Abstract group 629856.le: $C_3^5.(S_3\times C_3^2:\GL(2,3))$

• Label: 629856.le
• Order: \( 2^{5} \cdot 3^{9} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $81$
• Trans deg.: $81$
• Rank: $2$

Group information

Description:	$C_3^5.(S_3\times C_3^2:\GL(2,3))$
Order:	\(629856\)\(\medspace = 2^{5} \cdot 3^{9} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_3^5.(S_3\times C_3^2:\GL(2,3))$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \)
Composition factors:	$C_2$ x 5, $C_3$ x 9
Derived length:	$5$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	9	12	18	24
Elements	1	3483	16280	8748	296136	87480	42768	69984	34992	69984	629856
Conjugacy classes	1	5	28	2	37	4	7	6	3	8	101
Divisions	1	5	28	2	37	2	5	5	2	3	90
Autjugacy classes	1	5	28	2	37	4	7	6	3	8	101

Minimal presentations

Permutation degree:	$81$
Transitive degree:	$81$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	72	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid a^{2}=b^{6}=d^{12}=e^{3}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $81$ $\langle(1,2,3)(4,12,8)(5,9,7)(6,13,11)(10,20,27)(14,32,18)(17,31,39)(19,26,33) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^3.C_3^4.D_6)$ . $S_4$	$(C_3^5.C_3:S_3.S_4)$ . $S_3$ (2)	$(C_3^4:C_6)$ . $(S_3\wr S_3)$	$(C_3^5.C_3:S_3.A_4)$ . $D_6$ (2)	all 20
Aut. group:	$\Aut(C_3^4:C_3)$	$\Aut(C_3^4:C_6)$	$\Aut(C_3^4:C_6)$	$\Aut((C_3^2\times \He_3):S_3)$

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

Homology

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

Subgroups

There are 27 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:	not computed
Frattini:	a subgroup isomorphic to $C_3^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^4.C_3$

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 162 (order at least 3888) are shown, as well as all normal subgroups of any index.

Order 629856: $C_3^5.(S_3\times C_3^2:\GL(2,3))$

Order 314928: $C_3^5.C_3^2.C_6.A_4.C_2$, $C_3^5.C_3:S_3.A_4.S_3$, $C_3^4.C_3^3.Q_8.C_3.C_6$

Order 209952: $C_3^5.C_3^2.Q_8.D_6$, $C_3^4.C_3^4.D_4.C_2^2$

Order 157464: $C_3^4.C_3^3.Q_8.C_3^2$, $C_3^4.C_3^2.S_3^3$

Order 104976: $C_3^4.C_3^3.Q_8.S_3$ x 2, $C_3^5.C_3^3.\SD_{16}$, $C_3^5.C_3:S_3.S_4$, $C_3^4.C_3^4.\SD_{16}$, $C_3^4.C_3^4.C_8.C_2$, $C_3^4.C_3^3.Q_8.C_6$, $C_3^4.C_3^3.C_{24}.C_2$, $C_3^4.C_3^3.C_4.C_6.C_2$, $C_3^3.C_3^5.C_2^3.C_2$, $C_3^3.C_3^4.D_6.C_2^2$

Order 78732: $C_3^4.C_3^4.D_6$ x 2, $C_3^4.C_3^4.C_6.C_2$ x 2, $C_3^5.C_3^3.C_6.C_2$, $C_3^4.C_3^3.(C_6\times S_3)$, $C_3^3.C_3^5:D_6$

Order 69984: $C_3^4.F_9:D_6$ x 2, $C_3^5.Q_8.S_3^2$

Order 52488: $C_3^5.C_3:S_3.A_4$ x 2, $C_3^4.C_3^3.C_6.C_2^2$ x 2, $C_3^6.C_3:S_3.C_2^2$, $C_3^5.S_3^3$, $C_3^4:C_3.S_3^3$, $C_3^4.C_3^4.C_8$, $C_3^4.C_3^4.C_2^3$, $C_3^4.C_3^3.D_6.C_2$, $C_3^4.C_3^3.C_{12}.C_2$, $C_3^4.C_3^2.C_{12}.C_6$, $C_3^3.C_3^4.C_{12}.C_2$, $C_3^3.C_3^4.C_6.C_2^2$

Order 39366: $C_3^4.C_3^4.C_6$ x 3, $C_3^6.\He_3.C_2$, $C_3^4.\He_3.C_3.C_6$, $C_3^4.C_3^3.C_3.C_6$, $C_3.C_3^5.C_3^3.C_2$

