D₁₁₂₄₇ #36376541

order	22494
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₂₄₇
unambiguous semidirect product	C₃⋊D₃₇₄₉
unambiguous semidirect product	C₂₃⋊D₄₈₉
unambiguous semidirect product	C₁₆₃⋊D₆₉

Splittings

Fitting, socle, derived, Hall	C₃×C₂₃×C₁₆₃ ⋊ C₂
Hall	C₂₃×C₁₆₃ ⋊ D₃
Hall	C₃×C₁₆₃ ⋊ D₂₃
Sylow	C₁₆₃ ⋊ D₆₉
Hall	C₃×C₂₃ ⋊ D₁₆₃
Sylow	C₂₃ ⋊ D₄₈₉
Sylow	C₃ ⋊ D₃₇₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₂₃×C₁₆₃	C₂
LCS	G ▷ C₃×C₂₃×C₁₆₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 11247

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

Polycyclic presentation:

2,3,23,163|b^a=b^2,c^a=c^22,d^a=d^162

Conjugacy classes & character table

5625 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22494, 9);

D11247 #36376541