C₇×#12804 #13190020

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12804

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₇
prime radical	C₂³×C₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

432 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3,4,5,6,7)(9,11)(12,13,17,22,18,23,15,27)(14,19,24,34,26,35,28,42)(16,21,29,37,31,38,30,20)(25,33,40,32,41,43,39,36);(8,9,10,11)(12,14)(13,19)(15,28)(16,25)(17,24)(18,26)(20,36)(21,33)(22,34)(23,35)(27,42)(29,40)(30,39)(31,41)(32,37)(38,43);(12,15,18,17)(13,20,23,37)(14,24,26,28)(16,30,31,29)(19,32,35,36)(21,22,38,27)(25,40,41,39)(33,42,43,34)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5792);