C₇×#11844 #13189060

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₇×#11844

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₇×#392	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#117 ◁ G

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

normal subgroups only

This group has 432 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)(10,20)(11,25,28,14)(12,18)(13,19)(15,23)(16,26,31,39)(17,29)(21,35)(22,32,27,33)(24,34)(30,37,36,38);(9,14)(10,21)(11,26)(15,32)(16,18)(17,31)(19,25)(20,30)(22,27)(23,24)(28,39)(33,35)(34,36)(37,38);(8,10,13,24)(9,15,19,35)(11,27,28,22)(12,23,29,21)(14,30,25,36)(16,32,31,33)(17,34,18,20)(26,37,39,38)

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, 4832);