C₇²×B(4,𝔽₂) #72246133

order	3136
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇² · Schur multiplier (H₂) = C₂⁴×C₇

Aliases

	subgroup	G/H
center, socle	C₂×C₇²	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₇²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇² ◁ C₂³×C₇² ◁ G

990 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 270 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 392

A minimal faithful permutation representation — 3 generators on 22 points:

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

784 conjugacy classes. The character table is too large to display.

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,4,6)(3,5,7,8)(9,10,11,12,13,14,15), (1,3)(2,5)(4,7)(6,8)(16,17,18,19,20,21,22), (5,8));