C₇×#7425 #13184641

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7425

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1347
commutator	C₂×C₁₆	C₂×C₄×C₇
Frattini, cosocle	C₂×M₃₂	C₂²×C₇
prime radical	C₂³×C₇	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂×C₇×M₃₂ ◁ G

136 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,12,19,16,21,15,42)(10,22,24,56,28,57,27,63)(11,29,32,64,35,55,36,65)(13,38,37,54,14,40,31,20)(17,43,44,59,47,60,46,23)(18,48,50,33,52,67,51,34)(25,61,39,70,26,62,41,49)(30,53,66,69,58,71,68,45);(8,10,16,28)(9,17,21,47)(11,30,35,58)(12,27,15,24)(13,39,14,41)(18,49,52,70)(19,44,42,46)(20,53,54,71)(22,55,57,29)(23,33,59,34)(25,31,26,37)(32,68,36,66)(38,69,40,45)(43,48,60,67)(50,61,51,62)(56,65,63,64)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 413);