C₇×#11794 #13189010

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#11794

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

378 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 448

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

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4782);