8	2	64	152	0	368	0	0	594
Irr. rational chars.	8	2	16	8	16	2	48	122	222

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	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:	${\langle a, b, c, d, e, f \mid b^{4}=c^{18}=d^{9}=e^{9}=f^{9}=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,5,15,30,27,16,3,6,14,28,26,17,2,4,13,29,25,18)(7,35,20,23,32,12,9,36,19,24,31,10,8,34,21,22,33,11) \!\cdots\! \rangle$
Transitive group:	36T20377				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_9:D_9^3)$ $\,\rtimes\,$ $C_2$	$(C_9^4:D_4)$ $\,\rtimes\,$ $C_2$	$C_9^4$ $\,\rtimes\,$ $(C_2\times D_4)$	$(C_9^2\wr C_2)$ $\,\rtimes\,$ $D_4$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_9^4.C_2)$ . $C_2^3$	$(C_9^4.C_2^2)$ . $C_2^2$	$C_3^4$ . $(C_3^4:(C_2\times D_4))$	$(C_3^2\times C_9^2)$ . $(S_3^2:C_2^2)$	all 6

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 4058427 subgroups in 9707 conjugacy classes, 31 normal (7 characteristic).

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

Special subgroups

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

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 104976: $C_9^4:(C_2\times D_4)$

Order 52488: $C_9^4:D_4$ x 4, $C_9:D_9^3$ x 2, $C_9^4:(C_2\times C_4)$

Order 26244: $C_9^4.C_2^2$ x 5, $C_9\wr C_2^2$ x 4, $C_9^4.C_4$ x 2

Order 17496: $C_3:D_9^3$ x 4

Order 13122: $C_9^2\wr C_2$ x 6, $C_9^4.C_2$

Order 11664: $C_9^2:(S_3^2:C_2^2)$ x 2

Order 8748: $C_9^3:D_6$ x 12, $C_9^3.D_6$ x 8, $(C_3\times C_9^3).C_2^2$ x 8, $C_3\times C_9:D_9^2$ x 4

Order 6561: $C_9^4$

Order 5832: $C_9^2:\SOPlus(4,2)$ x 12, $C_3^3.S_3^3$ x 8, $D_9^3$ x 4, $(C_3^2\times C_9^2):(C_2\times C_4)$ x 2

Order 4374: $C_9^3:S_3$ x 16, $C_9^3:C_6$ x 16, $C_9^3.S_3$ x 10

Order 2916: $C_9^3.C_2^2$ x 36, $(C_3\times C_9^2).D_6$ x 24, $C_9\times D_9^2$ x 12, $C_9^2:S_3^2$ x 12, $C_3^2:D_9^2$ x 12, $C_9^2.C_3:S_3.C_2$ x 10, $C_9^2.S_3^2$ x 8, $(C_3\times C_9^2).D_6$ x 6, $C_9:D_9^2$ x 4, $(C_3^3\times C_9).D_6$ x 2

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

Order 1944: $S_3\times D_9^2$ x 12, $C_9:S_3^3$ x 4

Order 1458: $C_9^3:C_2$ x 150, $(C_3\times C_9)\wr C_2$ x 52, $D_9\times C_9^2$ x 36, $C_9^2:C_{18}$ x 36, $(C_3\times C_9^2):S_3$ x 30, $C_3\times C_9^2:C_6$ x 6, $C_9\times C_3^2:C_{18}$ x 6

Order 1296: $D_9^2:C_2^2$ x 2, $C_3^4:(C_2\times D_4)$

Order 972: $C_3:D_9^2$ x 132, $C_9^2:D_6$ x 24, $C_3^3:D_{18}$ x 16, $C_9^2:D_6$ x 16, $C_3^3:D_{18}$ x 12, $C_3:D_9^2$ x 12, $C_3\times D_9^2$ x 12, $C_9\times C_3:S_3^2$ x 4

