C₂²×#36163 #72335693

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#36163

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

108345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7708420, C₂²×#7630949, C₂²×#7670207
maximal normal subgroup of	C₂²×#7708420, C₂²×#7630949, C₂²×#7670207
maximal quotient of	C₂²×#612912, C₂²×#519290

GAP

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

gap> G := Group((17,18,21,19)(20,22,24,23), (5,6,7,8)(10,13)(11,15)(18,19)(20,24), (5,6,7,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,22,24,23), (3,4), (5,6)(7,8)(17,20)(18,22)(19,23)(21,24), (1,2)(3,4), (9,11)(10,14)(12,15)(13,16));

C22×#36163 #72335693