C₂×#153148 #72283345

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#153148

Subgroups

	subgroup	G/H
center, socle	C₂²	#13346
commutator, Frattini, cosocle	#532	C₂⁴
prime radical	C₂×#33546	C₂
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3647 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);(3,4,9,13)(5,10,18,17)(6,19)(7,8,24,12)(11,27)(14,29)(15,33,26,16)(20,34)(21,22)(23,25)(28,31)(30,32);(3,5,9,18)(4,10,13,17)(6,20,23,32)(7,16,24,33)(8,15,12,26)(11,28,22,14)(19,31,25,29)(21,34,27,30);(3,6)(4,11)(5,14)(7,21)(8,25)(9,23)(10,20)(12,19)(13,22)(15,30)(16,29)(17,32)(18,28)(24,27)(26,34)(31,33)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231252
maximal normal subgroup of	C₂×#72231252

GAP

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

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

C2×#153148 #72283345