Dic₄₀₅₃ #30633849

order	16212
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₅₃
unambiguous semidirect product	C₃⋊Dic₁₃₅₁
unambiguous semidirect product	C₇⋊Dic₅₇₉

Splittings

derived, Hall	C₃×C₇×C₁₉₃ ⋊ C₄
Hall	C₇×C₁₉₃ ⋊ Dic₃
Hall	C₃×C₁₉₃ ⋊ Dic₇
Sylow	C₁₉₃ ⋊ Dic₂₁
Hall	C₃×C₇ ⋊ Dic₁₉₃
Sylow	C₇ ⋊ Dic₅₇₉
Sylow	C₃ ⋊ Dic₁₃₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₀₅₃
commutator, hypocenter	C₃×C₇×C₁₉₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₇×C₁₉₃	C₂
LCS	G ▷ C₃×C₇×C₁₉₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 207
transitive degree	μ_t(G) = 16212

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

Polycyclic presentation:

2,3,7,193,2|a^2=e,b^a=b^2,c^a=c^6,d^a=d^192

Conjugacy classes & character table

4056 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₁₀₆
maximal normal subgroup of	Dic₈₁₀₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16212, 12);

Dic4053 #30633849