D₁₂₀₉₉ #37564094

order	24198
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₀₉₉
unambiguous semidirect product	C₃⋊D₄₀₃₃
unambiguous semidirect product	C₃₇⋊D₃₂₇
unambiguous semidirect product	C₁₀₉⋊D₁₁₁

Splittings

Fitting, socle, derived, Hall	C₃×C₃₇×C₁₀₉ ⋊ C₂
Hall	C₃₇×C₁₀₉ ⋊ D₃
Hall	C₃×C₁₀₉ ⋊ D₃₇
Hall	C₃×C₃₇ ⋊ D₁₀₉
Sylow	C₁₀₉ ⋊ D₁₁₁
Sylow	C₃₇ ⋊ D₃₂₇
Sylow	C₃ ⋊ D₄₀₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₃₇×C₁₀₉	C₂
LCS	G ▷ C₃×C₃₇×C₁₀₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 149
transitive degree	μ_t(G) = 12099

A minimal faithful permutation representation — 2 generators on 149 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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149);(2,3)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)

Polycyclic presentation:

2,3,37,109|b^a=b^2,c^a=c^36,d^a=d^108

Conjugacy classes & character table

6051 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24198, 12);

D12099 #37564094