C₇×#13315 #13190531

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13315

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 75
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 75 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,19,25)(14,26,31,57)(15,32,36,54)(16,37,41,52)(17,24,18,23)(20,43,47,28)(21,42,50,35)(22,39,53,40)(27,58,60,75)(29,51,30,56)(33,55,62,73)(34,65,68,49)(38,45,61,46)(44,70,71,64)(48,66,72,59)(63,69,74,67);(12,14,19,31)(13,20,25,47)(15,33,36,62)(16,38,41,61)(17,30,18,29)(21,48,50,72)(22,51,53,56)(23,46,24,45)(26,52,57,37)(27,42,60,35)(28,39,43,40)(32,44,54,71)(34,66,68,59)(49,73,65,55)(58,74,75,63)(64,69,70,67);(9,11)(12,15)(13,21)(14,27)(16,34)(17,42)(18,35)(19,36)(20,44)(22,49)(23,54)(24,32)(25,50)(26,55)(28,59)(29,62)(30,33)(31,60)(37,63)(38,64)(39,69)(40,67)(41,68)(43,66)(45,72)(46,48)(47,71)(51,58)(52,74)(53,65)(56,75)(57,73)(61,70)

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, 6303);