C₇×#12877 #13190093

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×#12877

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

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

normal subgroups only

This group has 366 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)(10,20,23,15)(11,25)(12,29)(13,35)(14,17)(16,27)(18,33)(19,32)(21,48,49,67)(22,52,30,62)(24,39,54,47)(26,57)(28,60)(31,45)(34,61)(36,44)(37,50,46,59)(38,55,43,64)(40,65)(41,58)(42,66)(51,69,56,71)(53,68,63,70);(8,10)(9,15)(11,21)(12,30)(13,24)(14,23)(16,37)(17,20)(18,43)(19,39)(22,34)(25,48)(26,49)(27,59)(28,51)(29,52)(31,63)(32,54)(33,64)(35,47)(36,56)(38,42)(40,68)(41,46)(44,71)(45,70)(50,58)(53,65)(55,66)(57,67)(60,69)(61,62);(8,11,13,36,14,26,32,28)(9,16,35,40,17,41,19,45)(10,21,24,56,23,49,54,51)(12,31,33,27,34,65,66,58)(15,37,47,68,20,46,39,70)(18,44,61,57,42,60,29,25)(22,53,55,50,30,63,64,59)(38,69,52,48,43,71,62,67)

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, 5865);