C₂×#128714 #72320587

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#128714

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×#102
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	#8202
prime radical	C₂²×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₃₂ ◁ C₂×C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1649 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224954, C₂×#72224963, C₂×#72224959
maximal normal subgroup of	C₂×#72224954, C₂×#72224963, C₂×#72224959

GAP

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

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

C2×#128714 #72320587