C₂×#152136 #72256452

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#152136

Subgroups

	subgroup	G/H
center, socle	C₂²	#12707
commutator	#117	C₂³×C₄
Frattini, cosocle	#535	C₂⁴
prime radical	C₂×#61643	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#3484 ◁ G
LECS	G ▷ #535 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4921 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);(5,13)(10,26)(14,27)(16,28)(17,18)(19,34)(29,33)(31,32);(3,4)(5,14,17,29)(6,19,20,34)(7,22,15,11)(8,12)(9,16,25,32)(10,21)(13,27,18,33)(23,28,24,31)(26,30);(3,5,15,17,8,13,7,18)(4,9,11,25,12,23,22,24)(6,16,30,31,20,28,21,32)(10,14,19,29,26,27,34,33)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216447
maximal normal subgroup of	C₂×#72216447

GAP

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

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

C2×#152136 #72256452