C₇×#12266 #13189482

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12148
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#12266

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₇×#2670	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₂²⋊C₄ ◁ G

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

normal subgroups only

This group has 592 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

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

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, 5254);