Dic₄₈₃₃ #72216331

order	19332
automorphism	Aut(G) order 30969864
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₈₃₃

Splittings

derived, Hall	C₂₇×C₁₇₉ ⋊ C₄
Sylow	C₁₇₉ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₁₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₈₃₃
commutator, hypocenter	C₂₇×C₁₇₉	C₄
Frattini	C₂×C₉	D₅₃₇
Fitting, cosocle	C₂×C₂₇×C₁₇₉	C₂
socle, prime radical	C₂×C₃×C₁₇₉	D₉
LCS	G ▷ C₂₇×C₁₇₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 210
transitive degree	μ_t(G) = 19332

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

Polycyclic presentation:

2^2,3,179,3^2|a^2=b,c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^178,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

4836 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₆₆₆, Dic₁₄₄₉₉
maximal normal subgroup of	Dic₉₆₆₆
maximal quotient of	Dic₁₄₄₉₉

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, 179, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^178);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4833 #72216331