C₇×#12268 #13189484

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12148
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12268

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₂⁵×C₇	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₂²⋊C₄ ◁ G

458 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 70 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)(8,9,14,19)(10,20,22,30)(11,16,25,35)(12,27,26,17)(13,18,29,34)(15,28,32,21)(23,36,37,33)(24,31,38,39);(8,10,13,23)(9,15,18,31)(11,24,26,21)(12,28,25,38)(14,22,29,37)(16,33,17,30)(19,32,34,39)(20,35,36,27);(8,11,14,25)(9,16,34,27)(10,21,22,28)(12,13,26,29)(15,30,39,36)(17,19,35,18)(20,31,33,32)(23,24,37,38)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5256);