Dic₇₇₆₁ #44526798

order	31044
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₇₆₁
unambiguous semidirect product	C₃⋊Dic₂₅₈₇
unambiguous semidirect product	C₁₉₉⋊Dic₃₉

Splittings

derived, Hall	C₃×C₁₃×C₁₉₉ ⋊ C₄
Hall	C₁₃×C₁₉₉ ⋊ Dic₃
Hall	C₃×C₁₉₉ ⋊ Dic₁₃
Sylow	C₁₉₉ ⋊ Dic₃₉
Hall	C₃×C₁₃ ⋊ Dic₁₉₉
Sylow	C₁₃ ⋊ Dic₅₉₇
Sylow	C₃ ⋊ Dic₂₅₈₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₇₆₁
commutator, hypocenter	C₃×C₁₃×C₁₉₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₃×C₁₉₉	C₂
LCS	G ▷ C₃×C₁₃×C₁₉₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 219
transitive degree	μ_t(G) = 31044

A minimal faithful permutation representation — 2 generators on 219 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219);(1,3,2,4)(6,7)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,219)(23,218)(24,217)(25,216)(26,215)(27,214)(28,213)(29,212)(30,211)(31,210)(32,209)(33,208)(34,207)(35,206)(36,205)(37,204)(38,203)(39,202)(40,201)(41,200)(42,199)(43,198)(44,197)(45,196)(46,195)(47,194)(48,193)(49,192)(50,191)(51,190)(52,189)(53,188)(54,187)(55,186)(56,185)(57,184)(58,183)(59,182)(60,181)(61,180)(62,179)(63,178)(64,177)(65,176)(66,175)(67,174)(68,173)(69,172)(70,171)(71,170)(72,169)(73,168)(74,167)(75,166)(76,165)(77,164)(78,163)(79,162)(80,161)(81,160)(82,159)(83,158)(84,157)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)

Polycyclic presentation:

2,3,13,199,2|a^2=e,b^a=b^2,c^a=c^12,d^a=d^198

Conjugacy classes & character table

7764 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₅₂₂
maximal normal subgroup of	Dic₁₅₅₂₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31044, 12);

Dic7761 #44526798