Finite Group Databasebeta

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#157×#453 #72227604

order2304
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C3

Aliases

direct product#157×#453

Splittings

FittingC32×#453 ⋊ C4
SylowC32 ⋊ C4×#453

Subgroups

subgroupG/H
centerC2#120×#157
commutatorC2×C32×C4C23×C4
FrattiniC2×C4C23×#157
FittingC32×#453C4
socleC2×C32C4×#120
prime radicalC3⋊D3×#453C2
cosocleC2×C4×C3⋊D3C24
hypercenter#453#157
hypocenterC32C4×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

2422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#453×PSU(3,𝔽2)
maximal normal subgroup of#453×PSU(3,𝔽2)

GAP

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

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