Abstract group 629856.ka: $C_3^5.S_3^2\wr C_2$

• Label: 629856.ka
• Order: \( 2^{5} \cdot 3^{9} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^5.S_3^2\wr C_2$
Order:	\(629856\)\(\medspace = 2^{5} \cdot 3^{9} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^7.S_3\wr C_2^2$, of order \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \)
Composition factors:	$C_2$ x 5, $C_3$ x 9
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	10971	242	96228	59742	19440	34992	303264	104976	629856
Conjugacy classes	1	10	10	3	44	127	2	175	6	378
Divisions	1	10	10	3	38	65	2	103	2	234
Autjugacy classes	1	7	7	2	25	30	1	58	1	132

Dimension	1	2	4	8	12	16	24	32	36	48	72	96	144	288
Irr. complex chars.	8	22	56	34	40	14	54	1	0	114	0	35	0	0	378
Irr. rational chars.	8	10	30	30	16	16	36	5	8	24	6	5	30	10	234

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	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{18}=d^{18}=e^{18}=f^{9}=[c,e]=[c,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,4,8,23,14,28,20,11,25,18,31,36,3,6,9,24,15,29,19,10,26,17,32,34,2,5,7,22,13,30,21,12,27,16,33,35) \!\cdots\! \rangle$
Transitive group:	36T33149				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:	$D_9^4$ . $S_3$	$(C_9:D_9^3)$ . $D_6$	$C_3^4$ . $(S_3^4:S_3)$	$D_9^2$ . $(D_9^2:S_3)$ (2)	all 32

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 64 normal subgroups (22 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 $C_3^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^4.C_3$

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 54 (order at least 11664) are shown, as well as all normal subgroups of any index.

Order 629856: $C_3^5.S_3^2\wr C_2$

Order 314928: $C_9^4.C_6:D_4$ x 2, $C_9^4.C_6.D_4$ x 2, $C_9^4.C_6.C_2^3$, $C_9^4.C_6.D_4$, $C_9^4.C_6.C_2^3$

Order 209952: $D_9^2\wr C_2$

Order 157464: $C_9^4.C_6.C_2^2$ x 7, $C_9^3.C_9:C_3.C_2^3$ x 2, $C_9^2\wr C_2.D_6$ x 2, $C_9^2\wr C_2.D_6$ x 2, $C_3^5.C_3\wr D_4$ x 2, $C_9^4.D_6.C_2$, $C_9^4.C_3:D_4$, $C_9^4.C_6:C_4$, $C_9^4:(C_2\times D_6)$

Order 104976: $D_9^2.D_9^2$ x 2, $C_9^2\wr C_2.D_4$ x 2, $C_9^2:(C_6^2:D_{18})$ x 2, $D_9^4$, $C_9^3.C_6^2.C_2^2$, $C_3^3.C_3^5.C_2^4$, $C_9^2\wr C_2.D_4$, $C_9^4:(C_2\times D_4)$

Order 78732: $C_9^4.C_6.C_2$ x 9, $(C_3\times C_9^3).C_6^2$ x 8, $C_9^2.C_3^3.C_6^2$ x 4, $C_3^4.C_3^4.C_{12}$ x 2, $C_9^4:D_6$ x 2, $C_3^5.C_3^4.C_4$, $C_3^4.C_3^4.D_6$

Order 69984: $C_3^3.S_3^2\wr C_2$ x 2

Order 52488: $C_3^3.C_3^5.C_2^3$ x 10, $C_9^3.C_6^2.C_2$ x 8, $C_9^3.C_3^2.C_2^3$ x 5, $C_9^4.C_2^3$ x 4, $C_9^2.C_3^3.C_6.C_2^2$ x 4, $C_3^4.C_3^4.C_2^3$ x 3, $D_9^2\times C_9:D_9$ x 2, $C_9^3.C_9.C_2^3$ x 2, $C_3^2.C_3^5.C_6.C_2^2$ x 2, $(C_3\times C_9^3).C_6.C_2^2$ x 2, $C_9^4:D_4$ x 2, $C_9^4.(C_2\times C_4)$ x 2, $C_9^4:D_4$ x 2, $C_9:D_9^3$ x 2, $C_3^4.C_3\wr D_4$ x 2, $C_9\wr D_4$ x 2, $C_9^2.C_3^3.C_6.C_2^2$ x 2, $C_9^3:C_6.D_6$ x 2, $C_9^3:C_6.D_6$ x 2, $C_9^2:C_3^2:D_{36}$ x 2, $C_3^5.S_3^3$ x 2, $C_9^3:C_6.D_6$ x 2, $D_9^2\times C_9\times D_9$, $C_9\times D_9\times D_9^2$, $C_9^4:(C_2\times C_4)$

