C₇×C₃⋊(C₂₉⋊D₂₉) #47977378

order	35322
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₂₉

Aliases

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

Splittings

Fitting, socle, Hall	C₃×C₇×C₂₉² ⋊ C₂
derived, Hall	C₃×C₂₉² ⋊ C₂×C₇
Hall	C₇×C₂₉² ⋊ D₃
Sylow	C₂₉² ⋊ D₃×C₇
Hall	C₃×C₇ ⋊ C₂₉⋊D₂₉
Sylow	C₃ ⋊ C₇×C₂₉⋊D₂₉

Subgroups

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

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:

(1,2,3,4,5,6,7)(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);(8,9,10)(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

8841 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35322, 82);

C7×C3⋊(C29⋊D29) #47977378