Dic₆₆₀₁ #40995987

order	26404
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₆₀₁
unambiguous semidirect product	C₇⋊Dic₉₄₃
unambiguous semidirect product	C₂₃⋊Dic₂₈₇

Splittings

derived, Hall	C₇×C₂₃×C₄₁ ⋊ C₄
Hall	C₂₃×C₄₁ ⋊ Dic₇
Hall	C₇×C₄₁ ⋊ Dic₂₃
Hall	C₇×C₂₃ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₆₁
Sylow	C₂₃ ⋊ Dic₂₈₇
Sylow	C₇ ⋊ Dic₉₄₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₆₀₁
commutator, hypocenter	C₇×C₂₃×C₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₃×C₄₁	C₂
LCS	G ▷ C₇×C₂₃×C₄₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 26404

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)(6,11)(7,10)(8,9)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,7,23,41,2|a^2=e,b^a=b^6,c^a=c^22,d^a=d^40

Conjugacy classes & character table

6604 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₂₀₂
maximal normal subgroup of	Dic₁₃₂₀₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26404, 8);

Dic6601 #40995987