Dic₆₇₂₁ #41001873

order	26884
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₇₂₁
unambiguous semidirect product	C₁₁⋊Dic₆₁₁
unambiguous semidirect product	C₄₇⋊Dic₁₄₃

Splittings

derived, Hall	C₁₁×C₁₃×C₄₇ ⋊ C₄
Hall	C₁₃×C₄₇ ⋊ Dic₁₁
Hall	C₁₁×C₄₇ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₅₁₇
Sylow	C₁₁ ⋊ Dic₆₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₇₂₁
commutator, hypocenter	C₁₁×C₁₃×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₄₇	C₂
LCS	G ▷ C₁₁×C₁₃×C₄₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 26884

A minimal faithful permutation representation — 2 generators on 75 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);(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)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,11,13,47,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^46

Conjugacy classes & character table

6724 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₄₄₂
maximal normal subgroup of	Dic₁₃₄₄₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26884, 8);

Dic6721 #41001873