D₁₂₂₅₇ #37589199

order	24514
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₅₇
unambiguous semidirect product	C₇⋊D₁₇₅₁
unambiguous semidirect product	C₁₇⋊D₇₂₁
unambiguous semidirect product	C₁₀₃⋊D₁₁₉

Splittings

Fitting, socle, derived, Hall	C₇×C₁₇×C₁₀₃ ⋊ C₂
Hall	C₁₇×C₁₀₃ ⋊ D₇
Hall	C₇×C₁₀₃ ⋊ D₁₇
Hall	C₇×C₁₇ ⋊ D₁₀₃
Sylow	C₁₀₃ ⋊ D₁₁₉
Sylow	C₁₇ ⋊ D₇₂₁
Sylow	C₇ ⋊ D₁₇₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₇×C₁₀₃	C₂
LCS	G ▷ C₇×C₁₇×C₁₀₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 12257

A minimal faithful permutation representation — 2 generators on 127 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(2,7)(3,6)(4,5)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(26,127)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2,7,17,103|b^a=b^6,c^a=c^16,d^a=d^102

Conjugacy classes & character table

6130 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24514, 9);

D12257 #37589199