Order 729: $C_9^3$ x 150, $C_3^2\times C_9^2$ x 30

Order 648: $D_9\wr C_2$ x 28, $D_9\times S_3^2$ x 12, $D_9\times D_{18}$ x 8, $C_3^4:D_4$ x 4, $C_3:S_3^3$ x 2, $C_2\times C_9^2:C_4$ x 2, $C_3^4:(C_2\times C_4)$

Order 486: $C_9^2:S_3$ x 650, $C_9^2:C_6$ x 132, $C_9^2:S_3$ x 132, $S_3\times C_9^2$ x 16, $C_3^3:C_{18}$ x 16, $C_3^3:D_9$ x 16, $C_9^2:C_6$ x 16, $C_3^3:D_9$ x 10

Order 324: $D_9^2$ x 222, $C_9^2:C_4$ x 58, $C_9:S_3^2$ x 52, $C_3^2:D_{18}$ x 36, $C_3^2:D_{18}$ x 24, $C_9\times D_{18}$ x 12, $C_9:S_3^2$ x 12, $C_9\times S_3^2$ x 12, $C_9:D_{18}$ x 6, $C_3^2:S_3^2$ x 5, $C_3\wr C_2^2$ x 4, $C_3^4:C_4$ x 2

Order 243: $C_3\times C_9^2$ x 650, $C_3^3\times C_9$ x 10

Order 216: $S_3\times D_{18}$ x 12, $S_3^3$ x 4

Order 162: $C_9:D_9$ x 1436, $C_9\times D_9$ x 372, $C_3^2:D_9$ x 230, $C_3^2:D_9$ x 52, $C_3^2:C_{18}$ x 52, $C_3^2:C_{18}$ x 36, $C_3^2\times D_9$ x 36, $C_3^2\wr C_2$ x 6, $C_9\times C_{18}$ x 6, $C_3^3:S_3$

Order 144: $S_3^2:C_2^2$ x 2

Order 108: $S_3\times D_9$ x 156, $C_3:S_3^2$ x 16, $C_3:D_{18}$ x 16, $C_3\times S_3^2$ x 12, $S_3\times C_{18}$ x 12, $C_3\times D_{18}$ x 12, $C_3:S_3^2$ x 4

Order 81: $C_9^2$ x 1430, $C_3^2\times C_9$ x 230, $C_3^4$

Order 72: $\SOPlus(4,2)$ x 12, $S_3\times D_6$ x 8, $C_2\times D_{18}$ x 4, $C_2\times C_3^2:C_4$ x 2

Order 54: $C_3:D_9$ x 666, $S_3\times C_9$ x 132, $C_3\times D_9$ x 132, $C_3\times C_{18}$ x 16, $C_3^2:C_6$ x 16, $S_3\times C_3^2$ x 16, $C_3^2:S_3$ x 10

Order 36: $D_{18}$ x 48, $S_3^2$ x 46, $C_6\times S_3$ x 12, $C_3^2:C_4$ x 10, $C_6:S_3$ x 6, $C_2\times C_{18}$ x 4

Order 27: $C_3\times C_9$ x 650, $C_3^3$ x 10

Order 24: $C_2\times D_6$ x 4

Order 18: $D_9$ x 186, $C_3\times S_3$ x 52, $C_3:S_3$ x 36, $C_{18}$ x 36, $C_3\times C_6$ x 6

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 28, $C_2\times C_6$ x 4

Order 9: $C_9$ x 150, $C_3^2$ x 30

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 26, $C_6$ x 16

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 10

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 104976: $C_9^4:(C_2\times D_4)$

Order 52488: $C_9^4:D_4$, $C_9^4:(C_2\times C_4)$, $C_9:D_9^3$

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

Order 17496: $C_3:D_9^3$

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

Order 11664: $C_9^2:(S_3^2:C_2^2)$

Order 8748: $C_9^3.D_6$ x 2, $C_9^3:D_6$ x 2, $(C_3\times C_9^3).C_2^2$, $C_3\times C_9:D_9^2$

