Dic₃₆₁₉ #28407913

order	14476
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₆₁₉
unambiguous semidirect product	C₇⋊Dic₅₁₇
unambiguous semidirect product	C₁₁⋊Dic₃₂₉
unambiguous semidirect product	C₄₇⋊Dic₇₇

Splittings

derived, Hall	C₇×C₁₁×C₄₇ ⋊ C₄
Hall	C₁₁×C₄₇ ⋊ Dic₇
Hall	C₇×C₄₇ ⋊ Dic₁₁
Hall	C₇×C₁₁ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₇₇
Sylow	C₁₁ ⋊ Dic₃₂₉
Sylow	C₇ ⋊ Dic₅₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₆₁₉
commutator, hypocenter	C₇×C₁₁×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₁×C₄₇	C₂
LCS	G ▷ C₇×C₁₁×C₄₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 14476

A minimal faithful permutation representation — 2 generators on 69 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,7,11,47,2|a^2=e,b^a=b^6,c^a=c^10,d^a=d^46

Conjugacy classes & character table

3622 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₂₃₈
maximal normal subgroup of	Dic₇₂₃₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14476, 8);

Dic3619 #28407913