C₂²×#7670220 #72301040

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7670220

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6891 conjugacy classes of subgroups; only the 448488 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232033
maximal normal subgroup of	C₂²×#72232033
maximal quotient of	C₂²×#120×#121, C₂²×#72279977

GAP

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

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

C22×#7670220 #72301040