C₂×#130292 #72280258

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

Aliases

direct product

C₂×#130292

Subgroups

	subgroup	G/H
center, socle	C₂²	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
prime radical	C₂×#62283	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4198 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:

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

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280405, C₂×#72231624, C₂×#72231596, …
maximal normal subgroup of	#72280405, C₂×#72231624, C₂×#72231596, …

GAP

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

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

C2×#130292 #72280258