Abstract group 559872.fk: $C_3^6.C_2^5:S_4$

• Label: 559872.fk
• Order: \( 2^{8} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $26$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^6.C_2^5:S_4$
Order:	\(559872\)\(\medspace = 2^{8} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_2^6.C_6.C_2^4$, of order \(4478976\)\(\medspace = 2^{11} \cdot 3^{7} \)
Composition factors:	$C_2$ x 8, $C_3$ x 7
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
Elements	1	6927	11096	40176	116760	82944	239760	62208	559872
Conjugacy classes	1	14	26	13	302	5	71	2	434
Divisions	1	14	24	13	233	5	53	2	345

Minimal presentations

Permutation degree:	$26$
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	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid c^{2}=d^{6}=e^{6}=f^{6}=g^{6}=h^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(2,4,7,3,5,8)(6,9)(11,13,15,16)(14,17)(19,21,23,26,24,20)(22,25), (1,2,3)(4,6,5,8,9,7)(10,11,12,13,14,16)(15,17)(18,19,20,22,24,26,25,23,21)\rangle$
Transitive group:	36T30562	36T30563			more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_3^6.C_2^5)$ . $S_4$ (3)	$C_3^6$ . $(C_2^5:S_4)$	$(C_3^6.C_2^6)$ . $D_6$	$C_3^3$ . $(C_6:D_6^2:S_4)$	all 22

Elements of the group are displayed as permutations of degree 26.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 29 normal subgroups (27 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 $C_2^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 64 (order at least 8748) are shown, as well as all normal subgroups of any index.

Order 559872: $C_3^6.C_2^5:S_4$

Order 279936: $C_3^6.C_2^6.S_3$, $C_3^6.C_2^4:S_4$, $C_3^6.C_2^5:A_4$

Order 186624: $C_3^6.C_2^6.C_2^2$

Order 139968: $C_3^6.C_2^6.C_3$, $C_3^3.C_6^3:S_4$, $C_3^3.C_6^3:S_4$, $C_6^4.C_3:S_3^2$

Order 93312: $C_3^6.C_2^5.C_2^2$ x 10, $C_3^6.C_2^6.C_2$ x 3, $C_3^5.C_2^4.C_6.C_2^2$, $(C_3^3\times C_6^3).C_2^3.C_2$

Order 69984: $C_3^6.C_2^4.S_3$ x 4, $C_3^5:D_6.S_4$ x 4, $C_3^6.C_2^3:A_4$ x 2, $C_6^4.C_3^3.C_2$, $C_3^6.C_2^3:A_4$, $C_2^4.C_3\wr S_3$, $C_6^4.(S_3\times C_3^2)$

Order 62208: $C_3^5.C_2^6.C_2^2$ x 2

Order 46656: $C_3^6.C_2^5.C_2$ x 23, $C_3^6.C_2^4.C_2^2$ x 11, $C_3^6.C_2\wr C_2^2$ x 6, $C_3^6.D_4^2$ x 4, $(C_3^4\times C_6^2).C_2^4$ x 3, $C_3^6.C_2^3.C_2^3$ x 2, $(C_3^4\times C_6^2).C_4^2$ x 2, $(C_3^2\times C_6^4).C_2^2$ x 2, $C_6^4.S_3^2$ x 2, $C_3^6.C_2^6$, $C_3^5.D_6.C_2^4$, $C_3^5.D_4:D_{12}$, $C_3^5.C_3:D_4:D_4$, $C_3^5.(C_6\times C_2^2.D_4)$, $C_3^5.(C_2^4\times C_6).C_2$, $C_3^4.C_6^2.C_4^2$, $C_3^4.C_6^2.C_2^4$, $C_3^4.C_6^2.C_2^3.C_2$, $C_3^4.C_6^2.C_2.C_2^3$, $(C_3^5\times C_6).C_2^4.C_2$, $(C_3^4\times C_6^2).C_2^3.C_2$, $C_6^4.S_3^2$, $C_6^4.S_3^2$

Order 34992: $C_3^5:D_6.A_4$ x 9, $C_3^6:(C_2\times S_4)$ x 2, $C_6^4.C_3^3$, $C_3^5.(S_3\times S_4)$

Order 31104: $C_3^5.C_2^5.C_2^2$ x 24, $C_3^5.C_2^6.C_2$ x 7, $C_3^4.C_2^4.C_6.C_2^2$, $(C_3\times C_6^3).C_6.C_2^3$

