C₇×#12878 #13190094

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12878

Subgroups

	subgroup	G/H
center	C₄×C₇	#418
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1824
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

362 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,13,20,15,18,14,37)(10,21,24,42,25,16,36,41)(11,26,29,58,27,59,38,48)(12,30,33,46,35,62,34,19)(17,32,43,60,45,66,44,28)(22,50,53,71,51,67,54,69)(23,55,57,47,31,63,65,40)(39,56,68,52,49,64,70,61);(8,10)(9,16)(11,22)(12,31)(13,36)(14,24)(15,25)(17,39)(18,21)(19,47)(20,42)(23,35)(26,50)(27,51)(28,61)(29,54)(30,55)(32,64)(33,57)(34,65)(37,41)(38,53)(40,46)(43,70)(44,68)(45,49)(48,71)(52,60)(56,66)(58,69)(59,67)(62,63);(8,11,14,38,15,27,13,29)(9,17,37,44,18,45,20,43)(10,22,24,53,25,51,36,54)(12,32,34,28,35,66,33,60)(16,39,41,68,21,49,42,70)(19,48,62,59,46,58,30,26)(23,56,57,52,31,64,65,61)(40,69,55,50,47,71,63,67)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5866);