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G = C34⋊10C6  order 486 = 2·35

10th semidirect product of C34 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C34⋊10C6, C34⋊11S3, C3⋊(He3⋊4S3), He3⋊4(C3⋊S3), C33⋊8(C3⋊S3), (C3×He3)⋊22S3, C33⋊15(C3×S3), C34⋊C2⋊3C3, (C32×He3)⋊6C2, C32⋊6(C32⋊C6), C32⋊1(C33⋊C2), C32⋊3(C3×C3⋊S3), C3.2(C3×C33⋊C2), SmallGroup(486,242)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C34 — C34⋊10C6
Chief series
C1 — C3 — C32 — C33 — C34 — C32×He3 — C34⋊10C6
Lower central
C34 — C34⋊10C6
Upper central
C1

Generators and relations for C34⋊10C6
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, eae-1=ad-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 4004 in 456 conjugacy classes, 80 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, He3, He3, C33, C33, C33, C32⋊C6, C3×C3⋊S3, C33⋊C2, C3×He3, C3×He3, C34, C34, He3⋊4S3, C3×C33⋊C2, C34⋊C2, C32×He3, C34⋊10C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, C33⋊C2, He3⋊4S3, C3×C33⋊C2, C34⋊10C6

Smallest permutation representation of C34⋊10C6
►On 81 points
Generators in S81
(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)

(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)(55 73 64)(56 74 65)(57 75 66)(58 76 67)(59 77 68)(60 78 69)(61 79 70)(62 80 71)(63 81 72)

(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)

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

(1 28 56)(2 29 55)(3 30 57)(4 31 59)(5 32 58)(6 33 60)(7 34 62)(8 35 61)(9 36 63)(10 37 65)(11 38 64)(12 39 66)(13 40 68)(14 41 67)(15 42 69)(16 43 71)(17 44 70)(18 45 72)(19 46 74)(20 47 73)(21 48 75)(22 49 77)(23 50 76)(24 51 78)(25 52 80)(26 53 79)(27 54 81)

(2 3)(4 7)(5 9)(6 8)(10 19)(11 21)(12 20)(13 25)(14 27)(15 26)(16 22)(17 24)(18 23)(28 56)(29 57)(30 55)(31 62)(32 63)(33 61)(34 59)(35 60)(36 58)(37 74)(38 75)(39 73)(40 80)(41 81)(42 79)(43 77)(44 78)(45 76)(46 65)(47 66)(48 64)(49 71)(50 72)(51 70)(52 68)(53 69)(54 67)
 
G:=sub<Sym(81)| (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), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45)(55,73,64)(56,74,65)(57,75,66)(58,76,67)(59,77,68)(60,78,69)(61,79,70)(62,80,71)(63,81,72), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(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), (1,28,56)(2,29,55)(3,30,57)(4,31,59)(5,32,58)(6,33,60)(7,34,62)(8,35,61)(9,36,63)(10,37,65)(11,38,64)(12,39,66)(13,40,68)(14,41,67)(15,42,69)(16,43,71)(17,44,70)(18,45,72)(19,46,74)(20,47,73)(21,48,75)(22,49,77)(23,50,76)(24,51,78)(25,52,80)(26,53,79)(27,54,81), (2,3)(4,7)(5,9)(6,8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,56)(29,57)(30,55)(31,62)(32,63)(33,61)(34,59)(35,60)(36,58)(37,74)(38,75)(39,73)(40,80)(41,81)(42,79)(43,77)(44,78)(45,76)(46,65)(47,66)(48,64)(49,71)(50,72)(51,70)(52,68)(53,69)(54,67)>;
 
G:=Group( (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), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45)(55,73,64)(56,74,65)(57,75,66)(58,76,67)(59,77,68)(60,78,69)(61,79,70)(62,80,71)(63,81,72), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(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), (1,28,56)(2,29,55)(3,30,57)(4,31,59)(5,32,58)(6,33,60)(7,34,62)(8,35,61)(9,36,63)(10,37,65)(11,38,64)(12,39,66)(13,40,68)(14,41,67)(15,42,69)(16,43,71)(17,44,70)(18,45,72)(19,46,74)(20,47,73)(21,48,75)(22,49,77)(23,50,76)(24,51,78)(25,52,80)(26,53,79)(27,54,81), (2,3)(4,7)(5,9)(6,8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,56)(29,57)(30,55)(31,62)(32,63)(33,61)(34,59)(35,60)(36,58)(37,74)(38,75)(39,73)(40,80)(41,81)(42,79)(43,77)(44,78)(45,76)(46,65)(47,66)(48,64)(49,71)(50,72)(51,70)(52,68)(53,69)(54,67) );
 
