Dic₆₁₄₉ #37589697

order	24596
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₁₄₉
unambiguous semidirect product	C₁₁⋊Dic₅₅₉
unambiguous semidirect product	C₄₃⋊Dic₁₄₃

Splittings

derived, Hall	C₁₁×C₁₃×C₄₃ ⋊ C₄
Hall	C₁₃×C₄₃ ⋊ Dic₁₁
Hall	C₁₁×C₄₃ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₄₇₃
Sylow	C₁₁ ⋊ Dic₅₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₁₄₉
commutator, hypocenter	C₁₁×C₁₃×C₄₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₄₃	C₂
LCS	G ▷ C₁₁×C₁₃×C₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 24596

A minimal faithful permutation representation — 2 generators on 71 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2,11,13,43,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^42

Conjugacy classes & character table

6152 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₂₉₈
maximal normal subgroup of	Dic₁₂₂₉₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24596, 8);

Dic6149 #37589697