C₂³×#1864 #72279372

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂³×#1864

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂×D₄
commutator	C₂×C₄	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

71349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#21452, #72306144, #72306155
maximal normal subgroup of	C₂³×#21452, #72306144, #72306155
maximal quotient of	C₂³×#12956

GAP

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

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

C23×#1864 #72279372