G=PermutationGroup([[(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)], [(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(28,46,37),(29,47,38),(30,48,39),(31,49,40),(32,50,41),(33,51,42),(34,52,43),(35,53,44),(36,54,45),(55,73,64),(56,74,65),(57,75,66),(58,76,67),(59,77,68),(60,78,69),(61,79,70),(62,80,71),(63,81,72)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78)], [(1,3,2),(4,6,5),(7,9,8),(10,12,11),(13,15,14),(16,18,17),(19,21,20),(22,24,23),(25,27,26),(28,30,29),(31,33,32),(34,36,35),(37,39,38),(40,42,41),(43,45,44),(46,48,47),(49,51,50),(52,54,53),(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)], [(1,28,56),(2,29,55),(3,30,57),(4,31,59),(5,32,58),(6,33,60),(7,34,62),(8,35,61),(9,36,63),(10,37,65),(11,38,64),(12,39,66),(13,40,68),(14,41,67),(15,42,69),(16,43,71),(17,44,70),(18,45,72),(19,46,74),(20,47,73),(21,48,75),(22,49,77),(23,50,76),(24,51,78),(25,52,80),(26,53,79),(27,54,81)], [(2,3),(4,7),(5,9),(6,8),(10,19),(11,21),(12,20),(13,25),(14,27),(15,26),(16,22),(17,24),(18,23),(28,56),(29,57),(30,55),(31,62),(32,63),(33,61),(34,59),(35,60),(36,58),(37,74),(38,75),(39,73),(40,80),(41,81),(42,79),(43,77),(44,78),(45,76),(46,65),(47,66),(48,64),(49,71),(50,72),(51,70),(52,68),(53,69),(54,67)]])
 

54 conjugacy classes

class 1  2 3A···3M3N3O3P···3AX6A6B
order123···3333···366
size1812···2336···68181

54 irreducible representations

dim11112226
type+++++
imageC1C2C3C6S3S3C3×S3C32⋊C6
kernelC34⋊10C6C32×He3C34⋊C2C34C3×He3C34C33C32
# reps1122121269

Matrix representation of C34⋊10C6 ►in GL10(𝔽7)

2000000000
0200000000
0040000000
0004000000
0000100000
0000010000
0000000600
0000001600
0000546001
0000061666
,
1100000000
4500000000
0051000000
0041000000
0000060000
0000160000
0000000600
0000001600
0000420166
0000211610
,
1100000000
4500000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
0000610000
0000600000
0000006100
0000006000
0000350601
0000566166
,
1000000000
0100000000
0016000000
0035000000
0000546661
0000530056
0000100000
0000010000
0000061161
0000530650
,
2100000000
4500000000
0026000000
0035000000
0000610000
0000010000
0000061165
0000530056
0000250611
0000550155

G:=sub<GL(10,GF(7))| [2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,5,0,0,0,0,0,0,1,0,0,4,6,0,0,0,0,0,0,0,1,6,1,0,0,0,0,0,0,6,6,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,6],[1,4,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,0,0,5,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,2,0,0,0,0,6,6,0,0,2,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,6,6,1,6,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,6,0],[1,4,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,6,0,0,3,5,0,0,0,0,1,0,0,0,5,6,0,0,0,0,0,0,6,6,0,6,0,0,0,0,0,0,1,0,6,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,6],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,6,5,0,0,0,0,0,0,0,0,0,0,5,5,1,0,0,5,0,0,0,0,4,3,0,1,6,3,0,0,0,0,6,0,0,0,1,0,0,0,0,0,6,0,0,0,1,6,0,0,0,0,6,5,0,0,6,5,0,0,0,0,1,6,0,0,1,0],[2,4,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,6,5,0,0,0,0,0,0,0,0,0,0,6,0,0,5,2,5,0,0,0,0,1,1,6,3,5,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,6,1,0,0,0,0,0,0,6,5,1,5,0,0,0,0,0,0,5,6,1,5] >;
 

C34⋊10C6 in GAP, Magma, Sage, TeX 

C_3^4\rtimes_{10}C_6
 
% in TeX
 
G:=Group("C3^4:10C6");
 
// GroupNames label
 
G:=SmallGroup(486,242);
 
// by ID
 
G=gap.SmallGroup(486,242);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,1520,867,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*d^-1,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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