Dic₇₀₀₇ #43242152

order	28028
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₀₀₇
unambiguous semidirect product	C₁₁⋊Dic₆₃₇
unambiguous semidirect product	C₄₉⋊Dic₁₄₃

Splittings

derived, Hall	C₁₁×C₁₃×C₄₉ ⋊ C₄
Hall	C₁₃×C₄₉ ⋊ Dic₁₁
Hall	C₁₁×C₄₉ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₄₉
Sylow	C₄₉ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₅₃₉
Sylow	C₁₁ ⋊ Dic₆₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₀₀₇
commutator, hypocenter	C₁₁×C₁₃×C₄₉	C₄
Frattini	C₂×C₇	D₁₀₀₁
Fitting, cosocle	C₂×C₁₁×C₁₃×C₄₉	C₂
socle, prime radical	C₂×C₇×C₁₁×C₁₃	D₇
LCS	G ▷ C₁₁×C₁₃×C₄₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 28028

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)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(55,64)(56,63)(57,62)(58,61)(59,60)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2,7,11,13,2,7|a^2=e,b^7=f^6,b^a=b^6f,c^a=c^10,d^a=d^12,f^a=f^6

Conjugacy classes & character table

7010 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₀₁₄
maximal normal subgroup of	Dic₁₄₀₁₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28028, 8);

Dic7007 #43242152