Order 39366: $C_9^2.C_3^3.C_3.C_6$ x 4, $C_9^3.C_3^2.C_6$ x 2, $C_9.C_3^4.C_3^3.C_2$ x 2, $C_9^4.C_6$, $C_3^5.C_3^4.C_2$

Order 34992: $(C_3\times D_9^2):D_{18}$ x 2, $C_3^5.C_6^2:C_4$ x 2, $(C_3^3\times S_3^2).S_3^2$ x 2, $D_9^2:(C_3^3:C_4)$ x 2, $C_3^5.S_3^2:C_4$ x 2, $(C_3\times D_9^2).S_3^2$ x 2, $C_9^2:(S_3^2:D_6)$ x 2, $S_3\times D_9\times D_9^2$, $D_9^2\times S_3\times D_9$, $C_9^3.C_6.C_2^3$, $C_9^2.C_3^3.C_2^4$, $C_3^4.C_3^3.C_2^4$, $C_3^2.C_3^5.C_2^4$, $C_2\times C_9^3.C_6.C_2^2$, $(C_3.(C_3.C_3^3)):C_2^2\times S_3^2$

Order 26244: $C_9^3.(C_6\times S_3)$ x 19, $C_9^2.C_3^3.C_6.C_2$ x 17, $C_9^4.C_2^2$ x 12, $C_9^2.C_9:C_9.C_2^2$ x 10, $C_3^3.C_3^5.C_2^2$ x 9, $C_9^3.C_{18}.C_2$ x 8, $C_3^3.C_3^4.C_6.C_2$ x 8, $C_9^3.C_6^2$ x 7, $C_3^4.C_3^4.C_2^2$ x 7, $C_3^2.C_3^5.C_6.C_2$ x 7, $(C_3\times C_9^3).C_6.C_2$ x 7, $C_9^2.C_3^2.C_6^2$ x 4, $C_9.C_3^4.C_6^2$ x 4, $C_9\wr C_2^2$ x 4, $C_3\times (C_3\times C_9^2).C_6^2$ x 3, $C_9^2\times D_9^2$ x 2, $C_9\times D_9\times C_9:D_9$ x 2, $C_3^4.C_3^3.D_6$ x 2, $C_3^4.C_3\wr C_4$ x 2, $C_9\wr C_4$ x 2, $(C_3\times C_9^3).D_6$ x 2, $C_3^4.C_3\wr C_4$ x 2, $C_9^3:C_6.S_3$ x 2, $(C_3\times C_9^3):D_6$ x 2, $(C_3\times C_9^3):D_6$ x 2, $(C_3\times C_9^3):D_6$ x 2, $S_3\times (C_3\times C_9).C_3^3.C_6$, $C_9^4.C_4$, $C_9\times D_9\times C_9\times D_9$, $C_9\times C_9^3.C_2^2$, $C_3^5.C_3^2.D_6$, $C_3\times C_9^3.C_6.C_2$, $C_3.C_3^5.C_6^2$, $(C_9\times D_9)^2$, $(C_9:D_9)^2$, $(C_3^2\times C_9^2).C_6^2$

