C₂×#346283 #72340955

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

direct product

C₂×#346283

Subgroups

	subgroup	G/H
center, socle	C₂³	#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#127	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5073 conjugacy classes of subgroups; only the 501 normal subgroups are shown.

normal subgroups only

This group has 5073 conjugacy classes of subgroups — too many to draw. Only its 501 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72227559
maximal subgroup of	C₂×#72227559, #72341114
maximal normal subgroup of	C₂×#72227559, #72341114

GAP

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

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