Dic₂₉₈₅ #26045896

order	11940
automorphism	Aut(G) order 9456480
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₉₈₅
unambiguous semidirect product	C₃⋊Dic₉₉₅
unambiguous semidirect product	C₁₉₉⋊Dic₁₅

Splittings

derived, Hall	C₃×C₅×C₁₉₉ ⋊ C₄
Hall	C₅×C₁₉₉ ⋊ Dic₃
Hall	C₃×C₁₉₉ ⋊ Dic₅
Sylow	C₁₉₉ ⋊ Dic₁₅
Hall	C₃×C₅ ⋊ Dic₁₉₉
Sylow	C₅ ⋊ Dic₅₉₇
Sylow	C₃ ⋊ Dic₉₉₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₉₈₅
commutator, hypocenter	C₃×C₅×C₁₉₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₉₉	C₂
LCS	G ▷ C₃×C₅×C₁₉₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 211
transitive degree	μ_t(G) = 11940

A minimal faithful permutation representation — 2 generators on 211 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,211)(15,210)(16,209)(17,208)(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:

2,3,5,199,2|a^2=e,b^a=b^2,c^a=c^4,d^a=d^198

Conjugacy classes & character table

2988 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₉₅₅, C₃⋊((C₁₉₉⋊C₃)⋊Dic₅), C₃⋊Dic₂₉₈₅, …
maximal normal subgroup of	C₃⋊((C₁₉₉⋊C₃)⋊Dic₅), C₃×Dic₂₉₈₅, Dic₅₉₇₀
maximal quotient of	Dic₈₉₅₅, C₃⋊Dic₂₉₈₅, C₃×Dic₂₉₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11940, 9);

Dic2985 #26045896