C₂²×#7631422 #72325879

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

Aliases

direct product

C₂²×#7631422

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

1482464 conjugacy classes of subgroups.

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232072
maximal normal subgroup of	C₂²×#72232072

GAP

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

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

C22×#7631422 #72325879