C₂×#139749 #72268265

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#139749

Subgroups

	subgroup	G/H
center, socle	C₂²	#9495
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂×#3484	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233445, C₂×#72233461, C₂×#72233454
maximal normal subgroup of	C₂×#72233445, C₂×#72233461, C₂×#72233454

GAP

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

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

C2×#139749 #72268265