Dic₂₇₅₉ #26010674

order	11036
automorphism	Aut(G) order 14567520
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₅₉
unambiguous semidirect product	C₃₁⋊Dic₈₉

Splittings

derived, Hall	C₃₁×C₈₉ ⋊ C₄
Sylow	C₈₉ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₇₅₉
commutator, hypocenter	C₃₁×C₈₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₁×C₈₉	C₂
LCS	G ▷ C₃₁×C₈₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 124
transitive degree	μ_t(G) = 11036

A minimal faithful permutation representation — 2 generators on 124 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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124);(1,3,2,4)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Polycyclic presentation:

2,31,89,2|a^2=d,b^a=b^30,c^a=c^88

Conjugacy classes & character table

2762 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₇₅₉, Dic₈₂₇₇, (C₃₁⋊C₃)⋊Dic₈₉, …
maximal normal subgroup of	C₃×Dic₂₇₅₉, (C₃₁⋊C₃)⋊Dic₈₉, Dic₅₅₁₈
maximal quotient of	C₃×Dic₂₇₅₉, Dic₈₂₇₇, Dic₁₃₇₉₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11036, 4);

Dic2759 #26010674