D₇₂₆₇ #28409922

order	14534
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₂₆₇
unambiguous semidirect product	C₄₃⋊D₁₆₉
unambiguous semidirect product	C₁₆₉⋊D₄₃

Splittings

Fitting, derived, Hall	C₄₃×C₁₆₉ ⋊ C₂
Sylow	C₁₆₉ ⋊ D₄₃
Sylow	C₄₃ ⋊ D₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₃×C₁₆₉	C₂
Frattini	C₁₃	D₅₅₉
socle	C₁₃×C₄₃	D₁₃
LCS	G ▷ C₄₃×C₁₆₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 212
transitive degree	μ_t(G) = 7267

A minimal faithful permutation representation — 2 generators on 212 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)(170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212);(2,169)(3,168)(4,167)(5,166)(6,165)(7,164)(8,163)(9,162)(10,161)(11,160)(12,159)(13,158)(14,157)(15,156)(16,155)(17,154)(18,153)(19,152)(20,151)(21,150)(22,149)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(171,212)(172,211)(173,210)(174,209)(175,208)(176,207)(177,206)(178,205)(179,204)(180,203)(181,202)(182,201)(183,200)(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)(191,192)

Polycyclic presentation:

2,13,43,13|b^13=d^12,b^a=b^12d,c^a=c^42,d^a=d^12

Conjugacy classes & character table

3635 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₃×D₁₆₉, C₂×D₇₂₆₇
maximal normal subgroup of	D₄₃×D₁₆₉, C₂×D₇₂₆₇
maximal quotient of	Dic₇₂₆₇, C₂×D₇₂₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14534, 4);

D7267 #28409922