Dic₇₅₆₅ #44475433

order	30260
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₅₆₅

Splittings

derived, Hall	C₅×C₁₇×C₈₉ ⋊ C₄
Hall	C₁₇×C₈₉ ⋊ Dic₅
Hall	C₅×C₈₉ ⋊ Dic₁₇
Sylow	C₈₉ ⋊ Dic₈₅
Hall	C₅×C₁₇ ⋊ Dic₈₉
Sylow	C₁₇ ⋊ Dic₄₄₅
Sylow	C₅ ⋊ Dic₁₅₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₅₆₅
commutator, hypocenter	C₅×C₁₇×C₈₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₇×C₈₉	C₂
LCS	G ▷ C₅×C₁₇×C₈₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 30260

A minimal faithful permutation representation — 2 generators on 115 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3,2,4)(6,9)(7,8)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2,5,17,89,2|a^2=e,b^a=b^4,c^a=c^16,d^a=d^88

Conjugacy classes & character table

7568 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₁₃₀
maximal normal subgroup of	Dic₁₅₁₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30260, 8);

Dic7565 #44475433