C₂²×#612617 #72290837

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂²×#612617

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁷×D₄	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

8332824 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72233691
maximal normal subgroup of	C₂²×#72233691

GAP

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

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

C22×#612617 #72290837