C₇×#11478 #13188694

order	1792
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#11478

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#99
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
socle	C₂²×C₇	C₂×#99
prime radical	C₂³×C₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

428 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 134 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)(8,9,11,14,12,15,10,13)(16,17)(18,26)(19,23)(20,24,29,30)(21,25)(22,27,31,28);(16,18,21,19)(17,22,25,31)(20,28,29,27)(23,30,26,24);(8,10,12,11)(9,13,15,14)(16,19,21,18)(17,23,25,26)(20,28,29,27)(22,30,31,24)

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, 4466);