C₇×#11914 #13189130

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11914

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1530
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#97	C₂³×C₇
prime radical	C₂³×C₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#405 ◁ G

272 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 94 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)(8,9,12,18,13,19,33,49,14,20,32,50,35,17,44,46)(10,21,23,52,24,15,36,38,25,39,57,47,34,40,30,63)(11,26,28,58,29,43,61,48,27,45,31,65,59,16,41,42)(22,37,54,67,55,66,64,70,53,68,56,69,62,51,60,71);(8,10,14,25)(9,15,20,40)(11,22,27,53)(12,30,32,36)(13,34,35,24)(16,37,43,68)(17,39,19,21)(18,47,50,52)(23,33,57,44)(26,51,45,66)(28,60,31,64)(29,62,59,55)(38,46,63,49)(41,54,61,56)(42,70,48,71)(58,67,65,69);(8,11,14,27)(9,16)(10,22,25,53)(12,31,32,28)(13,29,35,59)(15,37)(17,45)(18,48)(19,26)(20,43)(21,51)(23,56,57,54)(24,55,34,62)(30,64,36,60)(33,41,44,61)(38,69)(39,66)(40,68)(42,50)(46,58)(47,71)(49,65)(52,70)(63,67)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4902);