Abstract group 1679616.hr: $C_6^4.(C_3^2\times C_6^2):C_4$

• Label: 1679616.hr
• Order: \( 2^{8} \cdot 3^{8} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{3} \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^4.(C_3^2\times C_6^2):C_4$
Order:	\(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	Group of order \(362797056\)\(\medspace = 2^{11} \cdot 3^{11} \)
Composition factors:	$C_2$ x 8, $C_3$ 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	8	9	12	18	24
Elements	1	11727	9800	174960	213048	139968	49248	559872	241056	279936	1679616
Conjugacy classes	1	12	45	11	1605	4	30	58	306	8	2080
Divisions	1	12	38	9	988	2	28	28	228	2	1336
Autjugacy classes	1	10	13	5	173	1	7	15	32	1	258

Minimal presentations

Permutation degree:	$30$
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	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid d^{6}=e^{6}=f^{9}=g^{6}=h^{6}=[e,g]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,4,3,6)(2,7,11,17,8,15,18,12,16,10,13,5)(9,14)(19,21,26,23)(20,24,25,22,29,30,27,28) \!\cdots\! \rangle$
Transitive group:	36T40648				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:	$(C_6^4.C_3^4.C_2)$ . $D_4$ (2)	$(C_6^4.C_3^3.D_6)$ . $C_2^2$	$(C_6^4.C_3^4.C_2^3)$ . $C_2$	$(C_6^4.C_3^4.C_2^2)$ . $C_4$ (2)	all 38

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{6}^{3}$
Commutator length:	$2$

Subgroups

There are 74 normal subgroups (46 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 $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_3^3\times C_6$
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 32 (order at least 52488) are shown, as well as all normal subgroups of any index.

Order 1679616: $C_6^4.(C_3^2\times C_6^2):C_4$

Order 839808: $C_2\times C_6^4.C_3^4:C_4$ x 2, $C_6^4.C_3^4.C_2^3$

Order 419904: $C_6^4.C_3^3.D_6$ x 4, $C_6^4.C_3^4:C_4$ x 2, $C_6^4.C_3^4.C_2^2$

Order 279936: $C_6^4.C_3^3.C_2^3$ x 19, $C_6^2.C_3^4.C_6.C_2^4$

Order 209952: $C_6^4.C_3^4.C_2$ x 5, $C_3^4.(C_3^3\times A_4).C_2^3$

Order 186624: $C_6^4.C_6^2:C_4$ x 4, $C_6^4.C_6^2:C_4$ x 2, $C_6^4.C_6^2:C_4$ x 2, $(C_3\times C_6^4).C_6.C_2^3$, $C_6^4.C_6^2:C_4$

Order 139968: $C_6^4.C_3^2.D_6$ x 90, $C_6^4.C_3^3.C_2^2$ x 43, $C_6^2.C_3^4.C_6.C_2^3$ x 12, $C_6^2.C_3^5.C_2^4$, $C_6^2.C_3^5.C_2.C_2^3$, $C_2\times C_6^2.C_3^5.C_2^3$

Order 104976: $(C_3^4\times C_6^2).C_3.D_6$ x 6, $C_6^4.C_3^4$, $(C_3^4\times C_6^2).C_6^2$, $C_3^6.C_6^2:C_4$

