Dic₉₄₉₉ #49184368

order	37996
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₄₉₉
unambiguous semidirect product	C₇⋊Dic₁₃₅₇
unambiguous semidirect product	C₂₃⋊Dic₄₁₃
unambiguous semidirect product	C₅₉⋊Dic₁₆₁

Splittings

derived, Hall	C₇×C₂₃×C₅₉ ⋊ C₄
Hall	C₂₃×C₅₉ ⋊ Dic₇
Hall	C₇×C₅₉ ⋊ Dic₂₃
Hall	C₇×C₂₃ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₁₆₁
Sylow	C₂₃ ⋊ Dic₄₁₃
Sylow	C₇ ⋊ Dic₁₃₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₄₉₉
commutator, hypocenter	C₇×C₂₃×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₃×C₅₉	C₂
LCS	G ▷ C₇×C₂₃×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 37996

A minimal faithful permutation representation — 2 generators on 93 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2,7,23,59,2|a^2=e,b^a=b^6,c^a=c^22,d^a=d^58

Conjugacy classes & character table

9502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37996, 8);

Dic9499 #49184368