C₇×#33564 #13210780

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇×#33564

Subgroups

	subgroup	G/H
center	C₄×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₇
socle	C₂×C₇	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

1112 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 218 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)(10,20)(11,17)(12,25)(13,16)(14,19)(15,21)(18,29)(22,36)(23,31)(24,34)(26,33)(27,35)(28,37)(30,38)(32,39);(8,10)(9,15)(11,21)(12,26)(13,28)(14,23)(16,30)(17,31)(18,32)(19,33)(20,25)(22,29)(24,39)(27,38)(34,37)(35,36);(9,16)(13,29)(15,30)(17,34)(18,19)(20,36)(22,28)(24,27)(25,35)(31,37)(32,33)(38,39);(8,11,14,12)(9,17,19,25)(10,21,23,26)(13,24,29,27)(15,31,33,20)(16,34,18,35)(22,38,28,39)(30,37,32,36)

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, 26552);