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C24×#382 #72325641

order1024
nilpotency class2
automorphism?
homologyAbelianization (H1) = C26×C4 · Schur multiplier (H2) = C220×C4

Aliases

direct productC24×#382

Subgroups

subgroupG/H
center, socle, prime radicalC27C23
commutatorC22C26×C4
Frattini, cosocleC23C27
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C27 ◁ G
LECSG ▷ C23 ▷ C1

184469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 1024

A minimal faithful permutation representation — 7 generators on 24 points:

(10,12)(13,14,16,15)(18,21)(19,23);(10,12)(13,14,16,15)(17,18)(19,22)(20,21)(23,24);(3,4);(5,6);(9,10)(11,12)(13,15,16,14)(17,19,20,23)(18,22,21,24);(7,8);(1,2)(3,4)(5,6)(7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC23×#15670, C24×#2535, C23×#15669, …
maximal normal subgroup ofC23×#15670, C24×#2535, C23×#15669, …

GAP

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

gap> G := Group((10,12)(13,14,16,15)(18,21)(19,23), (10,12)(13,14,16,15)(17,18)(19,22)(20,21)(23,24), (3,4), (5,6), (9,10)(11,12)(13,15,16,14)(17,19,20,23)(18,22,21,24), (7,8), (1,2)(3,4)(5,6)(7,8));