D₁₂₀₂₅ #37562549

order	24050
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₀₂₅
unambiguous semidirect product	C₁₃⋊D₉₂₅
unambiguous semidirect product	C₂₅⋊D₄₈₁
unambiguous semidirect product	C₃₇⋊D₃₂₅

Splittings

Fitting, derived, Hall	C₁₃×C₂₅×C₃₇ ⋊ C₂
Hall	C₂₅×C₃₇ ⋊ D₁₃
Hall	C₁₃×C₃₇ ⋊ D₂₅
Hall	C₁₃×C₂₅ ⋊ D₃₇
Sylow	C₃₇ ⋊ D₃₂₅
Sylow	C₂₅ ⋊ D₄₈₁
Sylow	C₁₃ ⋊ D₉₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₂₅×C₃₇	C₂
Frattini	C₅	D₂₄₀₅
socle	C₅×C₁₃×C₃₇	D₅
LCS	G ▷ C₁₃×C₂₅×C₃₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 12025

A minimal faithful permutation representation — 2 generators on 75 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2,5,13,37,5|b^5=e^4,b^a=b^4e,c^a=c^12,d^a=d^36,e^a=e^4

Conjugacy classes & character table

6014 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24050, 8);

D12025 #37562549