Order 93312: $C_6^2.C_3^3.C_6.C_2^4$ x 36, $C_3^3.A_4^2.C_6.C_2^2$ x 21, $C_6^4.C_6^2.C_2$ x 15, $C_6^2.C_6^2.C_6^2.C_2$ x 9, $(C_2\times C_6^3).C_3^3.C_2^3$ x 8, $C_2\times C_3^4.A_4^2:C_4$ x 8, $C_3^5.(C_6\times D_4).C_2^3$ x 4, $C_2\times C_3^4.A_4^2:C_4$ x 4, $(C_2^3\times C_6^2).C_3\wr C_4$ x 4, $C_2^4.C_9^2.C_6^2.C_2$ x 3, $(C_3\times C_6^2).C_6^3.C_2^2$ x 3, $(C_3^3\times C_6^3).C_2^3.C_2$ x 2, $(C_2^3\times C_6^2).C_3\wr C_4$ x 2, $C_6^2.C_3^4.C_2.C_2^4$, $C_6^2.(A_4\times \He_3).C_2^3$, $C_2^4.C_3^4.C_6^2.C_2$, $(C_6^2\times A_4).C_3^3.C_2^3$, $(C_3^2\times C_6^4).C_2^3$, $(C_2^3\times C_6).C_3^3.C_6^2.C_2$

Order 69984: $C_6^4.C_3^3.C_2$ x 133, $C_3^4.C_6^2.D_6.C_2$ x 15, $C_3^4.(C_3\times A_4).D_6.C_2$ x 15, $C_6^2.C_3^4.C_6.C_2^2$ x 14, $C_6^2.C_3^5.C_2^3$ x 13, $C_6^2.C_3^5.C_4.C_2$ x 4, $C_6^2.C_3^4.C_{12}.C_2$ x 4, $C_2\times C_6^2.C_3^5.C_2^2$ x 4, $(C_3^4\times C_6^2).D_6.C_2$ x 3, $C_6^2.C_3^5.C_2^2.C_2$ x 2, $C_6^2.C_3^3.C_3^2.C_2^3$ x 2, $(C_3\times C_6^2).C_3^4.C_2^3$ x 2, $C_3^5.C_6^2.C_2^3$

Order 62208: $(C_2\times C_6^3).C_3:S_3.D_4$ x 4, $C_6^2.A_4^2:C_{12}$ x 2, $(C_3\times C_6^2).A_4^2:C_4$ x 2, $(C_3\times C_6^2).A_4^2:C_4$ x 2, $C_6^4.C_6.C_2^3$, $(C_3\times C_6^4).C_2^3.C_2$, $C_6^2.A_4^2:C_{12}$, $(C_3\times C_6^2):(A_4^2:C_4)$

Order 52488: $C_3^4.(C_3^3\times A_4).C_2$ x 7, $C_2\times (C_3\wr C_3)^2:C_4$ x 2, $C_3^6.C_3.D_6.C_2$

Order 46656: $C_6^4.C_6^2$ x 9, $C_6^6$

Order 23328: $C_6^4.C_3.C_6$ x 9, $C_3\times C_6^5$

Order 11664: $C_6^4.C_3^2$ x 9, $C_3^2\times C_6^4$

Order 5184: $C_2^2\times C_6^4$

Order 2916: $C_3^4\times C_6^2$

Order 2592: $C_2\times C_6^4$

Order 1728: $C_2^3\times C_6^3$ x 2

Order 1458: $C_3^5\times C_6$

Order 1296: $C_6^4$

Order 864: $C_2^2\times C_6^3$ x 2

Order 729: $C_3^6$

Order 576: $C_2^4\times C_6^2$

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

Order 324: $C_3^2\times C_6^2$

Order 288: $C_2^3\times C_6^2$

Order 162: $C_3^3\times C_6$

Order 144: $C_2^2\times C_6^2$

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

Order 81: $C_3^4$

Order 64: $C_2^6$

Order 54: $C_3^2\times C_6$ x 2

Order 36: $C_6^2$

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 18: $C_3\times C_6$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1679616: $C_6^4.(C_3^2\times C_6^2):C_4$ ($C_1$)

Order 839808: $C_2\times C_6^4.C_3^4:C_4$ ($C_2$) x 2, $C_6^4.C_3^4.C_2^3$ ($C_2$)

Order 419904: $C_6^4.C_3^4.C_2^2$ ($C_4$), $C_6^4.C_3^3.D_6$ ($C_2^2$), $C_6^4.C_3^3.D_6$ ($C_4$)

