Dic₅₄₅₁ #36345514

order	21804
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₄₅₁
unambiguous semidirect product	C₃⋊Dic₁₈₁₇
unambiguous semidirect product	C₂₃⋊Dic₂₃₇
unambiguous semidirect product	C₇₉⋊Dic₆₉

Splittings

derived, Hall	C₃×C₂₃×C₇₉ ⋊ C₄
Hall	C₂₃×C₇₉ ⋊ Dic₃
Hall	C₃×C₇₉ ⋊ Dic₂₃
Sylow	C₇₉ ⋊ Dic₆₉
Hall	C₃×C₂₃ ⋊ Dic₇₉
Sylow	C₂₃ ⋊ Dic₂₃₇
Sylow	C₃ ⋊ Dic₁₈₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₄₅₁
commutator, hypocenter	C₃×C₂₃×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₂₃×C₇₉	C₂
LCS	G ▷ C₃×C₂₃×C₇₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 21804

A minimal faithful permutation representation — 2 generators on 109 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);(1,3,2,4)(6,7)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2,3,23,79,2|a^2=e,b^a=b^2,c^a=c^22,d^a=d^78

Conjugacy classes & character table

5454 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₉₀₂, Dic₁₆₃₅₃
maximal normal subgroup of	Dic₁₀₉₀₂
maximal quotient of	Dic₁₆₃₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21804, 9);

Dic5451 #36345514