Dic₆₉₇₅ #42124271

order	27900
automorphism	Aut(G) order 50220000
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₉₇₅
unambiguous semidirect product	C₉⋊Dic₇₇₅
unambiguous semidirect product	C₃₁⋊Dic₂₂₅

Splittings

derived, Hall	C₉×C₂₅×C₃₁ ⋊ C₄
Hall	C₂₅×C₃₁ ⋊ Dic₉
Hall	C₉×C₃₁ ⋊ Dic₂₅
Hall	C₉×C₂₅ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₂₂₅
Sylow	C₂₅ ⋊ Dic₂₇₉
Sylow	C₉ ⋊ Dic₇₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₉₇₅
commutator, hypocenter	C₉×C₂₅×C₃₁	C₄
Frattini	C₂×C₃×C₅	D₄₆₅
Fitting, cosocle	C₂×C₉×C₂₅×C₃₁	C₂
socle, prime radical	C₂×C₃×C₅×C₃₁	D₁₅
LCS	G ▷ C₉×C₂₅×C₃₁
UCS	C₁ ◁ C₂

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 27900

A minimal faithful permutation representation — 2 generators on 69 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,3,5,31,2,3,5|a^2=e,b^3=f^2,c^5=g^4,b^a=b^2f,c^a=c^4g,d^a=d^30,f^a=f^2,g^a=g^4

Conjugacy classes & character table

6978 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₉₅₀
maximal normal subgroup of	Dic₁₃₉₅₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 18);

Dic6975 #42124271