C₇×#10983 #13188199

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇×#10983

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

1318 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 31
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(16,17,20,25,21,24,28,31)(18,23,26,30,19,22,27,29);(9,11)(12,13)(14,15)(16,18,21,19)(17,22,24,23)(20,27,28,26)(25,30,31,29);(13,15)(16,19,21,18)(17,23,24,22)(20,26,28,27)(25,29,31,30)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3971);