C₇×#12884 #13190100

order	1792
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#12884

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

302 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,21,27,22,26,38,43,23,28,37,25,20,39,19,35)(18,29,32,44,33,41,30,42,34,45,46,47,31,24,36,40);(9,12)(13,15)(16,18,20,31,23,34,22,33)(17,24,26,29,28,41,39,45)(19,36,37,46,38,30,21,32)(25,42,35,47,27,40,43,44);(8,10,11,14)(9,13,12,15)(16,19,20,37,23,38,22,21)(17,25,26,35,28,27,39,43)(18,30,31,32,34,36,33,46)(24,40,29,44,41,42,45,47)

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5872);