C₇×#18997 #13196213

order	1792
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#18997

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×H(3,𝔽₄)	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 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 — 4 generators on 39 points:

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

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, 11985);