Dic₃₄₈₅ #28358274

order	13940
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₄₈₅

Splittings

derived, Hall	C₅×C₁₇×C₄₁ ⋊ C₄
Hall	C₁₇×C₄₁ ⋊ Dic₅
Hall	C₅×C₄₁ ⋊ Dic₁₇
Hall	C₅×C₁₇ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₈₅
Sylow	C₁₇ ⋊ Dic₂₀₅
Sylow	C₅ ⋊ Dic₆₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₄₈₅
commutator, hypocenter	C₅×C₁₇×C₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₇×C₄₁	C₂
LCS	G ▷ C₅×C₁₇×C₄₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 13940

A minimal faithful permutation representation — 2 generators on 67 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);(1,3,2,4)(6,9)(7,8)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(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:

2,5,17,41,2|a^2=e,b^a=b^4,c^a=c^16,d^a=d^40

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₉₇₀
maximal normal subgroup of	Dic₆₉₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13940, 9);

Dic3485 #28358274