C₇×#13643 #13190859

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#13643

Subgroups

	subgroup	G/H
center	C₇×C₈	#117
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	#2163
prime radical	C₂³×C₇	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₇×#346 ◁ G

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

normal subgroups only

This group has 164 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)(8,9,11,18,12,19,28,44,13,20,29,43,30,45,55,66,14,17,27,40,31,41,54,64,32,16,36,38,56,39,61,63)(10,21,23,42,24,46,49,65,25,37,50,59,51,62,68,71,22,15,26,34,47,35,52,60,48,33,53,57,67,58,69,70);(8,10)(9,15)(11,26)(12,24)(13,25)(14,22)(16,37)(17,21)(18,42)(19,35)(20,33)(23,27)(28,52)(29,53)(30,51)(31,47)(32,48)(34,40)(36,50)(38,57)(39,62)(41,46)(43,59)(44,65)(45,58)(49,54)(55,69)(56,67)(60,64)(61,68)(63,70)(66,71);(9,16,17,20)(10,22)(11,27)(15,33,21,37)(18,43,40,38)(19,39,41,45)(24,47)(25,48)(28,54)(29,36)(34,59,42,57)(35,58,46,62)(44,66,64,63)(51,67)(55,61)(60,71,65,70)

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6631);