C₇×#12171 #13189387

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12171

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₇×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 332 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)(10,20)(11,25)(12,29)(13,35)(14,19)(15,22)(16,27)(17,34)(18,32)(21,48)(23,55)(24,47)(26,59)(28,41)(30,44)(31,64)(33,45)(36,42)(37,53)(38,54)(39,66)(40,67)(43,52)(46,50)(49,56)(51,69)(57,62)(58,61)(60,68)(63,71)(65,70);(8,10,14,24)(9,15,19,38)(11,21,27,50)(12,30,33,66)(13,36,32,23)(16,37,25,56)(17,42,43,55)(18,44,35,39)(20,45,47,29)(22,52,54,34)(26,60,31,65)(28,63,61,51)(40,69,57,71)(41,70,58,68)(46,59,48,64)(49,67,53,62);(8,11,14,27)(9,16)(10,21,24,50)(12,31,33,26)(13,28,32,61)(15,37)(17,40)(18,41)(19,25)(20,46)(22,53,54,49)(23,51,36,63)(29,64)(30,65,66,60)(34,67,52,62)(35,58)(38,56)(39,68)(42,69)(43,57)(44,70)(45,59)(47,48)(55,71)

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, 5159);