C₇×#9007 #13186223

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9007

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#120
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#386
prime radical	C₇×#3150	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

502 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 47
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,13,15,12,10,11,14)(16,17)(18,27,30,38)(19,31)(20,25)(21,35)(22,26)(23,29,39,34)(24,32)(28,43,46,37)(33,40)(36,41)(42,44,47,45);(8,10,12,9)(11,14,13,15)(16,18,21,29,22,30,20,34)(17,23,35,27,26,39,25,38)(19,28,36,44,33,46,32,45)(24,37,31,42,41,43,40,47);(11,13)(14,15)(16,19)(17,24)(18,28)(20,32)(21,36)(22,33)(23,37)(25,40)(26,41)(27,42)(29,44)(30,46)(31,35)(34,45)(38,47)(39,43)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1995);