C₇×#12118 #13189334

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12118

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₇×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 392 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)(10,22)(11,27)(12,31)(13,20)(14,19)(15,21)(16,24)(17,29)(18,34)(23,51)(25,59)(26,50)(28,62)(30,47)(32,40)(33,43)(35,48)(36,44)(37,46)(38,45)(39,66)(41,57)(42,58)(49,53)(52,60)(54,71)(55,69)(56,67)(61,65)(63,68)(64,70);(8,10,15,26)(9,16,21,42)(11,23,30,55)(12,32,35,57)(13,25,14,38)(17,39,44,68)(18,45,46,59)(19,41,20,40)(22,48,50,31)(24,37,58,34)(27,60,47,67)(28,63,33,66)(29,53,36,54)(43,69,62,51)(49,70,71,61)(52,64,56,65);(8,11)(9,17)(10,23)(12,33)(13,36)(14,29)(15,30)(16,39)(18,43)(19,47)(20,27)(21,44)(22,49)(24,56)(25,54)(26,55)(28,35)(31,61)(32,66)(34,64)(37,65)(38,53)(40,60)(41,67)(42,68)(45,69)(46,62)(48,70)(50,71)(51,59)(52,58)(57,63)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5106);