C₇×#13447 #13190663

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

Aliases

direct product

C₇×#13447

Subgroups

	subgroup	G/H
center	C₄×C₇	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#2054
prime radical	C₇×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#405 ◁ G

386 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 106 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)(10,23)(11,29)(12,36)(13,39)(14,19)(15,41)(16,22)(17,25)(18,32)(20,38)(21,37)(24,43)(26,53)(27,45)(28,52)(30,54)(31,62)(33,51)(34,65)(35,59)(40,47)(42,44)(46,58)(48,60)(49,63)(50,64)(55,67)(56,70)(57,69)(61,71)(66,68);(8,10)(9,17)(11,30)(12,26)(13,40)(14,25)(15,42)(16,28)(18,48)(19,44)(20,52)(21,53)(22,47)(23,36)(24,33)(27,37)(29,43)(31,56)(32,57)(34,55)(35,61)(38,58)(39,45)(41,46)(49,70)(50,71)(51,67)(54,62)(59,69)(60,63)(64,66)(65,68);(8,11,12,31,37,63,13,32,16,35,38,64,15,34,14,33)(9,18,19,49,41,62,20,29,22,51,39,65,21,50,36,59)(10,24,25,55,42,66,58,61,28,57,40,60,27,56,26,30)(17,43,23,54,53,70,45,48,47,69,52,71,46,68,44,67)

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, 6435);