C₇×#11912 #13189128

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11912

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1530
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#97	C₂³×C₇
prime radical	C₇×#540	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#405 ◁ G

284 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 94 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)(8,9,11,17,12,14,26,28,13,18,25,34,27,16,23,33)(10,19,21,36,22,31,38,35,20,32,24,39,37,15,29,30);(9,14)(11,23)(12,27)(15,19)(16,18)(17,34)(21,29)(22,37)(24,38)(25,26)(30,35)(31,32);(8,10)(9,15,18,31)(11,24)(12,22)(13,20)(14,19,16,32)(17,35,34,30)(21,25)(23,38)(26,29)(27,37)(28,39,33,36)

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, 4900);