Order 34992: $C_3^4.C_3^3.\SD_{16}$ x 4, $\He_3^2:(S_3\times D_4)$ x 3, $C_3^5.F_9.C_2$ x 2, $C_3^5.Q_8.C_3.S_3$, $C_3^4.F_9:S_3$, $C_3^4.C_3^3.C_8.C_2$, $C_3^4.C_3^2.D_6.C_2^2$, $C_3^4.C_3^2.C_{24}.C_2$, $C_3^4.C_3^2.C_4.C_6.C_2$, $C_3^3.C_3^4.\SD_{16}$, $C_3^3.C_3^3.D_6.C_2^2$, $C_3^6:\GL(2,3)$, $C_3^5:(S_3\times \SL(2,3))$

Order 26244: $C_3^4.C_3^3.D_6$ x 10, $C_3^4.C_3^3.C_6.C_2$ x 7, $C_3^4.C_3^4.C_2^2$ x 4, $C_3^5.C_3^2.D_6$ x 2, $C_3^4.C_3^2.C_6.C_6$ x 2, $C_3^3.C_3^4.D_6$ x 2, $C_3^5.\He_3.C_2^2$, $C_3^5.C_3^3.C_2^2$, $C_3^4.\He_3.D_6$, $C_3^4.\He_3.C_6.C_2$, $C_3^3.\He_3.C_6^2$, $C_3^3.C_3^5.C_2^2$, $C_3^6.S_3^2$

Order 23328: $C_3^6.D_4.C_2^2$, $C_3^4.S_3\wr C_2.C_2^2$, $C_3^4:C_{24}:D_6$, $C_3^4:(S_3\times \GL(2,3))$, $C_3^5:(C_2\times \GL(2,3))$

Order 19683: $C_3^4.C_3^4.C_3$

Order 17496: $C_3^4:C_3.C_6.C_6.C_2$ x 4, $C_3^4.S_3^3$ x 4, $C_3^4:C_3.C_6^2.C_2$ x 3, $C_3^4.C_3^3.C_2^3$ x 3, $(C_3\times \He_3).S_3^3$ x 3, $C_3^5.F_9$ x 2, $C_3^5.C_3:S_3.C_2^2$ x 2, $C_3^4.C_3^3.Q_8$ x 2, $C_3^4.C_3^3.D_4$ x 2, $C_3^4.C_3^2.D_6.C_2$ x 2, $C_3^4.C_3:S_3.C_6.C_2$ x 2, $C_3^4.(C_3\times F_9)$ x 2, $C_3^3.C_3^4.C_2^3$ x 2, $C_3^5.S_3\wr C_2$, $C_3^4.C_3^2.C_{24}$, $C_3^4.(C_3\times D_6:S_3)$, $C_3^3.C_3^4.C_8$, $C_3^3.C_3^3.D_6.C_2$, $C_3^4:S_3^3$, $C_3^4.S_3^3$, $C_3^6:\SL(2,3)$

Order 13122: $C_3^4.C_3^3.C_6$ x 18, $C_3^4.C_3^4.C_2$ x 11, $C_3^4.\He_3.C_6$ x 4, $C_3^6.C_3.C_6$ x 3, $C_3^5.\He_3.C_2$ x 3, $C_3^3.C_3^4.C_6$ x 3, $C_3^4.C_9:C_3.C_6$ x 2, $C_3^4.\He_3:S_3$, $C_3^4.C_9.C_3.C_6$, $C_3^3.C_3^5.C_2$, $C_3^6:(C_3\times S_3)$

Order 11664: $C_3^6.\SD_{16}$ x 2, $C_3^4:C_3.D_6.C_2^2$ x 2, $C_3^4.F_9.C_2$ x 2, $C_3^4.D_6:S_3.C_2$ x 2, $C_3^4.C_3:S_3.D_4$ x 2, $C_3^3.S_3^3.C_2$ x 2, $C_3^6:\SD_{16}$ x 2, $C_3^5:\GL(2,3)$ x 2, $\He_3^2:(C_2\times D_4)$ x 2, $C_3^6.C_8.C_2$, $C_3^4:C_3.C_{24}.C_2$, $C_3^4:C_3.C_6.D_4$, $C_3^4.C_{12}.C_6.C_2$, $C_3^4.C_3^2.\SD_{16}$, $C_3^4:C_3:\GL(2,3)$, $C_3^4:(C_3\times \GL(2,3))$, $C_3^4:(S_3\times \SL(2,3))$, $C_3^6:(C_2\times D_4)$, $C_3^5:(C_2\times \SL(2,3))$, $C_3^5:(S_3\times Q_8)$, $C_3^5:\GL(2,3)$

