C₇×#13362 #13190578

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13362

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇
prime radical	C₇×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

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

normal subgroups only

This group has 670 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)(8,9,15,20)(10,21,25,47)(11,26,31,51)(12,32,36,18)(13,37,14,19)(16,33,43,23)(17,30,46,29)(22,54,56,70)(24,59,38,53)(27,52,60,71)(28,63,34,61)(35,49,39,50)(40,64,69,62)(41,57,42,58)(44,55,68,67)(45,65,48,66);(8,10)(9,16)(11,27)(12,33)(13,24)(14,38)(15,25)(17,44)(18,47)(19,41)(20,43)(21,32)(22,30)(23,36)(26,40)(28,64)(29,56)(31,60)(34,62)(35,58)(37,42)(39,57)(45,54)(46,68)(48,70)(49,59)(50,53)(51,69)(52,63)(55,65)(61,71)(66,67);(8,11,15,31)(9,17,20,46)(10,22,25,56)(12,34,36,28)(13,30,14,29)(16,40,43,69)(18,48,32,45)(19,51,37,26)(21,52,47,71)(23,55,33,67)(24,60,38,27)(35,65,39,66)(41,44,42,68)(49,61,50,63)(53,70,59,54)(57,64,58,62)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6350);