Dic₄₉₂₁ #34088964

order	19684
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₉₂₁
unambiguous semidirect product	C₇⋊Dic₇₀₃
unambiguous semidirect product	C₁₉⋊Dic₂₅₉

Splittings

derived, Hall	C₇×C₁₉×C₃₇ ⋊ C₄
Hall	C₁₉×C₃₇ ⋊ Dic₇
Hall	C₇×C₃₇ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₁₃₃
Sylow	C₁₉ ⋊ Dic₂₅₉
Sylow	C₇ ⋊ Dic₇₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₉₂₁
commutator, hypocenter	C₇×C₁₉×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₉×C₃₇	C₂
LCS	G ▷ C₇×C₁₉×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 19684

A minimal faithful permutation representation — 2 generators on 67 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2,7,19,37,2|a^2=e,b^a=b^6,c^a=c^18,d^a=d^36

Conjugacy classes & character table

4924 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₈₄₂, Dic₁₄₇₆₃
maximal normal subgroup of	Dic₉₈₄₂
maximal quotient of	Dic₁₄₇₆₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19684, 8);

Dic4921 #34088964