C₃×D₄₃₃₁ #39906994

order	25986
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₄₃₃₁
unambiguous semidirect product	C₇₁⋊(C₃×D₆₁)

Splittings

Fitting, socle, Hall	C₃×C₆₁×C₇₁ ⋊ C₂
derived, Hall	C₆₁×C₇₁ ⋊ C₂×C₃
Hall	C₃×C₇₁ ⋊ D₆₁
Hall	C₃×C₆₁ ⋊ D₇₁
Sylow	C₇₁ ⋊ C₃×D₆₁
Sylow	C₆₁ ⋊ C₃×D₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₄₃₃₁
commutator, cosocle, hypocenter	C₆₁×C₇₁	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₆₁×C₇₁	C₂
LCS	G ▷ C₆₁×C₇₁
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 12993

A minimal faithful permutation representation — 2 generators on 135 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)

Conjugacy classes & character table

6501 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25986, 5);

C3×D4331 #39906994