Order 6561: $C_9^4$

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

Order 4374: $C_9^3.S_3$ x 3, $C_9^3:S_3$ x 3, $C_9^3:C_6$ x 3

Order 2916: $C_9^3.C_2^2$ x 5, $(C_3\times C_9^2).D_6$ x 3, $C_9^2.C_3:S_3.C_2$ x 3, $(C_3\times C_9^2).D_6$ x 2, $C_9^2.S_3^2$ x 2, $C_9\times D_9^2$ x 2, $C_9^2:S_3^2$ x 2, $C_3^2:D_9^2$ x 2, $(C_3^3\times C_9).D_6$, $C_9:D_9^2$

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

Order 1944: $S_3\times D_9^2$ x 2, $C_9:S_3^3$

Order 1458: $C_9^3:C_2$ x 12, $(C_3\times C_9^2):S_3$ x 8, $(C_3\times C_9)\wr C_2$ x 7, $D_9\times C_9^2$ x 5, $C_9^2:C_{18}$ x 5, $C_3\times C_9^2:C_6$ x 2, $C_9\times C_3^2:C_{18}$ x 2

Order 1296: $C_3^4:(C_2\times D_4)$, $D_9^2:C_2^2$

Order 972: $C_3:D_9^2$ x 12, $C_3^3:D_{18}$ x 3, $C_9^2:D_6$ x 3, $C_9^2:D_6$ x 3, $C_3^3:D_{18}$ x 2, $C_3:D_9^2$ x 2, $C_3\times D_9^2$ x 2, $C_9\times C_3:S_3^2$

Order 729: $C_9^3$ x 12, $C_3^2\times C_9^2$ x 8

Order 648: $D_9\wr C_2$ x 3, $D_9\times S_3^2$ x 2, $D_9\times D_{18}$ x 2, $C_3:S_3^3$, $C_3^4:D_4$, $C_3^4:(C_2\times C_4)$, $C_2\times C_9^2:C_4$

Order 486: $C_9^2:S_3$ x 43, $C_9^2:C_6$ x 12, $C_9^2:S_3$ x 12, $S_3\times C_9^2$ x 3, $C_3^3:D_9$ x 3, $C_3^3:C_{18}$ x 3, $C_3^3:D_9$ x 3, $C_9^2:C_6$ x 3

Order 324: $D_9^2$ x 18, $C_9^2:C_4$ x 7, $C_9:S_3^2$ x 7, $C_3^2:D_{18}$ x 5, $C_3^2:D_{18}$ x 3, $C_9:D_{18}$ x 2, $C_9\times D_{18}$ x 2, $C_3^2:S_3^2$ x 2, $C_9:S_3^2$ x 2, $C_9\times S_3^2$ x 2, $C_3\wr C_2^2$, $C_3^4:C_4$

Order 243: $C_3\times C_9^2$ x 43, $C_3^3\times C_9$ x 3

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

Order 162: $C_9:D_9$ x 72, $C_3^2:D_9$ x 24, $C_9\times D_9$ x 22, $C_3^2:D_9$ x 7, $C_3^2:C_{18}$ x 7, $C_3^2:C_{18}$ x 5, $C_3^2\times D_9$ x 5, $C_3^2\wr C_2$ x 2, $C_9\times C_{18}$ x 2, $C_3^3:S_3$

Order 144: $S_3^2:C_2^2$

Order 108: $S_3\times D_9$ x 15, $C_3:S_3^2$ x 3, $C_3:D_{18}$ x 3, $C_3\times S_3^2$ x 2, $S_3\times C_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3:S_3^2$

Order 81: $C_9^2$ x 70, $C_3^2\times C_9$ x 24, $C_3^4$

Order 72: $S_3\times D_6$ x 2, $\SOPlus(4,2)$ x 2, $C_2\times C_3^2:C_4$, $C_2\times D_{18}$

Order 54: $C_3:D_9$ x 46, $S_3\times C_9$ x 12, $C_3\times D_9$ x 12, $C_3\times C_{18}$ x 3, $C_3^2:S_3$ x 3, $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3

