C₇×#13312 #13190528

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13312

Subgroups

	subgroup	G/H
center	C₄×C₇	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

520 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 114 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)(10,22)(11,28)(12,33)(13,20)(14,19)(15,21)(16,24)(17,31)(18,36)(23,52)(25,58)(26,46)(27,51)(29,61)(30,63)(32,44)(34,49)(35,48)(37,47)(38,45)(39,68)(40,56)(41,57)(42,53)(43,66)(50,54)(55,71)(59,62)(60,65)(64,69)(67,70);(8,10)(9,16)(11,29)(12,34)(13,25)(14,26)(15,27)(17,42)(18,46)(19,40)(20,49)(21,41)(22,33)(23,31)(24,36)(28,59)(30,64)(32,54)(35,57)(37,56)(38,55)(39,44)(43,50)(45,69)(47,51)(48,58)(52,60)(53,66)(61,63)(62,65)(67,68)(70,71);(8,11)(9,17)(10,23)(12,30)(13,38)(14,31)(15,32)(16,39)(18,43)(19,44)(20,28)(21,45)(22,50)(24,53)(25,29)(26,54)(27,55)(33,60)(34,62)(35,67)(36,65)(37,66)(40,69)(41,59)(42,49)(46,71)(47,70)(48,63)(51,61)(52,58)(56,68)(57,64)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6300);