Order 23328: $(S_3\times D_9)\wr C_2$ x 2

Order 19683: $C_9^4.C_3$

Order 17496: $C_9^3.C_6.C_2^2$ x 32, $C_3^3.C_3^4.C_2^3$ x 13, $C_3^2.C_3^5.C_2^3$ x 12, $C_9^2.C_3^3.C_2^3$ x 11, $(C_3\times C_9^3).C_2^3$ x 11, $C_2\times C_9^3.C_6.C_2$ x 7, $C_3^5.\SOPlus(4,2)$ x 5, $D_9^2\times C_3:D_9$ x 4, $C_3^4.C_3^3.C_2^3$ x 4, $C_2\times (C_3\times C_9^2).C_6^2$ x 4, $C_3:D_9^3$ x 4, $C_3.C_3^5.C_6.C_2^2$ x 3, $(C_3^3\times C_9).C_6^2.C_2$ x 3, $(C_3\times C_9^2).C_9.C_2^3$ x 3, $S_3\times D_9\times C_9\times D_9$ x 2, $S_3\times D_9\times C_9:D_9$ x 2, $C_9^2.C_9:C_3.C_2^3$ x 2, $C_9\times D_9\times S_3\times D_9$ x 2, $C_9.C_3^4.C_6.C_2^2$ x 2, $C_9\times D_9^2:S_3$ x 2, $C_9^3.D_{12}$ x 2, $C_9^3:(C_4\times S_3)$ x 2, $(C_3\times C_9^2):D_{36}$ x 2, $C_3\times D_9^2:D_9$ x 2, $C_3\times C_9^3:D_4$ x 2, $C_3^3.C_3\wr D_4$ x 2, $C_9^2:C_6.S_3^2$ x 2, $C_9^2:(C_2\times C_3^3:C_4)$ x 2, $C_3^3:C_{18}.S_3^2$ x 2, $C_3^4:C_9.D_{12}$ x 2, $C_3^4.S_3^3$ x 2, $C_3^5.\SOPlus(4,2)$ x 2, $C_3^5.\SOPlus(4,2)$ x 2, $S_3^2\times C_9:(D_9:C_3)$, $D_9^2\times C_9\times S_3$, $D_9^2\times C_3\times D_9$, $D_9:(C_9:C_3)\times S_3^2$, $C_9^3.D_6.C_2$, $C_9^2.(C_{18}\times S_3).C_2$, $C_9^2.(C_6\times C_{18}).C_2$, $C_9\times S_3\times D_9^2$, $C_3^3.(C_9\times S_3^2).C_2$, $C_3^3.(C_6\times \He_3).C_2^2$, $C_3\times D_9\times D_9^2$, $(C_3.(C_3.C_3^3)):C_2^2\times C_3\times S_3$, $(C_3.(C_3.C_3^3)):C_2^2\times C_3:S_3$, $C_3^4.S_3^3$, $C_3^4.S_3^3$, $C_3^5.\SOPlus(4,2)$

Order 13122: $C_9.C_3^4.C_3.C_6$ x 13, $C_9^2.C_3^3.C_6$ x 11, $C_9^2.C_9:C_9.C_2$ x 10, $C_9.(C_3\times C_9:C_9).C_6$ x 9, $(C_3\times C_9^3).C_6$ x 6, $C_3^5.C_3^3.C_2$ x 5, $C_3^2.C_3^4.C_3.C_6$ x 5, $(C_3\times C_9).C_9:\He_3.C_2$ x 5, $C_9^4.C_2$ x 4, $C_9^2.C_9:C_3.C_6$ x 4, $C_9.(C_3.C_9^2).C_6$ x 4, $C_3\times (C_3\times C_9).C_3^3.C_6$ x 4, $C_9^2\wr C_2$ x 4, $C_9^3.C_3^2.C_2$ x 3, $C_9.C_3^2:C_9.C_3.C_6$ x 3, $C_3\times C_9^2.C_3^2.C_6$ x 3, $C_3.C_9^2.C_9.C_6$ x 3, $C_9^3.C_{18}$ x 2, $C_9^3.C_3.C_6$ x 2, $C_9^2\times C_9\times D_9$ x 2, $C_3^4.C_3^2.C_3.C_6$, $C_3^3.C_3^4.C_6$, $C_3\times C_9.C_3^2:C_9.C_6$, $(C_3\times C_9).C_9^2.C_6$

Order 11664: $C_2\times D_9^2\times D_9$ x 2, $C_2\times C_9^2.C_3^2.C_2^3$ x 2, $(S_3\times D_9)^2$ x 2, $D_9^2:D_{18}$ x 2, $C_3^2.C_{18}^2:C_4$ x 2, $S_3^2:D_9^2$ x 2, $C_9^2:(C_6^2:C_4)$ x 2, $C_9^2:(S_3^2:C_4)$ x 2, $D_9^2:S_3^2$ x 2, $D_9^2:S_3^2$ x 2, $C_9^2:(S_3^2:C_2^2)$ x 2, $C_3^4:(D_4\times D_9)$ x 2, $S_3^2\times D_9^2$, $S_3\times D_9:C_3\times S_3^2$, $D_9^2\times S_3^2$, $D_6\times (C_3.(C_3.C_3^3)):C_2^2$, $C_9^2.C_3^2.C_2^4$, $C_3.C_3^5.C_2^4$, $C_2\times C_3^2.C_3^4.C_2^3$, $(C_3.(C_3.C_3^3)):C_2^2\times D_6$

