Dic₂₈₉₅ #26040683

order	11580
automorphism	Aut(G) order 8893440
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₈₉₅
unambiguous semidirect product	C₃⋊Dic₉₆₅

Splittings

derived, Hall	C₃×C₅×C₁₉₃ ⋊ C₄
Hall	C₅×C₁₉₃ ⋊ Dic₃
Hall	C₃×C₁₉₃ ⋊ Dic₅
Sylow	C₁₉₃ ⋊ Dic₁₅
Hall	C₃×C₅ ⋊ Dic₁₉₃
Sylow	C₅ ⋊ Dic₅₇₉
Sylow	C₃ ⋊ Dic₉₆₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₈₉₅
commutator, hypocenter	C₃×C₅×C₁₉₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₉₃	C₂
LCS	G ▷ C₃×C₅×C₁₉₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 205
transitive degree	μ_t(G) = 11580

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

Polycyclic presentation:

2,3,5,193,2|a^2=e,b^a=b^2,c^a=c^4,d^a=d^192

Conjugacy classes & character table

2898 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₆₈₅, C₃⋊(Dic₉₆₅⋊C₃), C₃⋊Dic₂₈₉₅, …
maximal normal subgroup of	C₃⋊(Dic₉₆₅⋊C₃), C₃×Dic₂₈₉₅, Dic₅₇₉₀
maximal quotient of	Dic₈₆₈₅, C₃⋊Dic₂₈₉₅, C₃×Dic₂₈₉₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11580, 9);

Dic2895 #26040683