D₁₂₀₃₃ #37562673

order	24066
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₀₃₃
unambiguous semidirect product	C₇⋊D₁₇₁₉
unambiguous semidirect product	C₉⋊D₁₃₃₇
unambiguous semidirect product	C₁₉₁⋊D₆₃

Splittings

Fitting, derived, Hall	C₇×C₉×C₁₉₁ ⋊ C₂
Hall	C₉×C₁₉₁ ⋊ D₇
Hall	C₇×C₁₉₁ ⋊ D₉
Sylow	C₁₉₁ ⋊ D₆₃
Hall	C₇×C₉ ⋊ D₁₉₁
Sylow	C₉ ⋊ D₁₃₃₇
Sylow	C₇ ⋊ D₁₇₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₁₉₁	C₂
Frattini	C₃	D₄₀₁₁
socle	C₃×C₇×C₁₉₁	D₃
LCS	G ▷ C₇×C₉×C₁₉₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 207
transitive degree	μ_t(G) = 12033

A minimal faithful permutation representation — 2 generators on 207 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207);(2,9)(3,8)(4,7)(5,6)(11,16)(12,15)(13,14)(18,207)(19,206)(20,205)(21,204)(22,203)(23,202)(24,201)(25,200)(26,199)(27,198)(28,197)(29,196)(30,195)(31,194)(32,193)(33,192)(34,191)(35,190)(36,189)(37,188)(38,187)(39,186)(40,185)(41,184)(42,183)(43,182)(44,181)(45,180)(46,179)(47,178)(48,177)(49,176)(50,175)(51,174)(52,173)(53,172)(54,171)(55,170)(56,169)(57,168)(58,167)(59,166)(60,165)(61,164)(62,163)(63,162)(64,161)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,144)(82,143)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)

Polycyclic presentation:

2,3,7,191,3|b^3=e^2,b^a=b^2e,c^a=c^6,d^a=d^190,e^a=e^2

Conjugacy classes & character table

6018 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24066, 9);

D12033 #37562673