Order 23328: $C_3^6.C_2^4.C_2$ x 78, $C_3^6.C_2^3.C_2^2$ x 13, $C_3^4.C_6^2.C_2^3$ x 12, $C_3^6.C_2^5$ x 11, $(C_3^3\times C_6^3).C_2^2$ x 10, $C_6^4.C_3.C_6$ x 8, $(C_3^4\times C_6^2).C_2^3$ x 8, $C_3^6.C_2^2:D_4$ x 6, $(C_3^5\times C_6).C_2^4$ x 6, $(C_3^5\times C_6).C_2^3.C_2$ x 6, $C_3^6.C_2.D_4.C_2$ x 5, $C_3^5.D_6.C_2^3$ x 5, $C_3^5.(C_6\times D_4).C_2$ x 5, $C_3^6.(C_4\times D_4)$ x 4, $C_3^5.C_4:D_{12}$ x 4, $C_3^5.(C_{12}\times D_4)$ x 4, $(C_3^2\times C_6^4).C_2$ x 4, $C_3^6.D_4.C_2^2$ x 3, $C_3^6.C_4:D_4$ x 2, $C_3^6.(C_4.D_4)$ x 2, $C_3^4.D_6:D_{12}$ x 2, $C_3^4.C_6^2.C_4.C_2$ x 2, $C_6^4.(C_3\times S_3)$ x 2, $C_6^4.(C_3\times C_6)$ x 2, $C_3^6.C_2^2:Q_8$, $C_3^6.C_2.D_4:C_2$, $C_3^5.D_6:Q_8$, $C_3^5.(D_6.D_4)$, $C_3^4.(C_6\times C_6.D_4)$, $(C_3^3\times C_6^2).S_4$, $C_6^4.(C_3\times S_3)$, $C_6^4.(C_3\times C_6)$, $C_6^4.(C_3\times C_6)$

Order 20736: $C_3^4.C_2^6.C_2^2$ x 2, $C_6:D_6^2:S_4$, $(S_3\times D_6^2).S_4$, $C_6^4.(C_2\times D_4)$

Order 17496: $C_3^6:S_4$ x 3, $C_3^6.S_4$ x 3, $C_3^6:(C_2\times A_4)$ x 3, $C_3^4.C_3^2:S_4$ x 2, $C_3^6.S_4$ x 2, $C_3^5.(C_6\times A_4)$

Order 15552: $C_3^5.C_2^5.C_2$ x 81, $C_3^5.C_2^4.C_2^2$ x 35, $C_3^5.C_2\wr C_2^2$ x 22, $(C_3\times C_6^4).C_2^2$ x 14, $(C_3^3\times C_6^2).C_2^4$ x 13, $C_3^5.D_4^2$ x 8, $C_3^5.C_2^3.C_2^3$ x 5, $C_3^4.D_4:D_{12}$ x 4, $C_3^4.(C_2^4\times C_6).C_2$ x 4, $C_3^5.C_2^6$ x 3, $C_3^4.D_6.C_2^4$ x 3, $C_3^4.(C_6\times C_2^2.D_4)$ x 3, $C_3^4.C_3:D_4:D_4$ x 2, $(C_3^3\times C_6^2).C_4^2$ x 2, $C_6^4.D_6$ x 2, $C_6^4.D_6$ x 2, $C_3^4.(C_2\times C_6).C_2^4$, $(C_3^4\times C_6).C_2^4.C_2$, $(C_3^3\times C_6).C_6.C_4^2$, $C_6^4:D_6$, $C_6^4:D_6$, $C_6^4:D_6$

Order 11664: $C_3^6.C_2^4$ x 86, $C_3^6.C_2^3.C_2$ x 67, $C_3^5.(C_6\times D_4)$ x 39, $C_3^5.D_6:C_4$ x 31, $C_3^6.D_4.C_2$ x 13, $(C_3^3\times C_6^3).C_2$ x 11, $C_3^4.C_6^2.C_2^2$ x 10, $(C_3^5\times C_6).C_2^2.C_2$ x 10, $C_6^4.C_3^2$ x 8, $C_3^4.(C_3\times D_6:C_4)$ x 8, $(C_3^5\times C_6).C_2^3$ x 8, $C_3^6.C_2^2.C_2^2$ x 5, $C_3^5.D_6.C_2^2$ x 4, $C_3^5.C_6.C_4.C_2$ x 4, $C_3^4:(S_3\times D_{12})$ x 4, $C_3^4.D_6:D_6$ x 3, $C_3^6.D_4:C_2$ x 2, $C_3^6.C_4^2$ x 2, $C_3^6.C_4:C_4$ x 2, $C_3^5.C_6.C_2^3$ x 2, $C_3^5.(C_3\times C_4:C_4)$ x 2, $C_2\times C_3^6.C_2^3$ x 2, $C_3^4.(C_6\times S_4)$ x 2, $C_3^6.(C_2\times D_4)$, $C_3^5.C_{12}.C_2^2$, $C_3^4.(C_3\times C_6.D_4)$, $C_3^2\times C_6^4$, $C_3^5:(C_2\times S_4)$, $C_3^5:(C_2\times S_4)$

