C₇×#8515 #13185731

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#8515

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#3567	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

644 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,9,12,19,15,10,21,25)(11,27,30,63,32,53,49,62)(13,35,33,45,38,61,50,56)(14,20,34,17,39,26,51,23)(16,40,42,67,22,52,41,29)(18,46,48,58,24,57,60,37)(28,64,31,47,66,71,68,59)(36,43,69,44,70,54,65,55);(8,10)(9,15)(11,22)(12,17)(13,35)(14,26)(16,32)(18,37)(19,34)(20,39)(21,23)(24,58)(25,51)(27,36)(28,41)(29,47)(30,44)(31,54)(33,60)(38,61)(40,62)(42,66)(43,68)(45,57)(46,56)(48,50)(49,55)(52,63)(53,70)(59,67)(64,69)(65,71);(8,11,14,31)(9,16,20,43)(10,22,26,54)(12,30,34,66)(13,36,37,29)(15,32,39,68)(17,44,19,42)(18,47,35,27)(21,49,51,28)(23,55,25,41)(24,59,61,53)(33,69,46,40)(38,70,58,67)(45,63,48,71)(50,65,57,52)(56,62,60,64)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1503);