Dic₄₁₈₇ #30660446

order	16748
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₁₈₇
unambiguous semidirect product	C₇₉⋊Dic₅₃

Splittings

derived, Hall	C₅₃×C₇₉ ⋊ C₄
Sylow	C₇₉ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₁₈₇
commutator, hypocenter	C₅₃×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅₃×C₇₉	C₂
LCS	G ▷ C₅₃×C₇₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 16748

A minimal faithful permutation representation — 2 generators on 136 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);(1,3,2,4)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)

Polycyclic presentation:

2,53,79,2|a^2=d,b^a=b^52,c^a=c^78

Conjugacy classes & character table

4190 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₃₇₄, Dic₁₂₅₆₁
maximal normal subgroup of	Dic₈₃₇₄
maximal quotient of	Dic₁₂₅₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16748, 4);

Dic4187 #30660446