C₇×#12466 #13189682

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12466

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

186 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 70 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)(8,9,16,23)(10,24,31,56)(11,32,36,48)(12,20,37,42)(13,19,25,51)(14,22,15,21)(17,41,47,27)(18,33,50,34)(26,58,61,70)(28,45,38,46)(29,44,30,52)(35,65,40,53)(39,54,57,49)(43,64,68,60)(55,62,67,59)(63,69,66,71);(8,10,25,28,14,29,12,27,16,31,13,38,15,30,37,41)(9,17,42,44,21,45,19,24,23,47,20,52,22,46,51,56)(11,26,57,63,35,60,34,62,36,61,39,66,40,64,33,59)(18,43,65,69,49,58,32,55,50,68,53,71,54,70,48,67);(8,11)(9,18)(10,26)(12,34)(13,39)(14,35)(15,40)(16,36)(17,43)(19,32)(20,53)(21,49)(22,54)(23,50)(24,55)(25,57)(27,62)(28,63)(29,60)(30,64)(31,61)(33,37)(38,66)(41,59)(42,65)(44,69)(45,58)(46,70)(47,68)(48,51)(52,71)(56,67)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5454);