Order 209952: $C_6^4.C_3^4.C_2$ ($D_4$) x 2, $C_6^4.C_3^4.C_2$ ($C_2\times C_4$)

Order 104976: $C_6^4.C_3^4$ ($C_2^2:C_4$)

Order 46656: $C_6^4.C_6^2$ ($C_3^2:C_4$) x 9, $C_6^6$ ($C_3^2:C_4$)

Order 23328: $C_6^4.C_3.C_6$ ($C_2\times C_3^2:C_4$) x 9, $C_3\times C_6^5$ ($C_2\times C_3^2:C_4$)

Order 11664: $C_6^4.C_3^2$ ($C_6^2:C_4$) x 9, $C_3^2\times C_6^4$ ($C_6^2:C_4$)

Order 5184: $C_2^2\times C_6^4$ ($C_3^4:C_4$)

Order 2916: $C_3^4\times C_6^2$ ($A_4^2:C_4$)

Order 2592: $C_2\times C_6^4$ ($C_2\times C_3^4:C_4$)

Order 1728: $C_2^3\times C_6^3$ ($C_3^3:C_3^2:C_4$), $C_2^3\times C_6^3$ ($C_3^3:C_3^2:C_4$)

Order 1458: $C_3^5\times C_6$ ($C_2\times A_4^2:C_4$)

Order 1296: $C_6^4$ ($(C_3^2\times C_6^2):C_4$)

Order 864: $C_2^2\times C_6^3$ ($C_6.C_3^4:C_4$), $C_2^2\times C_6^3$ ($C_6.C_3^4:C_4$)

Order 729: $C_3^6$ ($(C_2^2\times A_4^2):C_4$)

Order 576: $C_2^4\times C_6^2$ ($\He_3^2:C_4$)

Order 432: $C_2\times C_6^3$ ($C_3^3.(C_3:S_3.D_4)$) x 2

Order 324: $C_3^2\times C_6^2$ ($(C_3^2\times A_4^2):C_4$)

Order 288: $C_2^3\times C_6^2$ ($C_2\times \He_3^2:C_4$)

Order 162: $C_3^3\times C_6$ ($C_2^5:(C_3^4:C_4)$)

Order 144: $C_2^2\times C_6^2$ ($C_3^4:(C_6^2:C_4)$)

Order 108: $C_3\times C_6^2$ ($(C_3\times A_4^2).C_3^2.C_4$), $C_3\times C_6^2$ ($(C_2^3\times C_6).C_3^4.C_4$)

Order 81: $C_3^4$ ($A_4^2:(C_6^2:C_4)$)

Order 64: $C_2^6$ ($(C_3\wr C_3)^2:C_4$)

Order 54: $C_3^2\times C_6$ ($(C_2^3\times C_6).C_3^4.C_4.C_2$), $C_3^2\times C_6$ ($(C_2\times C_6^3).C_3:S_3.C_2^2$)

Order 36: $C_6^2$ ($C_3^4:(A_4^2:C_4)$)

Order 32: $C_2^5$ ($C_2\times (C_3\wr C_3)^2:C_4$)

Order 27: $C_3^3$ ($(C_3\times A_4^2).(C_3:S_3.D_4)$) x 2

Order 18: $C_3\times C_6$ ($C_2^5:(\He_3^2:C_4)$)

Order 16: $C_2^4$ ($C_3^6.C_6^2:C_4$)

Order 9: $C_3^2$ ($C_6^4.C_6^2:C_4$)

Order 4: $C_2^2$ ($C_6^4.C_3^4:C_4$)

Order 2: $C_2$ ($C_2\times C_6^4.C_3^4:C_4$)

Order 1: $C_1$ ($C_6^4.(C_3^2\times C_6^2):C_4$)

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 5 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $2080 \times 2080$ character table is not available for this group.

Rational character table

The $1336 \times 1336$ rational character table is not available for this group.