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C23×AGL(1,ℤ/25) #72275596

order4000
automorphism?
homologyAbelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26

Aliases

direct productC23×AGL(1,ℤ/25)

Splittings

FittingC23×5−1+2 ⋊ C4
derivedC25 ⋊ C23×C4×C5
Sylow5−1+2 ⋊ C23×C4

Subgroups

subgroupG/H
center, hypercenterC23AGL(1,ℤ/25)
commutator, hypocenterC25C23×C4×C5
FrattiniC5C23×C5×AGL(1,𝔽5)
FittingC23×5−1+2C4
socleC23×C5C5×AGL(1,𝔽5)
prime radicalC23×D25⋊C5C2
cosocleD25C24×C5
LCSG ▷ C25
UCSC1 ◁ C23

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 200

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×AGL(1,ℤ/25), C24×AGL(1,ℤ/25), C22⋊C4×AGL(1,ℤ/25), …
maximal normal subgroup ofC2×D4×AGL(1,ℤ/25), C24×AGL(1,ℤ/25), C22⋊C4×AGL(1,ℤ/25), …
maximal quotient ofC2×D4×AGL(1,ℤ/25), C24×AGL(1,ℤ/25), #41×AGL(1,ℤ/25), …

GAP

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

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