C₂²×C₇×#458 #13237609

order	1792
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₇ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×C₇×#458

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂²×C₇×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂⁵×C₇ ◁ G

5294 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.

normal subgroups only

This group has 5294 conjugacy classes of subgroups — too many to draw. Only its 1134 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) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 5 generators on 27 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 53381);