Order 8748: $C_3^4.C_3^2.D_6$ x 24, $C_3^4.C_3^3.C_2^2$ x 16, $C_3^3.C_3^4.C_2^2$ x 10, $C_3^4.\He_3.C_2^2$ x 8, $C_3^3.C_3^3.D_6$ x 8, $C_3^5:S_3^2$ x 8, $C_3.C_3^5.D_6$ x 7, $C_3^4:C_3.C_6.C_6$ x 6, $C_3.C_3^5.C_6.C_2$ x 6, $C_3^5.S_3^2$ x 4, $C_3^4:C_3.C_6^2$ x 4, $C_3^4.C_3:S_3.C_6$ x 4, $C_3^3.\He_3.D_6$ x 4, $C_3^3.\He_3.C_6.C_2$ x 3, $C_3^3.C_3^3.C_6.C_2$ x 3, $C_3^6.C_6.C_2$ x 2, $C_3^5.C_3.D_6$ x 2, $C_3^6:D_6$ x 2, $C_3^6:D_6$ x 2, $C_3^5.S_3^2$, $C_3^6:D_6$, $C_3^6:D_6$, $C_3^6:D_6$, $C_3^6:D_6$, $C_3^5.S_3^2$, $C_3^6.D_6$

Order 7776: $C_3^4:(S_3\times \SD_{16})$ x 2, $C_3^4:(S_3\times \SD_{16})$ x 2, $C_3^3:Q_8.S_3^2$, $C_3^4:(C_2\times \GL(2,3))$

Order 6561: $C_3^4.C_3^3.C_3$ x 2, $C_3^4.C_3^3.C_3$ x 2, $C_3^4.C_3^3.C_3$ x 2, $C_3^4.C_3^3.C_3$ x 2, $C_3^4.C_3^4$, $C_3^4.C_3^4$, $C_3^5.C_3^3$, $C_3^4.C_3^4$, $C_3^6.C_3^2$, $C_3^5.C_3^3$, $C_3^3.C_3^5$

Order 5832: $C_3^3.S_3^3$ x 10, $C_3^3.S_3^3$ x 7, $\He_3^2:D_4$ x 4, $\He_3^2:D_4$ x 4, $C_3^4:C_3.D_6.C_2$ x 3, $C_3^4:C_3.C_{12}.C_2$ x 3, $C_3^4.C_3^2.Q_8$ x 3, $\He_3^2:C_2^3$ x 3, $C_3^5:D_{12}$ x 3, $\He_3^2:(C_2\times C_4)$ x 3, $C_3^4:C_3.C_6.C_2^2$ x 2, $C_3^4.S_3\wr C_2$ x 2, $C_3^4.F_9$ x 2, $C_3^4.C_3:S_3.C_2^2$ x 2, $C_3^3:S_3^3$ x 2, $C_3^3:S_3^3$ x 2, $C_3^3.S_3^3$ x 2, $C_3^5:\SL(2,3)$ x 2, $C_3^6:Q_8$ x 2, $\He_3^2:C_2^3$ x 2, $C_3^6.C_8$, $C_3^4.D_6:S_3$, $C_3^4.C_{12}.C_6$, $C_3^4.C_3^2.C_8$, $C_3^4.(C_3\times D_{12})$, $C_3.C_3^5.C_2^3$, $C_3^5:C_{24}$, $C_3^6:(C_2\times C_4)$, $C_3^6:C_2^3$, $C_3^3:S_3^3$, $C_3^4:(C_3\times \SL(2,3))$, $C_3^3.S_3^3$

Order 4374: $C_3^4.C_3^3.C_2$ x 58, $C_3^3.C_3^4.C_2$ x 31, $C_3^4.\He_3.C_2$ x 29, $C_3^5.C_3.C_6$ x 21, $C_3^3.\He_3.C_6$ x 18, $C_3^3.C_3^3.C_6$ x 12, $C_3.C_3^5.C_6$ x 12, $C_3^3.\He_3:S_3$ x 7, $C_3^4.C_9.C_6$ x 4, $C_3^6.C_6$ x 2, $C_3^6:C_6$ x 2, $C_3^5.(C_3\times S_3)$ x 2, $C_3\wr S_3$ x 2, $\He_3.\He_3.C_6$, $C_3^6:C_6$, $C_3^5.(C_3\times S_3)$, $C_3^6.C_6$, $C_3^5:C_3:S_3$, $C_3^4:C_9:C_6$

