C₇²×D₄₇ #17987195

order	4606
isoclinism	isoclinic class of D₄₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₇² · Schur multiplier (H₂) = C₇

Aliases

direct product	C₇²×D₄₇
unambiguous semidirect product	C₄₇⋊(C₂×C₇²)

Splittings

Fitting, socle, Hall	C₇²×C₄₇ ⋊ C₂
derived, Sylow	C₄₇ ⋊ C₂×C₇²

Subgroups

	subgroup	G/H
center, hypercenter	C₇²	D₄₇
commutator, cosocle, hypocenter	C₄₇	C₂×C₇²
Frattini	C₁	G
Fitting, socle	C₇²×C₄₇	C₂
LCS	G ▷ C₄₇
UCS	C₁ ◁ C₇²

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 2303

A minimal faithful permutation representation — 2 generators on 61 points:

(1,2,3,4,5,6,7)(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);(8,9,10,11,12,13,14)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)

Conjugacy classes & character table

1225 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×ΓL(1,𝔽₈)×D₄₇, D₄₇×#3861, D₄₇×#3862, …
maximal normal subgroup of	C₇×ΓL(1,𝔽₈)×D₄₇, D₄₇×#3861, D₄₇×#3862, …
maximal quotient of	C₇²×D₁₄₁, C₃×C₇²×D₄₇, C₅×C₇²×D₄₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4606, 6);

C72×D47 #17987195