Dic₂₁₄₈ #72212482

order	8592
automorphism	Aut(G) order 6117504
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₁₄₈

Splittings

Hall	C₃×C₁₇₉ ⋊ Q₁₆
Sylow	C₁₇₉ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₇₁₆

Subgroups

	subgroup	G/H
center	C₂	D₂₁₄₈
commutator, cosocle	C₃×C₄×C₁₇₉	C₂²
Frattini	C₄	C₂×D₅₃₇
Fitting	C₃×C₈×C₁₇₉	C₂
socle, prime radical	C₂×C₃×C₁₇₉	D₄
hypercenter	C₈	D₅₃₇
hypocenter	C₃×C₁₇₉	Q₁₆
LCS	G ▷ C₃×C₄×C₁₇₉ ▷ C₂×C₃×C₁₇₉ ▷ C₃×C₁₇₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 198
transitive degree	μ_t(G) = 8592

A minimal faithful permutation representation — 2 generators on 198 points:

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

Polycyclic presentation:

2^4,3,179|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^178

Conjugacy classes & character table

2151 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₂₉₆, Dic₆₄₄₄, Dic₁₀₇₄₀, …
maximal normal subgroup of	Dic₄₂₉₆
maximal quotient of	Dic₆₄₄₄, Dic₁₀₇₄₀, Dic₁₅₀₃₆

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

Dic2148 #72212482