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C11×ΓL(1,𝔽25) #10587178

order528
isoclinismisoclinic class of Dic6
automorphismAut(G) = C24×C5×D3 (order 480) · Inn(G) = C2×D3 · Out(G) = C23×C5
homologyAbelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×ΓL(1,𝔽25)

Splittings

FittingC3×C8×C11 ⋊ C2
HallC3×C11 ⋊ M16
SylowC3 ⋊ C11×M16

Subgroups

subgroupG/H
centerC4×C11C2×D3
commutatorC2×C3C2×C4×C11
FrattiniC4C2×D3×C11
FittingC3×C8×C11C2
socleC2×C3×C11C2×C4
prime radicalC2×D3×C11C4
cosocleC3×C4C22×C11
hypercenterC8×C11D3
hypocenterC3C11×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C11 ◁ C8×C11

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 264

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×ΓL(1,𝔽25), #11760696, #11760702, …
maximal normal subgroup ofD11×ΓL(1,𝔽25), #11760696, #11760702, …
maximal quotient ofC11×Dic3⋊C8, C11×#815, C11×(C2×D3)⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(528, 48);