Dic₁₄₃₂₅ #72290932

order	57300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₂₅

Splittings

derived, Hall	C₃×C₂₅×C₁₉₁ ⋊ C₄
Hall	C₂₅×C₁₉₁ ⋊ Dic₃
Hall	C₃×C₁₉₁ ⋊ Dic₂₅
Sylow	C₁₉₁ ⋊ Dic₇₅
Hall	C₃×C₂₅ ⋊ Dic₁₉₁
Sylow	C₂₅ ⋊ Dic₅₇₃
Sylow	C₃ ⋊ Dic₄₇₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₃₂₅
commutator, hypocenter	C₃×C₂₅×C₁₉₁	C₄
Frattini	C₂×C₅	D₂₈₆₅
Fitting, cosocle	C₂×C₃×C₂₅×C₁₉₁	C₂
socle, prime radical	C₂×C₃×C₅×C₁₉₁	D₅
LCS	G ▷ C₃×C₂₅×C₁₉₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 223
transitive degree	μ_t(G) = 57300

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

Polycyclic presentation:

2^2,3,5,191,5|a^2=b,d^5=f,c^a=c^2,d^a=d^4f^4,e^a=e^190,f^a=f^4

Conjugacy classes & character table

14328 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 3, 5, 191, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4);;
gap> SetConjugate(coll, 5, 1, F.5^190);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14325 #72290932