D₇₅₂₅ #28418769

order	15050
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₅₂₅
unambiguous semidirect product	C₇⋊D₁₀₇₅
unambiguous semidirect product	C₂₅⋊D₃₀₁
unambiguous semidirect product	C₄₃⋊D₁₇₅

Splittings

Fitting, derived, Hall	C₇×C₂₅×C₄₃ ⋊ C₂
Hall	C₂₅×C₄₃ ⋊ D₇
Hall	C₇×C₄₃ ⋊ D₂₅
Hall	C₇×C₂₅ ⋊ D₄₃
Sylow	C₄₃ ⋊ D₁₇₅
Sylow	C₂₅ ⋊ D₃₀₁
Sylow	C₇ ⋊ D₁₀₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₅×C₄₃	C₂
Frattini	C₅	D₁₅₀₅
socle	C₅×C₇×C₄₃	D₅
LCS	G ▷ C₇×C₂₅×C₄₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 7525

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)(27,32)(28,31)(29,30)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,5,7,43,5|b^5=e^4,b^a=b^4e,c^a=c^6,d^a=d^42,e^a=e^4

Conjugacy classes & character table

3764 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₃×D₁₇₅, D₇×D₁₀₇₅, D₂₅×D₃₀₁, …
maximal normal subgroup of	D₄₃×D₁₇₅, D₇×D₁₀₇₅, D₂₅×D₃₀₁, …
maximal quotient of	Dic₇₅₂₅, C₂×D₇₅₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15050, 9);

D7525 #28418769