C₇×#11771 #13188987

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#11771

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1505
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₂³×C₇	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#405 ◁ G

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

normal subgroups only

This group has 292 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)(10,21,24,50)(11,17)(12,32,34,48)(13,19)(14,37)(15,20)(16,23)(18,35,46,36)(22,49,52,67)(25,45,26,56)(27,57,53,71)(28,59,60,69)(29,43)(30,63)(31,44)(33,47)(38,51)(39,54)(40,55)(41,58)(42,61,68,62)(64,66)(65,70);(8,10,15,24)(9,16,20,40)(11,27,31,53)(12,33,34,54)(13,26,14,25)(17,41,44,66)(18,45,46,56)(19,47,37,39)(21,48,50,32)(22,30,52,29)(23,35,55,36)(28,51,60,70)(38,63,65,43)(42,67,68,49)(57,69,71,59)(58,61,64,62);(8,11,14,30,15,31,13,29)(9,17,37,63,20,44,19,43)(10,22,26,53,24,52,25,27)(12,28,36,62,34,60,35,61)(16,38,47,66,40,65,39,41)(18,42,48,69,46,68,32,59)(21,49,56,71,50,67,45,57)(23,51,33,64,55,70,54,58)

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, 4759);