D₁₀₆₇₅ #36319444

order	21350
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₆₇₅
unambiguous semidirect product	C₇⋊D₁₅₂₅
unambiguous semidirect product	C₂₅⋊D₄₂₇
unambiguous semidirect product	C₆₁⋊D₁₇₅

Splittings

Fitting, derived, Hall	C₇×C₂₅×C₆₁ ⋊ C₂
Hall	C₂₅×C₆₁ ⋊ D₇
Hall	C₇×C₆₁ ⋊ D₂₅
Hall	C₇×C₂₅ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₁₇₅
Sylow	C₂₅ ⋊ D₄₂₇
Sylow	C₇ ⋊ D₁₅₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₅×C₆₁	C₂
Frattini	C₅	D₂₁₃₅
socle	C₅×C₇×C₆₁	D₅
LCS	G ▷ C₇×C₂₅×C₆₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 10675

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

Polycyclic presentation:

2,5,7,61,5|b^5=e^4,b^a=b^4e,c^a=c^6,d^a=d^60,e^a=e^4

Conjugacy classes & character table

5339 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₂₀₂₅
maximal quotient of	D₃₂₀₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21350, 9);

D10675 #36319444