C₂⁴×#3862 #72246052

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₇

Aliases

Fitting, socle, Hall	C₂⁴×C₇² ⋊ C₃
derived, Sylow	C₇² ⋊ C₂⁴×C₃

	subgroup	G/H
center, hypercenter	C₂⁴	#3862
commutator, cosocle, hypocenter	C₇²	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₇²	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂⁴

670 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 4 generators on 22 points:

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

304 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((7,8)(10,11,12)(13,15,14)(17,19,22)(18,21,20), (5,6)(9,10,12,14,11,13,15)(16,17,19,21,22,18,20), (3,4), (1,2));