Dic₇₆₆₇ #44501517

order	30668
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₆₆₇
unambiguous semidirect product	C₁₁⋊Dic₆₉₇

Splittings

derived, Hall	C₁₁×C₁₇×C₄₁ ⋊ C₄
Hall	C₁₇×C₄₁ ⋊ Dic₁₁
Hall	C₁₁×C₄₁ ⋊ Dic₁₇
Hall	C₁₁×C₁₇ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₈₇
Sylow	C₁₇ ⋊ Dic₄₅₁
Sylow	C₁₁ ⋊ Dic₆₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₆₆₇
commutator, hypocenter	C₁₁×C₁₇×C₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₇×C₄₁	C₂
LCS	G ▷ C₁₁×C₁₇×C₄₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 30668

A minimal faithful permutation representation — 2 generators on 73 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2,11,17,41,2|a^2=e,b^a=b^10,c^a=c^16,d^a=d^40

Conjugacy classes & character table

7670 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₃₃₄
maximal normal subgroup of	Dic₁₅₃₃₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30668, 8);

Dic7667 #44501517