Dic₇₆₅₇ #44499545

order	30628
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₆₅₇
unambiguous semidirect product	C₁₉⋊Dic₄₀₃
unambiguous semidirect product	C₃₁⋊Dic₂₄₇

Splittings

derived, Hall	C₁₃×C₁₉×C₃₁ ⋊ C₄
Hall	C₁₉×C₃₁ ⋊ Dic₁₃
Hall	C₁₃×C₃₁ ⋊ Dic₁₉
Hall	C₁₃×C₁₉ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₂₄₇
Sylow	C₁₉ ⋊ Dic₄₀₃
Sylow	C₁₃ ⋊ Dic₅₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₆₅₇
commutator, hypocenter	C₁₃×C₁₉×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₁₉×C₃₁	C₂
LCS	G ▷ C₁₃×C₁₉×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 30628

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)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,13,19,31,2|a^2=e,b^a=b^12,c^a=c^18,d^a=d^30

Conjugacy classes & character table

7660 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₃₁₄
maximal normal subgroup of	Dic₁₅₃₁₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30628, 8);

Dic7657 #44499545