Dic₇₅₁₁ #44473469

order	30044
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₅₁₁
unambiguous semidirect product	C₇⋊Dic₁₀₇₃

Splittings

derived, Hall	C₇×C₂₉×C₃₇ ⋊ C₄
Hall	C₂₉×C₃₇ ⋊ Dic₇
Hall	C₇×C₃₇ ⋊ Dic₂₉
Hall	C₇×C₂₉ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₂₀₃
Sylow	C₂₉ ⋊ Dic₂₅₉
Sylow	C₇ ⋊ Dic₁₀₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₅₁₁
commutator, hypocenter	C₇×C₂₉×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₉×C₃₇	C₂
LCS	G ▷ C₇×C₂₉×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 30044

A minimal faithful permutation representation — 2 generators on 77 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2,7,29,37,2|a^2=e,b^a=b^6,c^a=c^28,d^a=d^36

Conjugacy classes & character table

7514 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₀₂₂
maximal normal subgroup of	Dic₁₅₀₂₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30044, 9);

Dic7511 #44473469