C₃×C₇⋊(C₂₉⋊D₂₉) #47977375

order	35322
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂₉

Aliases

direct product	C₃×C₇⋊(C₂₉⋊D₂₉)
unambiguous semidirect product	C₂₉⋊(C₃×D₂₀₃)

Splittings

Fitting, socle, Hall	C₃×C₇×C₂₉² ⋊ C₂
derived, Hall	C₇×C₂₉² ⋊ C₂×C₃
Hall	C₃×C₂₉² ⋊ D₇
Sylow	C₂₉² ⋊ C₃×D₇
Hall	C₃×C₇ ⋊ C₂₉⋊D₂₉
Sylow	C₇ ⋊ C₃×C₂₉⋊D₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₇⋊(C₂₉⋊D₂₉)
commutator, cosocle, hypocenter	C₇×C₂₉²	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₂₉²	C₂
LCS	G ▷ C₇×C₂₉²
UCS	C₁ ◁ C₃

256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 17661

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3)(5,6)(7,8)(9,10)(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)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68);(4,5,7,9,10,8,6)(40,41,43,45,47,49,51,53,55,57,59,61,63,65,67,68,66,64,62,60,58,56,54,52,50,48,46,44,42);(11,12,14,16,18,20,22,24,26,28,30,32,34,36,38,39,37,35,33,31,29,27,25,23,21,19,17,15,13)

Conjugacy classes & character table

8835 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35322, 79);

C3×C7⋊(C29⋊D29) #47977375