C₇×#12221 #13189437

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12096
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12221

Subgroups

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

392 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,9,13,20)(10,22,25,39)(11,28,30,38)(12,32,23,18)(14,34,15,21)(16,31,37,24)(17,33,36,29)(19,26,35,27);(8,10,14,26)(9,16,34,28)(11,23,24,33)(12,31,29,30)(13,25,15,27)(17,35,32,22)(18,39,36,19)(20,37,21,38);(8,11)(9,17)(10,23)(12,25)(13,30)(14,24)(15,31)(16,35)(18,21)(19,37)(20,36)(22,28)(26,33)(27,29)(32,34)(38,39)

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, 5209);