Dic₉₃₇₃ #49154180

order	37492
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₃₇₃
unambiguous semidirect product	C₇⋊Dic₁₃₃₉
unambiguous semidirect product	C₁₀₃⋊Dic₉₁

Splittings

derived, Hall	C₇×C₁₃×C₁₀₃ ⋊ C₄
Hall	C₁₃×C₁₀₃ ⋊ Dic₇
Hall	C₇×C₁₀₃ ⋊ Dic₁₃
Sylow	C₁₀₃ ⋊ Dic₉₁
Hall	C₇×C₁₃ ⋊ Dic₁₀₃
Sylow	C₁₃ ⋊ Dic₇₂₁
Sylow	C₇ ⋊ Dic₁₃₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₃₇₃
commutator, hypocenter	C₇×C₁₃×C₁₀₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₃×C₁₀₃	C₂
LCS	G ▷ C₇×C₁₃×C₁₀₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 37492

A minimal faithful permutation representation — 2 generators on 127 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,127)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2,7,13,103,2|a^2=e,b^a=b^6,c^a=c^12,d^a=d^102

Conjugacy classes & character table

9376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37492, 8);

Dic9373 #49154180