C₃×#72213393 #72229159

order	13872
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×#72213393

Splittings

Fitting, socle, Hall	C₃×C₁₇² ⋊ C₁₆
derived, Sylow	C₁₇² ⋊ C₃×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#72213393
commutator, hypocenter	C₁₇²	C₃×C₁₆
Frattini	C₁	G
Fitting, socle	C₃×C₁₇²	C₁₆
prime radical	C₃×C₁₇⋊D₁₇	C₈
cosocle	#72213191	C₂×C₃
LCS	G ▷ C₁₇²
UCS	C₁ ◁ C₃

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 102

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72213403, D₃×#72213393, #72259142
maximal normal subgroup of	C₃×#72213403, D₃×#72213393, #72259142

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3)(4,5)(6,9,10,20,11,21,13,26,12,22,23,37,24,34,27,31)(7,14,8,19,15,25,17,32,16,29,18,35,30,28,33,36), (4,25,35,10,14,13,12,11,32,36,24,6,27,29,23,19,28)(5,9,26,33,34,8,15,7,20,37,16,30,18,21,17,31,22));

C3×#72213393 #72229159