C₂³×#1874 #72279432

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂³×#1874

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂×D₄
commutator	C₂×C₄	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

103909 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:

(1,2)(3,4)(5,6)(11,12)(13,14);(7,8,9,10)(16,19)(17,21)(20,22);(7,8)(9,10)(11,13,14,12)(15,16)(17,22)(18,20)(19,21);(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, #72306228, #72306232, …
maximal normal subgroup of	C₂³×#21452, #72306228, #72306232, …
maximal quotient of	C₂³×#13001

GAP

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

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

C23×#1874 #72279432