C₇×#13647 #13190863

order	1792
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13647

Subgroups

	subgroup	G/H
center	C₇×C₃₂	D₄
commutator	C₄	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	D₄×C₁₆
prime radical	C₇×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₃₂ ◁ C₂×C₇×C₃₂ ◁ G

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,9,13,19,14,20,35,49,12,21,32,50,33,52,63,70,15,22,36,51,37,53,65,71,34,18,45,47,64,48,39,66)(10,23,25,54,26,29,43,60,27,41,56,67,58,31,44,62,28,46,57,69,59,61,42,68,24,16,38,40,55,11,17,30);(8,10)(9,16)(11,20)(12,27)(13,25)(14,26)(15,28)(17,39)(18,46)(19,40)(21,23)(22,41)(24,34)(29,52)(30,49)(31,53)(32,56)(33,58)(35,43)(36,57)(37,59)(38,45)(42,65)(44,63)(47,69)(48,61)(50,54)(51,67)(55,64)(60,70)(62,71)(66,68);(8,11,13,30,14,23,35,54,12,29,32,60,33,41,63,67,15,31,36,62,37,46,65,69,34,61,45,68,64,16,39,40)(9,17,19,10,20,25,49,26,21,43,50,27,52,56,70,58,22,44,51,28,53,57,71,59,18,42,47,24,48,38,66,55)

Conjugacy classes & character table

784 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6635);