C₇×#9190 #13186406

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9190

Subgroups

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

436 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 448

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

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

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, 2178);