Order 10368: $C_3^4.C_2^5.C_2^2$ x 26, $C_3^4.C_2^6.C_2$ x 12, $(C_2\times C_6^3):S_4$ x 4, $C_6^3.C_6.C_2^3$ x 3, $C_3^3.(C_2^4\times C_6).C_2^2$ x 2, $C_6^2.D_6\wr C_2$ x 2, $C_6^2.D_6\wr C_2$ x 2, $C_6^4.D_4$ x 2, $C_3^4.C_2^4.C_2^3$, $(C_2\times C_6^3):S_4$, $(S_3\times D_6^2):A_4$, $C_6:D_6^2:A_4$, $C_6^3:C_6.D_4$, $C_6^2.D_6\wr C_2$, $C_6^2.D_6\wr C_2$, $C_6^3:C_6:D_4$, $C_6^4.(C_2\times C_4)$, $C_6^4.C_2^3$

Order 8748: $C_3^6.A_4$ x 7, $(C_3^4\times C_6^2).C_3$ x 3, $C_3^6:D_6$

Order 5832: $C_3^6.C_2^3$

Order 2916: $C_3^5:D_6$ x 2, $C_3^4\times C_6^2$

Order 864: $C_6^3.C_2^2$

Order 729: $C_3^6$

Order 432: $C_2\times C_6^3$ x 2

Order 216: $C_6^2:S_3$

Order 108: $C_3\times C_6^2$ x 2

Order 27: $C_3^3$ x 2

Order 16: $C_2^4$

Order 4: $C_2^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 559872: $C_3^6.C_2^5:S_4$ ($C_1$)

Order 279936: $C_3^6.C_2^6.S_3$ ($C_2$), $C_3^6.C_2^4:S_4$ ($C_2$), $C_3^6.C_2^5:A_4$ ($C_2$)

Order 139968: $C_3^6.C_2^6.C_3$ ($C_2^2$)

Order 93312: $C_3^6.C_2^6.C_2$ ($S_3$)

Order 46656: $C_3^6.C_2^6$ ($D_6$)

Order 23328: $C_3^6.C_2^5$ ($S_4$), $C_3^6.C_2^4.C_2$ ($S_4$), $(C_3^2\times C_6^4).C_2$ ($S_4$)

Order 11664: $C_3^6.C_2^4$ ($C_2\times S_4$) x 2, $C_3^2\times C_6^4$ ($C_2\times S_4$)

Order 5832: $C_3^6.C_2^3$ ($C_2^2:S_4$)

Order 2916: $C_3^5:D_6$ ($C_2^3:S_4$) x 2, $C_3^4\times C_6^2$ ($C_2^3:S_4$)

Order 864: $C_6^3.C_2^2$ ($C_3^3:S_4$)

Order 729: $C_3^6$ ($C_2^5:S_4$)

Order 432: $C_2\times C_6^3$ ($C_2\times C_3^3:S_4$), $C_2\times C_6^3$ ($S_3\wr S_3$)

Order 216: $C_6^2:S_3$ ($(C_3\times C_6^2):S_4$)

Order 108: $C_3\times C_6^2$ ($C_6^3:S_4$), $C_3\times C_6^2$ ($S_3^3:S_4$)

Order 27: $C_3^3$ ($C_6:D_6^2:S_4$), $C_3^3$ ($(S_3\times D_6^2).S_4$)

Order 16: $C_2^4$ ($C_3^6:(C_2\times S_4)$)

Order 4: $C_2^2$ ($C_3^6.C_2^4.D_6$)

Order 1: $C_1$ ($C_3^6.C_2^5:S_4$)

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 7 larger groups in the database.

Character theory

Complex character table

The $434 \times 434$ character table is not available for this group.

Rational character table

The $345 \times 345$ rational character table is not available for this group.