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C2×C4⋊C4×#120 #72302669

order1024
nilpotency class2
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C216

Aliases

direct productC2×C4⋊C4×#120

Subgroups

subgroupG/H
center, socleC25C25
commutatorC23C25×C4
Frattini, cosocleC24C26
prime radicalC23×#120C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

79085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

(1,2)(4,6)(7,8,9,10)(11,12)(13,14);(15,16,17,18);(4,6)(8,10);(7,8)(9,10);(3,4,5,6)(7,8,9,10)(11,13,12,14)(15,16)(17,18);(3,4)(5,6)(7,8)(9,10)

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×#72280204, C2×C4×D4×#120, C2×#72279803, …
maximal normal subgroup ofC2×#72280204, C2×C4×D4×#120, C2×#72279803, …
maximal quotient ofC2×#95×#120, C22×C4⋊C4×#120, C2×#72334309, …

GAP

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

gap> G := Group((1,2)(4,6)(7,8,9,10)(11,12)(13,14), (15,16,17,18), (4,6)(8,10), (7,8)(9,10), (3,4,5,6)(7,8,9,10)(11,13,12,14)(15,16)(17,18), (3,4)(5,6)(7,8)(9,10));