C₂×#181727 #72239954

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂×#181727

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂²×#95
prime radical	C₂⁶	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

11505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239957, #72239966, C₂×#72210000, …
maximal normal subgroup of	#72239957, #72239966, C₂×#72210000, …

GAP

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

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

C2×#181727 #72239954