C₇×#7381 #13184597

order	1792
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7381

Subgroups

	subgroup	G/H
center	C₇×C₁₆	D₈
commutator	C₈	C₂×C₇×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂²×C₇
socle	C₂×C₇	D₄⋊C₁₆
prime radical	C₇×#135	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₁₆ ◁ C₂×C₇×C₁₆ ◁ C₄×C₇×C₁₆ ◁ G

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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)(8,9,12,18,14,20,37,49,13,21,36,50,11,30,33,61,15,22,38,51,34,17,44,46,32,19,52,48,35,47,63,69)(10,23,25,54,27,43,58,67,28,45,59,68,31,53,62,70,29,55,60,71,24,16,39,40,26,42,57,66,56,41,64,65);(8,10)(9,16)(11,31)(12,25)(13,28)(14,27)(15,29)(17,45)(18,40)(19,53)(20,42)(21,41)(22,43)(23,30)(24,34)(26,32)(33,62)(35,56)(36,59)(37,58)(38,60)(39,44)(46,68)(47,55)(48,70)(49,66)(50,65)(51,67)(52,57)(54,61)(63,64)(69,71)

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 369);