C₂×#141837 #72258682

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

direct product

C₂×#141837

Subgroups

	subgroup	G/H
center, socle	C₂³	#1957
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 34 points:

(3,4)(5,12)(6,8)(7,10)(9,15)(11,14)(13,16)(17,18)(20,24)(21,28)(22,29)(25,27)(26,33)(30,34)(31,32);(4,8,10,11)(5,13)(9,12,17,16)(15,18)(19,20,21,25,29,31,34,32,23,27,30,24,22,26,28,33);(1,2);(3,5,14,18,7,13,6,15)(4,9,8,16,10,17,11,12)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213975, C₂×#72213965, C₂×#72214020
maximal normal subgroup of	C₂×#72213975, C₂×#72213965, C₂×#72214020

GAP

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

gap> G := Group((3,4)(5,12)(6,8)(7,10)(9,15)(11,14)(13,16)(17,18)(20,24)(21,28)(22,29)(25,27)(26,33)(30,34)(31,32), (4,8,10,11)(5,13)(9,12,17,16)(15,18)(19,20,21,25,29,31,34,32,23,27,30,24,22,26,28,33), (1,2), (3,5,14,18,7,13,6,15)(4,9,8,16,10,17,11,12));

C2×#141837 #72258682