C₄₇×Dic₂₁ #16841123

order	3948
isoclinism	isoclinic class of D₂₁
automorphism	?
homology	Abelianization (H₁) = C₄×C₄₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄₇×Dic₂₁
unambiguous semidirect product	C₃⋊(Dic₇×C₄₇)
unambiguous semidirect product	C₇⋊(Dic₃×C₄₇)

Splittings

derived, Hall	C₃×C₇ ⋊ C₄×C₄₇
Hall	C₃×C₇×C₄₇ ⋊ C₄
Hall	C₇×C₄₇ ⋊ Dic₃
Hall	C₃×C₄₇ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₃×C₄₇
Sylow	C₃ ⋊ Dic₇×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄₇	D₂₁
commutator, hypocenter	C₃×C₇	C₄×C₄₇
Frattini	C₂	D₂₁×C₄₇
Fitting, socle, prime radical	C₂×C₃×C₇×C₄₇	C₂
cosocle	C₂×C₃×C₇	C₂×C₄₇
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 3948

A minimal faithful permutation representation — 2 generators on 61 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);(48,50,49,51)(53,54)(56,61)(57,60)(58,59)

Conjugacy classes & character table

1128 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄₇×Dic₆₃, C₄₇×C₃⋊(ΓL(1,𝔽₈)⋊C₄), C₃×C₄₇×Dic₂₁, …
maximal normal subgroup of	C₄₇×C₃⋊(ΓL(1,𝔽₈)⋊C₄), C₃×C₄₇×Dic₂₁, C₅×C₄₇×Dic₂₁, …
maximal quotient of	C₄₇×Dic₆₃, C₃×C₄₇×Dic₂₁, C₄₇×C₃⋊Dic₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3948, 11);

C47×Dic21 #16841123