C₂×(C₂₉⋊D₂₉)⋊C₉ #44475588

order	30276
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₂₉

Aliases

direct product	C₂×(C₂₉⋊D₂₉)⋊C₉
unambiguous semidirect product	(C₂×C₂₉⋊D₂₉)⋊C₉

Splittings

derived, Sylow	C₂₉² ⋊ C₂²×C₉
Hall	C₂₉²⋊C₉ ⋊ C₂²
Hall	C₂×C₂₉⋊D₂₉ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₂₉⋊D₂₉)⋊C₃
commutator, hypocenter	C₂₉²	C₂²×C₉
Frattini	C₃	C₂×(C₂₉⋊D₂₉)⋊C₃
Fitting, socle	C₂×C₃×C₂₉²	C₂×C₃
prime radical	C₂×C₃×C₂₉⋊D₂₉	C₃
cosocle	C₃×C₂₉²	C₂²×C₃
LCS	G ▷ C₂₉²
UCS	C₁ ◁ C₂×C₃

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 522

A minimal faithful permutation representation — 2 generators on 98 points:

(3,7,11,5,10,6,8,4,9)(12,13,15)(14,30,31,18,19,43)(16,41,42,23,24,66)(17,22,20,27,28,34)(21,64,56,36,37,32)(25,44,78,45,46,77)(26,75,81,48,49,96)(29,40,63,52,53,73)(33,76,35,59,60,62)(38,90,79,67,68,47)(39,95,54,71,72,58)(50,87,92,61,69,88)(51,89,84,65,82,55)(57,74,97,85,83,91)(70,98,86,93,94,80);(1,2)(12,14,20,35,63,88,72,96,37,66,82,78,97,90,94,98,68,91,77,89,42,64,81,95,92,73,62,34,18)(13,16,25,17,26,47,29,50,80,54,59,85,56,30,55,84,19,32,57,33,58,86,61,52,79,48,27,45,23)(15,21,38,69,60,46,65,22,39,70,49,31,41,74,40,53,83,24,43,75,93,71,28,51,44,76,87,67,36)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30276, 51);

C2×(C29⋊D29)⋊C9 #44475588