C₃²×C₇×C₂₉⋊C₇ #27165331

order	12789
isoclinism	isoclinic class of C₂₉⋊C₇
automorphism	?
homology	Abelianization (H₁) = C₃²×C₇² · Schur multiplier (H₂) = C₃×C₇

Aliases

direct product	C₃²×C₇×C₂₉⋊C₇
unambiguous semidirect product	C₂₉⋊(C₃²×C₇²)
unambiguous semidirect product	(C₃×C₂₉)⋊(C₃×C₇²)
unambiguous semidirect product	(C₃²×C₂₉)⋊C₇²
unambiguous semidirect product	(C₃²×C₇×C₂₉)⋊C₇

Splittings

Fitting, socle	C₃²×C₇×C₂₉ ⋊ C₇
derived, Sylow	C₂₉ ⋊ C₃²×C₇²
Hall	C₃²×C₂₉ ⋊ C₇²

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₇	C₂₉⋊C₇
commutator, cosocle, hypocenter	C₂₉	C₃²×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₂₉	C₇
LCS	G ▷ C₂₉
UCS	C₁ ◁ C₃²×C₇

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 1827

A minimal faithful permutation representation — 2 generators on 42 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃²×C₇×D₂₉⋊C₇, C₃²×C₇⋊(D₂₉⋊C₇), C₃²×D₇×C₂₉⋊C₇, …
maximal normal subgroup of	C₃²×C₇×D₂₉⋊C₇, C₃²×C₇⋊(D₂₉⋊C₇), C₃²×D₇×C₂₉⋊C₇, …
maximal quotient of	C₂×C₃²×C₇×C₂₉⋊C₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12789, 16);

C32×C7×C29⋊C7 #27165331