C₂×#183694 #72321839

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#183694

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×C₄×D₄
prime radical	C₂⁵	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72318410, C₂×#72318357, C₂×#72318426
maximal normal subgroup of	C₂×#72318410, C₂×#72318357, C₂×#72318426

GAP

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

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

C2×#183694 #72321839