C₂²×#7626230 #72339117

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

Aliases

direct product

C₂²×#7626230

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁵×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1783768 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72236233
maximal normal subgroup of	C₂²×#72236233

GAP

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

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

C22×#7626230 #72339117