C₇×#11443 #13188659

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11443

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

382 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 142 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)(10,14,15,12)(11,13)(16,17,20,26,21,27,35,44,22,24,36,40,19,25,33,42)(18,28,30,45,29,38,46,43,31,41,34,47,32,23,37,39);(8,10,11,15)(9,12,13,14)(16,18)(17,23)(19,32)(20,34)(21,29)(22,31)(24,38)(25,41)(26,43)(27,28)(30,36)(33,46)(35,37)(39,40)(42,45)(44,47);(17,24)(23,38)(25,27)(26,40)(28,41)(39,43)(42,44)(45,47)

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, 4431);