C₂³×#34198 #72295996

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₄ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂³×#34198

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4190184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7670203, C₂²×#72279977
maximal normal subgroup of	C₂³×#7670203, C₂²×#72279977
maximal quotient of	C₂³×C₂²⋊C₄×#120

GAP

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

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

C23×#34198 #72295996