Order 36: $S_3^2$ x 8, $D_{18}$ x 7, $C_3^2:C_4$ x 3, $C_6:S_3$ x 2, $C_6\times S_3$ x 2, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 43, $C_3^3$ x 3

Order 24: $C_2\times D_6$

Order 18: $D_9$ x 17, $C_3:S_3$ x 10, $C_3\times S_3$ x 7, $C_{18}$ x 5, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 5, $C_2\times C_6$

Order 9: $C_9$ x 12, $C_3^2$ x 8

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

Order 6: $S_3$ x 6, $C_6$ x 3

Order 4: $C_2^2$ x 3, $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 104976: $C_9^4:(C_2\times D_4)$ ($C_1$)

Order 52488: $C_9^4:D_4$ ($C_2$) x 4, $C_9:D_9^3$ ($C_2$) x 2, $C_9^4:(C_2\times C_4)$ ($C_2$)

Order 26244: $C_9^4.C_2^2$ ($C_2^2$) x 5, $C_9^4.C_4$ ($C_2^2$) x 2

Order 13122: $C_9^2\wr C_2$ ($D_4$) x 2, $C_9^4.C_2$ ($C_2^3$)

Order 6561: $C_9^4$ ($C_2\times D_4$)

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

Order 729: $C_3^2\times C_9^2$ ($S_3^2:C_2^2$) x 2

Order 162: $C_9:D_9$ ($D_9\wr C_2$) x 2

Order 81: $C_9^2$ ($D_9^2:C_2^2$) x 2, $C_3^4$ ($C_3^4:(C_2\times D_4)$)

Order 9: $C_3^2$ ($C_9^2:(S_3^2:C_2^2)$) x 2

Order 1: $C_1$ ($C_9^4:(C_2\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 104976: $C_9^4:(C_2\times D_4)$ ($C_1$)

Order 52488: $C_9^4:D_4$ ($C_2$), $C_9^4:(C_2\times C_4)$ ($C_2$), $C_9:D_9^3$ ($C_2$)

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

Order 13122: $C_9^4.C_2$ ($C_2^3$), $C_9^2\wr C_2$ ($D_4$)

Order 6561: $C_9^4$ ($C_2\times D_4$)

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

Order 729: $C_3^2\times C_9^2$ ($S_3^2:C_2^2$)

Order 162: $C_9:D_9$ ($D_9\wr C_2$)

Order 81: $C_9^2$ ($D_9^2:C_2^2$), $C_3^4$ ($C_3^4:(C_2\times D_4)$)

Order 9: $C_3^2$ ($C_9^2:(S_3^2:C_2^2)$)

Order 1: $C_1$ ($C_9^4:(C_2\times D_4)$)

Series

Derived series

$C_9^4:(C_2\times D_4)$

$\rhd$

$C_9^4:(C_2\times D_4)$

$\rhd$

$C_9^4.C_2$

$\rhd$

$C_9^4.C_2$

$\rhd$

$C_9^4$

$\rhd$

$C_9^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_9^4:(C_2\times D_4)$

$\rhd$

$C_9^4:(C_2\times D_4)$

$\rhd$

$C_9^4:D_4$

$\rhd$

$C_9^4:D_4$

$\rhd$

$C_9^4.C_2^2$

$\rhd$

$C_9^4.C_2^2$

$\rhd$

$C_9^4.C_2$

$\rhd$

$C_9^4.C_2$

$\rhd$

$C_9^4$

$\rhd$

$C_9^4$

$\rhd$

$C_3^2\times C_9^2$

$\rhd$

$C_3^2\times C_9^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_9^4:(C_2\times D_4)$

$\rhd$

$C_9^4:(C_2\times D_4)$

$\rhd$

$C_9^4.C_2$

$\rhd$

$C_9^4.C_2$

$\rhd$

$C_9^4$

$\rhd$

$C_9^4$

Upper central series

$C_1$

$\lhd$

$C_1$

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 $594 \times 594$ character table is not available for this group.

Rational character table

See the $222 \times 222$ rational character table (warning: may be slow to load).