D₁₂₇₇₅ #38784372

order	25550
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₇₇₅
unambiguous semidirect product	C₇⋊D₁₈₂₅
unambiguous semidirect product	C₂₅⋊D₅₁₁
unambiguous semidirect product	C₇₃⋊D₁₇₅

Splittings

Fitting, derived, Hall	C₇×C₂₅×C₇₃ ⋊ C₂
Hall	C₂₅×C₇₃ ⋊ D₇
Hall	C₇×C₇₃ ⋊ D₂₅
Hall	C₇×C₂₅ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₇₅
Sylow	C₂₅ ⋊ D₅₁₁
Sylow	C₇ ⋊ D₁₈₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₅×C₇₃	C₂
Frattini	C₅	D₂₅₅₅
socle	C₅×C₇×C₇₃	D₅
LCS	G ▷ C₇×C₂₅×C₇₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 105
transitive degree	μ_t(G) = 12775

A minimal faithful permutation representation — 2 generators on 105 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(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)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2,5,7,73,5|b^5=e^4,b^a=b^4e,c^a=c^6,d^a=d^72,e^a=e^4

Conjugacy classes & character table

6389 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₂₇₇₅
maximal normal subgroup of	C₂×D₁₂₇₇₅
maximal quotient of	Dic₁₂₇₇₅, C₂×D₁₂₇₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25550, 8);

D12775 #38784372