Dic₃₆₈₉ #28414221

order	14756
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₆₈₉
unambiguous semidirect product	C₇⋊Dic₅₂₇
unambiguous semidirect product	C₃₁⋊Dic₁₁₉

Splittings

derived, Hall	C₇×C₁₇×C₃₁ ⋊ C₄
Hall	C₁₇×C₃₁ ⋊ Dic₇
Hall	C₇×C₃₁ ⋊ Dic₁₇
Hall	C₇×C₁₇ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₁₉
Sylow	C₁₇ ⋊ Dic₂₁₇
Sylow	C₇ ⋊ Dic₅₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₆₈₉
commutator, hypocenter	C₇×C₁₇×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₃₁	C₂
LCS	G ▷ C₇×C₁₇×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 14756

A minimal faithful permutation representation — 2 generators on 59 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2,7,17,31,2|a^2=e,b^a=b^6,c^a=c^16,d^a=d^30

Conjugacy classes & character table

3692 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₃₇₈
maximal normal subgroup of	Dic₇₃₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14756, 8);

Dic3689 #28414221