Order 3888: $C_3^3:(C_2\times F_9)$ x 3, $C_3^2:(S_3\times F_9)$ x 3, $C_3\wr C_4:D_6$ x 3, $C_3^4:(S_3\times D_4)$ x 3, $C_3^2:F_9:S_3$ x 2, $C_3^3:F_9:C_2$ x 2, $C_3^4:(C_3\times \SD_{16})$ x 2, $C_3^4:(S_3\times Q_8)$ x 2, $C_3^3:F_9:C_2$ x 2, $C_3^4:(S_3\times Q_8)$ x 2, $C_3^5:\SD_{16}$ x 2, $C_3^5:\SD_{16}$ x 2, $C_3^5:\SD_{16}$ x 2, $C_3^5:\SD_{16}$ x 2, $C_3^3:Q_8.C_3.S_3$, $C_3^4:\GL(2,3)$, $C_3^3:(S_3\times \SL(2,3))$, $C_3^4:\GL(2,3)$, $C_3^4:(C_2\times \SL(2,3))$, $C_3^4:\GL(2,3)$, $C_3^4:\GL(2,3)$

Order 2187: $C_3^3.C_3^4$

Order 1458: $C_3^5:S_3$

Order 729: $C_3^6$

Order 486: $C_3^4:C_6$

Order 243: $C_3^5$, $C_3^4:C_3$

Order 162: $C_3^3:S_3$

Order 81: $C_3^4$

Order 27: $C_3^3$

Order 9: $C_3^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 629856: $C_3^5.(S_3\times C_3^2:\GL(2,3))$ ($C_1$)

Order 314928: $C_3^5.C_3^2.C_6.A_4.C_2$ ($C_2$), $C_3^5.C_3:S_3.A_4.S_3$ ($C_2$), $C_3^4.C_3^3.Q_8.C_3.C_6$ ($C_2$)

Order 157464: $C_3^4.C_3^3.Q_8.C_3^2$ ($C_2^2$)

Order 104976: $C_3^5.C_3:S_3.S_4$ ($S_3$), $C_3^3.C_3^4.D_6.C_2^2$ ($S_3$)

Order 52488: $C_3^5.C_3:S_3.A_4$ ($D_6$), $C_3^4.C_3^3.C_{12}.C_2$ ($D_6$)

Order 26244: $C_3^3.C_3^4.D_6$ ($S_4$)

Order 17496: $C_3^4.C_3^3.Q_8$ ($S_3^2$)

Order 13122: $C_3^6.C_3.C_6$ ($\GL(2,3)$), $C_3^4.C_3^4.C_2$ ($C_2\times S_4$), $C_3^4.C_3^4.C_2$ ($\GL(2,3)$)

Order 6561: $C_3^3.C_3^5$ ($C_2\times \GL(2,3)$)

Order 4374: $C_3^4.C_3^3.C_2$ ($S_3\times S_4$)

Order 2187: $C_3^3.C_3^4$ ($S_3\times \GL(2,3)$)

Order 1458: $C_3^5:S_3$ ($C_3^2:\GL(2,3)$)

Order 729: $C_3^6$ ($C_2\times C_3^2:\GL(2,3)$)

Order 486: $C_3^4:C_6$ ($S_3\wr S_3$)

Order 243: $C_3^5$ ($S_3\times C_3^2:\GL(2,3)$), $C_3^4:C_3$ ($C_3^3:(C_2\times \GL(2,3))$)

Order 162: $C_3^3:S_3$ ($C_3^3:(S_3\times S_4)$)

Order 81: $C_3^4$ ($C_3^3:Q_8.S_3^2$)

Order 27: $C_3^3$ ($C_3^5:(C_2\times \GL(2,3))$)

Order 9: $C_3^2$ ($C_3^5.Q_8.S_3^2$)

Order 1: $C_1$ ($C_3^5.(S_3\times C_3^2:\GL(2,3))$)

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

Character theory

Complex character table

The $101 \times 101$ character table is not available for this group.

Rational character table

The $90 \times 90$ rational character table is not available for this group.