Dic₅₉₂₁ #37557013

order	23684
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₉₂₁
unambiguous semidirect product	C₃₁⋊Dic₁₉₁
unambiguous semidirect product	C₁₉₁⋊Dic₃₁

Splittings

derived, Hall	C₃₁×C₁₉₁ ⋊ C₄
Sylow	C₁₉₁ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₉₂₁
commutator, hypocenter	C₃₁×C₁₉₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₁×C₁₉₁	C₂
LCS	G ▷ C₃₁×C₁₉₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 226
transitive degree	μ_t(G) = 23684

A minimal faithful permutation representation — 2 generators on 226 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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226);(1,3,2,4)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(37,226)(38,225)(39,224)(40,223)(41,222)(42,221)(43,220)(44,219)(45,218)(46,217)(47,216)(48,215)(49,214)(50,213)(51,212)(52,211)(53,210)(54,209)(55,208)(56,207)(57,206)(58,205)(59,204)(60,203)(61,202)(62,201)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,168)(96,167)(97,166)(98,165)(99,164)(100,163)(101,162)(102,161)(103,160)(104,159)(105,158)(106,157)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)

Polycyclic presentation:

2,31,191,2|a^2=d,b^a=b^30,c^a=c^190

Conjugacy classes & character table

5924 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₈₄₂
maximal normal subgroup of	Dic₁₁₈₄₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23684, 4);

Dic5921 #37557013