C₇×#11311 #13188527

order	1792
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#11311

Subgroups

	subgroup	G/H
center	C₄×C₇	#396
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂×C₇	#1443
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂³×C₄×C₇ ◁ G

532 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 202 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)(11,24,28,16)(12,18)(13,19)(14,17)(15,22)(21,32,36,37)(23,35)(25,30,38,39)(26,33,27,34)(29,31);(8,10,14,22,13,23,12,29)(9,15,17,35,19,31,18,20)(11,25,26,21,28,38,27,36)(16,32,34,39,24,37,33,30);(8,11,12,26,13,28,14,27)(9,16,17,33,19,24,18,34)(10,21,22,25,23,36,29,38)(15,30,20,32,31,39,35,37)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4299);