Dic₇₅₆₇ #44475495

order	30268
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₅₆₇
unambiguous semidirect product	C₇⋊Dic₁₀₈₁
unambiguous semidirect product	C₂₃⋊Dic₃₂₉
unambiguous semidirect product	C₄₇⋊Dic₁₆₁

Splittings

derived, Hall	C₇×C₂₃×C₄₇ ⋊ C₄
Hall	C₂₃×C₄₇ ⋊ Dic₇
Hall	C₇×C₄₇ ⋊ Dic₂₃
Hall	C₇×C₂₃ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₆₁
Sylow	C₂₃ ⋊ Dic₃₂₉
Sylow	C₇ ⋊ Dic₁₀₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₅₆₇
commutator, hypocenter	C₇×C₂₃×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₃×C₄₇	C₂
LCS	G ▷ C₇×C₂₃×C₄₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 30268

A minimal faithful permutation representation — 2 generators on 81 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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(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)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2,7,23,47,2|a^2=e,b^a=b^6,c^a=c^22,d^a=d^46

Conjugacy classes & character table

7570 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₁₃₄
maximal normal subgroup of	Dic₁₅₁₃₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30268, 9);

Dic7567 #44475495