C₂²×#7849737 #72325874

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

Aliases

direct product

C₂²×#7849737

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 16758 conjugacy classes of subgroups; only the 445520 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:

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

Conjugacy classes & character table

404 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((9,10)(11,13)(12,14)(15,16)(18,21)(22,23), (10,11)(14,15)(17,18)(19,22)(20,21)(23,24), (10,11)(12,16), (3,4), (6,8)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24), (1,2)(3,4), (9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,24)(21,23), (5,6)(7,8)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,24)(21,23));

C22×#7849737 #72325874