C₇×#11989 #13189205

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11989

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1551
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3176	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#405 ◁ G

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

normal subgroups only

This group has 292 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)(8,9,11,17,12,16,24,32,13,18,25,33,26,14,23,27)(10,19,21,34,22,15,28,29,20,31,35,39,36,30,37,38);(9,14)(11,23)(12,26)(15,19)(16,18)(17,33)(21,37)(22,36)(24,25)(28,35)(29,38)(30,31);(8,10)(9,15,18,30)(11,21)(12,22)(13,20)(14,19,16,31)(17,29,33,38)(23,37)(24,28)(25,35)(26,36)(27,34,32,39)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4977);