C₇×#13612 #13190828

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13612

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2127
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#536 ◁ G

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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)(8,9,12,18,14,20,32,47)(10,21,23,45,25,38,52,64)(11,16,28,41,27,42,56,69)(13,19,34,46,35,17,31,44)(15,30,37,33,39,51,65,24)(22,48,49,67,26,53,54,70)(29,40,60,66,57,43,59,68)(36,61,62,55,63,58,71,50);(8,10,14,25)(9,15,20,39)(11,26,27,22)(12,30,32,51)(13,33,35,24)(16,36,42,63)(17,38,19,21)(18,45,47,64)(23,31,52,34)(28,58,56,61)(29,50,57,55)(37,46,65,44)(40,48,43,53)(41,67,69,70)(49,60,54,59)(62,66,71,68);(8,11)(9,16)(10,22)(12,28)(13,29)(14,27)(15,36)(17,43)(18,41)(19,40)(20,42)(21,48)(23,49)(24,50)(25,26)(30,61)(31,59)(32,56)(33,55)(34,60)(35,57)(37,62)(38,53)(39,63)(44,68)(45,67)(46,66)(47,69)(51,58)(52,54)(64,70)(65,71)

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, 6600);