Dic₁₃₅₇₅ #72289697

order	54300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₇₅

Splittings

derived, Hall	C₃×C₂₅×C₁₈₁ ⋊ C₄
Hall	C₂₅×C₁₈₁ ⋊ Dic₃
Hall	C₃×C₁₈₁ ⋊ Dic₂₅
Sylow	C₁₈₁ ⋊ Dic₇₅
Hall	C₃×C₂₅ ⋊ Dic₁₈₁
Sylow	C₂₅ ⋊ Dic₅₄₃
Sylow	C₃ ⋊ Dic₄₅₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₅₇₅
commutator, hypocenter	C₃×C₂₅×C₁₈₁	C₄
Frattini	C₂×C₅	D₂₇₁₅
Fitting, cosocle	C₂×C₃×C₂₅×C₁₈₁	C₂
socle, prime radical	C₂×C₃×C₅×C₁₈₁	D₅
LCS	G ▷ C₃×C₂₅×C₁₈₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 213
transitive degree	μ_t(G) = 54300

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

Polycyclic presentation:

2^2,3,5,181,5|a^2=b,d^5=f,c^a=c^2,d^a=d^4f^4,e^a=e^180,f^a=f^4

Conjugacy classes & character table

13578 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, 181, 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.5^180);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13575 #72289697