Dic₅₇₅₇ #37527792

order	23028
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₇₅₇
unambiguous semidirect product	C₃⋊Dic₁₉₁₉
unambiguous semidirect product	C₁₉⋊Dic₃₀₃

Splittings

derived, Hall	C₃×C₁₉×C₁₀₁ ⋊ C₄
Hall	C₁₉×C₁₀₁ ⋊ Dic₃
Hall	C₃×C₁₀₁ ⋊ Dic₁₉
Sylow	C₁₀₁ ⋊ Dic₅₇
Hall	C₃×C₁₉ ⋊ Dic₁₀₁
Sylow	C₁₉ ⋊ Dic₃₀₃
Sylow	C₃ ⋊ Dic₁₉₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇₅₇
commutator, hypocenter	C₃×C₁₉×C₁₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₉×C₁₀₁	C₂
LCS	G ▷ C₃×C₁₉×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 23028

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)(6,7)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2,3,19,101,2|a^2=e,b^a=b^2,c^a=c^18,d^a=d^100

Conjugacy classes & character table

5760 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₅₁₄
maximal normal subgroup of	Dic₁₁₅₁₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23028, 9);

Dic5757 #37527792