Dic₇₉₈₇ #44543972

order	31948
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₉₈₇
unambiguous semidirect product	C₄₉⋊Dic₁₆₃
unambiguous semidirect product	C₁₆₃⋊Dic₄₉

Splittings

derived, Hall	C₄₉×C₁₆₃ ⋊ C₄
Sylow	C₁₆₃ ⋊ Dic₄₉
Sylow	C₄₉ ⋊ Dic₁₆₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₉₈₇
commutator, hypocenter	C₄₉×C₁₆₃	C₄
Frattini	C₂×C₇	D₁₁₄₁
Fitting, cosocle	C₂×C₄₉×C₁₆₃	C₂
socle, prime radical	C₂×C₇×C₁₆₃	D₇
LCS	G ▷ C₄₉×C₁₆₃
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 216
transitive degree	μ_t(G) = 31948

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

Polycyclic presentation:

2,7,163,2,7|a^2=d,b^7=e^6,b^a=b^6e,c^a=c^162,e^a=e^6

Conjugacy classes & character table

7990 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₉₇₄
maximal normal subgroup of	Dic₁₅₉₇₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31948, 4);

Dic7987 #44543972