C₇²×C₃₁⋊C₃ #17986948

order	4557
isoclinism	isoclinic class of C₃₁⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₇² · Schur multiplier (H₂) = C₇

Aliases

direct product	C₇²×C₃₁⋊C₃
unambiguous semidirect product	C₃₁⋊(C₃×C₇²)

Splittings

Fitting, socle, Hall	C₇²×C₃₁ ⋊ C₃
derived, Sylow	C₃₁ ⋊ C₃×C₇²

Subgroups

	subgroup	G/H
center, hypercenter	C₇²	C₃₁⋊C₃
commutator, cosocle, hypocenter	C₃₁	C₃×C₇²
Frattini	C₁	G
Fitting, socle	C₇²×C₃₁	C₃
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₇²

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1519

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5,6,7)(16,17,19)(18,21,26)(20,24,23)(22,28,35)(25,31,33)(27,29,30)(32,38,42)(34,39,40)(36,43,37)(41,44,45);(8,9,10,11,12,13,14)(15,16,18,22,29,19,23,30,37,43,26,33,40,28,36,35,42,45,38,21,27,34,41,44,31,17,20,25,32,39,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×ΓL(1,𝔽₈)×C₃₁⋊C₃, C₃₁⋊C₃×#3861, C₃₁⋊C₃×#3862, …
maximal normal subgroup of	C₇×ΓL(1,𝔽₈)×C₃₁⋊C₃, C₃₁⋊C₃×#3861, C₃₁⋊C₃×#3862, …
maximal quotient of	C₇²×C₃₁⋊C₉, C₃×C₇²×C₃₁⋊C₃, C₇²×C₂²⋊(C₃₁⋊C₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4557, 7);

C72×C31⋊C3 #17986948