Abstract group 839808.qk: $C_3^8:D_8.C_8$

• Label: 839808.qk
• Order: \( 2^{7} \cdot 3^{8} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^8:D_8.C_8$
Order:	\(839808\)\(\medspace = 2^{7} \cdot 3^{8} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	Group of order \(33592320\)\(\medspace = 2^{10} \cdot 3^{8} \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_3$ 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	12	16
Elements	1	7371	6560	105300	64800	209952	25920	419904	839808
Conjugacy classes	1	4	60	9	15	14	10	28	141
Divisions	1	4	60	6	15	6	5	5	102
Autjugacy classes	1	3	5	5	2	7	1	8	32

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	32	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{8}=d^{3}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,27,2,23,9,22,3,19,4,24,6,25,8,26,5,20)(7,21)(10,34,13,28,11,29,12,33,17,35,14,32,16,31,18,30) \!\cdots\! \rangle$
Transitive group:	36T34926				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:	$(C_3^8:D_8)$ . $C_8$	$(C_3^8:Q_{16})$ . $C_8$	$(C_3^8.C_{16})$ . $D_4$ (2)	$(C_3^8.\SD_{16})$ . $C_8$ (2)	all 30

Elements of the group are displayed as permutations of degree 36.

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 47 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 $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8$

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 27 (order at least 31104) are shown, as well as all normal subgroups of any index.

Order 839808: $C_3^8:D_8.C_8$

Order 419904: $C_3^8:C_8.D_4$ x 2, $C_3^8:D_4.C_8$ x 2, $C_3^8.C_4^2.C_2^2$, $C_3^8.C_4.C_4^2$, $C_3^8.(C_8.C_8)$

Order 209952: $C_3^8:\OD_{32}$ x 4, $C_3^8:(C_2\times C_{16})$ x 3, $C_3^8:D_4.C_4$ x 2, $C_3^8:C_4\wr C_2$ x 2, $C_3^8.C_4^2.C_2$, $C_3^8.D_8:C_2$, $C_3^8:(C_2\times C_{16})$, $C_3^8:C_8.C_4$

Order 104976: $C_3^8.C_{16}$ x 4, $C_3^8:\OD_{16}$ x 4, $C_3^8.\SD_{16}$ x 2, $C_3^8.C_8.C_2$ x 2, $C_3^8.C_4.C_2^2$ x 2, $C_3^8.D_4:C_2$ x 2, $C_3^8.C_4^2$, $C_3^8:Q_{16}$, $C_3^8:D_8$, $C_3^8:C_{16}$

Order 52488: $C_3^8:C_8$ x 4, $C_3^8:D_4$ x 4, $C_3^8:(C_2\times C_4)$ x 3, $C_3^8.C_8$ x 2, $C_3^8:Q_8$ x 2, $C_3^8.C_4.C_2$

Order 26244: $C_3^8.C_4$ x 2, $C_3^7.D_6$

Order 13122: $C_3^8.C_2$

Order 6561: $C_3^8$

Order 128: $C_{16}.D_4$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 839808: $C_3^8:D_8.C_8$ ($C_1$)

Order 419904: $C_3^8:C_8.D_4$ ($C_2$) x 2, $C_3^8:D_4.C_8$ ($C_2$) x 2, $C_3^8.C_4^2.C_2^2$ ($C_2$), $C_3^8.C_4.C_4^2$ ($C_2$), $C_3^8.(C_8.C_8)$ ($C_2$)

Order 209952: $C_3^8:\OD_{32}$ ($C_2^2$) x 2, $C_3^8:D_4.C_4$ ($C_2^2$) x 2, $C_3^8:C_4\wr C_2$ ($C_4$) x 2, $C_3^8.C_4^2.C_2$ ($C_2^2$), $C_3^8:(C_2\times C_{16})$ ($C_2^2$), $C_3^8.D_8:C_2$ ($C_4$), $C_3^8:(C_2\times C_{16})$ ($C_2^2$), $C_3^8:C_8.C_4$ ($C_4$)

Order 104976: $C_3^8.\SD_{16}$ ($C_8$) x 2, $C_3^8.C_{16}$ ($D_4$) x 2, $C_3^8.D_4:C_2$ ($C_2\times C_4$) x 2, $C_3^8:\OD_{16}$ ($C_2\times C_4$) x 2, $C_3^8.C_4^2$ ($C_2\times C_4$), $C_3^8.C_4.C_2^2$ ($C_2^3$), $C_3^8.C_4.C_2^2$ ($C_2\times C_4$), $C_3^8:Q_{16}$ ($C_8$), $C_3^8:D_8$ ($C_8$)

Order 52488: $C_3^8:Q_8$ ($C_2\times C_8$) x 2, $C_3^8:D_4$ ($C_2\times C_8$) x 2, $C_3^8.C_8$ ($C_2\times C_8$), $C_3^8.C_8$ ($C_2\times D_4$), $C_3^8:C_8$ ($C_2\times C_8$), $C_3^8:C_8$ ($D_4:C_2$), $C_3^8:(C_2\times C_4)$ ($C_2^2\times C_4$)

Order 26244: $C_3^8.C_4$ ($C_2^2\times C_8$), $C_3^8.C_4$ ($C_4\times D_4$), $C_3^7.D_6$ ($\OD_{16}:C_2$)

Order 13122: $C_3^8.C_2$ ($C_8\times D_4$)

Order 6561: $C_3^8$ ($C_{16}.D_4$)

Order 1: $C_1$ ($C_3^8:D_8.C_8$)

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

The $141 \times 141$ character table is not available for this group.

Rational character table

The $102 \times 102$ rational character table is not available for this group.