C₇×#11542 #13188758

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11542

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₂⁵×C₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#117 ◁ G

460 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 4530);