C₇×C₃⋊((C₁₇⋊D₁₇)⋊C₃) #49138798

order	36414
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁₇

Aliases

direct product	C₇×C₃⋊((C₁₇⋊D₁₇)⋊C₃)
unambiguous semidirect product	C₃⋊(C₇×(C₁₇⋊D₁₇)⋊C₃)
unambiguous semidirect product	(C₃⋊(C₁₇⋊D₁₇))⋊(C₃×C₇)

Splittings

Fitting, socle	C₃×C₇×C₁₇² ⋊ C₂×C₃
derived	C₃×C₁₇² ⋊ C₂×C₃×C₇
Hall	C₃×C₇×C₁₇²⋊C₃ ⋊ C₂
Hall	C₃×C₁₇²⋊C₃ ⋊ C₂×C₇
Hall	C₇×C₁₇² ⋊ C₃×D₃
Sylow	C₁₇² ⋊ C₃×D₃×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₃⋊((C₁₇⋊D₁₇)⋊C₃)
commutator, cosocle, hypocenter	C₃×C₁₇²	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₇²	C₂×C₃
LCS	G ▷ C₃×C₁₇²
UCS	C₁ ◁ C₇

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 1071

A minimal faithful permutation representation — 2 generators on 61 points:

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

Conjugacy classes & character table

1071 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36414, 93);

C7×C3⋊((C17⋊D17)⋊C3) #49138798