C₂×#6343786 #72238264

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#6343786

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂⁵×C₄
Frattini, cosocle	C₄²	C₂⁶
socle	C₂³	C₂³×C₂²⋊C₄
prime radical	C₂²×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

28665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 34 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72209982, C₂×#72209983
maximal normal subgroup of	C₂×#72209982, C₂×#72209983

GAP

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

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

C2×#6343786 #72238264