C₂²×#7626568 #72325876

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₄ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7626568

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1579008 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232072, C₂²×#72232065
maximal normal subgroup of	C₂²×#72232072, C₂²×#72232065

GAP

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

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

C22×#7626568 #72325876