C₇×#11840 #13189056

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#11840

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

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

normal subgroups only

This group has 446 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)(11,24,27,14)(12,18)(13,19)(15,21)(16,25,30,39)(17,28)(22,34,26,31)(23,32)(29,38,35,36)(33,37);(9,14)(10,21)(11,25)(15,31)(16,18)(17,30)(19,24)(20,35)(23,37)(27,39)(29,32)(33,34);(8,10,13,23)(9,15,19,33)(11,26,27,22)(12,21,28,37)(14,29,24,35)(16,34,30,31)(17,32,18,20)(25,38,39,36)

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, 4828);