Dic₃₃₃₅ #27211103

order	13340
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₃₃₅
unambiguous semidirect product	C₂₃⋊Dic₁₄₅

Splittings

derived, Hall	C₅×C₂₃×C₂₉ ⋊ C₄
Hall	C₂₃×C₂₉ ⋊ Dic₅
Hall	C₅×C₂₉ ⋊ Dic₂₃
Hall	C₅×C₂₃ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ Dic₁₁₅
Sylow	C₂₃ ⋊ Dic₁₄₅
Sylow	C₅ ⋊ Dic₆₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₃₃₅
commutator, hypocenter	C₅×C₂₃×C₂₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₃×C₂₉	C₂
LCS	G ▷ C₅×C₂₃×C₂₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 13340

A minimal faithful permutation representation — 2 generators on 61 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);(1,3,2,4)(6,9)(7,8)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2,5,23,29,2|a^2=e,b^a=b^4,c^a=c^22,d^a=d^28

Conjugacy classes & character table

3338 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₆₇₀
maximal normal subgroup of	Dic₆₆₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13340, 8);

Dic3335 #27211103