C₇×#62849 #13240065

order	1792
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₇ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₇×#62849

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
prime radical	C₂⁴×C₇	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

6578 conjugacy classes of subgroups; only the 6270 normal subgroups are shown.

normal subgroups only

This group has 6578 conjugacy classes of subgroups — too many to draw. Only its 6270 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 — 6 generators on 47 points:

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

Conjugacy classes & character table

574 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 55837);