C₇×#11538 #13188754

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11538

Subgroups

	subgroup	G/H
center	C₄×C₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₇
socle	C₂×C₇	#1466
prime radical	C₇×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₇×#405 ◁ G

334 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 102 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)(8,9,11,16,12,17,24,32,13,14,23,27,25,28,36,39)(10,18,20,34,21,26,37,31,19,33,22,38,35,15,29,30);(9,14)(10,19)(17,28)(20,22)(21,35)(29,37)(30,31)(34,38);(8,10,12,21,13,19,25,35)(9,15,28,33,14,26,17,18)(11,22,24,29,23,20,36,37)(16,31,39,34,27,30,32,38)

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, 4526);