C₂⁴×#64423 #72260008

order	4704
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, hypercenter	C₂⁴	#64423
commutator, cosocle, hypocenter	C₇²	C₂⁵×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₇²	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂⁴

3394 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 5 generators on 22 points:

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

272 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)(16,17,21)(18,22,20), (5,6)(9,12,10,14,11,15,13), (3,4), (1,2), (17,19)(18,20)(21,22));