Dic₃₈₁₁ #29503791

order	15244
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₈₁₁
unambiguous semidirect product	C₁₀₃⋊Dic₃₇

Splittings

derived, Hall	C₃₇×C₁₀₃ ⋊ C₄
Sylow	C₁₀₃ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₁₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₈₁₁
commutator, hypocenter	C₃₇×C₁₀₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₇×C₁₀₃	C₂
LCS	G ▷ C₃₇×C₁₀₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 15244

A minimal faithful permutation representation — 2 generators on 144 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);(1,3,2,4)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)

Polycyclic presentation:

2,37,103,2|a^2=d,b^a=b^36,c^a=c^102

Conjugacy classes & character table

3814 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₆₂₂
maximal normal subgroup of	Dic₇₆₂₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15244, 4);

Dic3811 #29503791