Order 8748: $C_3^2\times (C_3.(C_3.C_3^3)):C_2^2$ x 2

Order 6561: $C_9^4$

Order 4374: $C_3^2\times C_9:(D_9:C_3)$ x 2

Order 2916: $C_3^2\times D_9^2$ x 2

Order 2187: $C_3^2\times C_3.(C_3.C_3^3)$ x 2

Order 1458: $C_3\times C_9^2:C_6$ x 2

Order 729: $C_3^2\times C_9^2$ x 2

Order 324: $D_9^2$ x 2

Order 243: $C_3^5$

Order 162: $C_9:D_9$ x 2

Order 81: $C_9^2$ x 2, $C_3^4$

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 629856: $C_3^5.S_3^2\wr C_2$ ($C_1$)

Order 314928: $C_9^4.C_6:D_4$ ($C_2$) x 2, $C_9^4.C_6.D_4$ ($C_2$) x 2, $C_9^4.C_6.C_2^3$ ($C_2$), $C_9^4.C_6.D_4$ ($C_2$), $C_9^4.C_6.C_2^3$ ($C_2$)

Order 157464: $C_9^4.C_6.C_2^2$ ($C_2^2$) x 3, $C_9^2\wr C_2.D_6$ ($C_2^2$) x 2, $C_9^4.C_6:C_4$ ($C_2^2$), $C_9^4:(C_2\times D_6)$ ($C_2^2$)

Order 104976: $D_9^4$ ($S_3$)

Order 78732: $C_9^4.C_6.C_2$ ($D_4$) x 4, $C_9^2.C_3^3.C_6^2$ ($D_4$) x 2, $C_9^4.C_6.C_2$ ($C_2^3$)

Order 52488: $D_9^2\times C_9:D_9$ ($D_6$) x 2, $C_9:D_9^3$ ($D_6$)

Order 39366: $C_9^2.C_3^3.C_3.C_6$ ($C_2\times D_4$) x 2, $C_9^4.C_6$ ($C_2\times D_4$)

Order 26244: $C_9^4.C_2^2$ ($C_3:D_4$) x 4, $C_9^2\times D_9^2$ ($C_3:D_4$) x 2, $(C_9:D_9)^2$ ($C_2\times D_6$)

Order 19683: $C_9^4.C_3$ ($C_2^2\wr C_2$)

Order 13122: $C_9^2\wr C_2$ ($C_6:D_4$) x 2, $C_9^4.C_2$ ($C_6:D_4$)

Order 8748: $C_3^2\times (C_3.(C_3.C_3^3)):C_2^2$ ($\SOPlus(4,2)$) x 2

Order 6561: $C_9^4$ ($C_2^4:S_3$)

Order 4374: $C_3^2\times C_9:(D_9:C_3)$ ($S_3^2:C_2^2$) x 2

Order 2916: $C_3^2\times D_9^2$ ($S_3^2:S_3$) x 2

Order 2187: $C_3^2\times C_3.(C_3.C_3^3)$ ($D_6\wr C_2$) x 2

Order 1458: $C_3\times C_9^2:C_6$ ($S_3^2:D_6$) x 2

Order 729: $C_3^2\times C_9^2$ ($D_6^2:S_3$) x 2

Order 324: $D_9^2$ ($D_9^2:S_3$) x 2

Order 243: $C_3^5$ ($S_3^2\wr C_2$)

Order 162: $C_9:D_9$ ($D_9^2:D_6$) x 2

Order 81: $C_9^2$ ($(D_{18}\times D_{18}):S_3$) x 2, $C_3^4$ ($S_3^4:S_3$)

Order 9: $C_3^2$ ($C_3^3.S_3^2\wr C_2$) x 2

Order 1: $C_1$ ($C_3^5.S_3^2\wr C_2$)

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 $378 \times 378$ 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 $234 \times 234$ rational character table (warning: may be slow to load).