C₂×#512045 #72307282

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#512045

Subgroups

	subgroup	G/H
center	C₂×C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁵
socle	C₂²	#32976
prime radical	C₂×#3483	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

4996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307300, #72307386, C₂×#72231593, …
maximal normal subgroup of	#72307300, #72307386, C₂×#72231593, …

GAP

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

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

C2×#512045 #72307282