C₂×#133995 #72235786

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

Aliases

direct product

C₂×#133995

Subgroups

	subgroup	G/H
center, socle	C₂³	#1782
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	C₂³×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#2297 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

4313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235795
maximal normal subgroup of	#72235795
maximal quotient of	C₂×#72235853

GAP

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

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

C2×#133995 #72235786