Aliases: C62.16C32, (C3×A4)⋊C9, (C3×C9)⋊3A4, (C6×C18)⋊3C3, C3.5(C9×A4), C3.3(C9⋊A4), (C2×C6).4He3, C3.1(C32⋊A4), (C32×A4).1C3, C22⋊1(C32⋊C9), C32.18(C3×A4), (C2×C6).23- 1+2, (C3×C3.A4)⋊1C3, (C2×C6).4(C3×C9), SmallGroup(324,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.16C32
G = < a,b,c,d | a6=b6=c3=1, d3=a2, ab=ba, cac-1=ab3, ad=da, cbc-1=a3b4, bd=db, dcd-1=b2c >
Subgroups: 205 in 56 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, C3, C22, C6, C9, C32, C32, A4, C2×C6, C2×C6, C18, C3×C6, C3×C9, C3×C9, C33, C3.A4, C2×C18, C3×A4, C3×A4, C62, C3×C18, C32⋊C9, C3×C3.A4, C6×C18, C32×A4, C62.16C32
Quotients: C1, C3, C9, C32, A4, C3×C9, He3, 3- 1+2, C3×A4, C32⋊C9, C9×A4, C9⋊A4, C32⋊A4, C62.16C32
►On 108 points
Generators in S108
(1 54 4 48 7 51)(2 46 5 49 8 52)(3 47 6 50 9 53)(10 29 13 32 16 35)(11 30 14 33 17 36)(12 31 15 34 18 28)(19 56 22 59 25 62)(20 57 23 60 26 63)(21 58 24 61 27 55)(37 76 40 79 43 73)(38 77 41 80 44 74)(39 78 42 81 45 75)(64 91 67 94 70 97)(65 92 68 95 71 98)(66 93 69 96 72 99)(82 107 85 101 88 104)(83 108 86 102 89 105)(84 100 87 103 90 106)
(1 42 63 12 90 66)(2 43 55 13 82 67)(3 44 56 14 83 68)(4 45 57 15 84 69)(5 37 58 16 85 70)(6 38 59 17 86 71)(7 39 60 18 87 72)(8 40 61 10 88 64)(9 41 62 11 89 65)(19 30 105 92 53 80)(20 31 106 93 54 81)(21 32 107 94 46 73)(22 33 108 95 47 74)(23 34 100 96 48 75)(24 35 101 97 49 76)(25 36 102 98 50 77)(26 28 103 99 51 78)(27 29 104 91 52 79)
(1 4 7)(2 58 88)(3 86 62)(5 61 82)(6 89 56)(8 55 85)(9 83 59)(10 97 104)(11 77 19)(12 28 54)(13 91 107)(14 80 22)(15 31 48)(16 94 101)(17 74 25)(18 34 51)(20 66 99)(21 43 29)(23 69 93)(24 37 32)(26 72 96)(27 40 35)(30 108 68)(33 102 71)(36 105 65)(38 95 50)(39 75 103)(41 98 53)(42 78 106)(44 92 47)(45 81 100)(46 67 79)(49 70 73)(52 64 76)(57 60 63)(84 87 90)
(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)
G:=sub<Sym(108)| (1,54,4,48,7,51)(2,46,5,49,8,52)(3,47,6,50,9,53)(10,29,13,32,16,35)(11,30,14,33,17,36)(12,31,15,34,18,28)(19,56,22,59,25,62)(20,57,23,60,26,63)(21,58,24,61,27,55)(37,76,40,79,43,73)(38,77,41,80,44,74)(39,78,42,81,45,75)(64,91,67,94,70,97)(65,92,68,95,71,98)(66,93,69,96,72,99)(82,107,85,101,88,104)(83,108,86,102,89,105)(84,100,87,103,90,106), (1,42,63,12,90,66)(2,43,55,13,82,67)(3,44,56,14,83,68)(4,45,57,15,84,69)(5,37,58,16,85,70)(6,38,59,17,86,71)(7,39,60,18,87,72)(8,40,61,10,88,64)(9,41,62,11,89,65)(19,30,105,92,53,80)(20,31,106,93,54,81)(21,32,107,94,46,73)(22,33,108,95,47,74)(23,34,100,96,48,75)(24,35,101,97,49,76)(25,36,102,98,50,77)(26,28,103,99,51,78)(27,29,104,91,52,79), (1,4,7)(2,58,88)(3,86,62)(5,61,82)(6,89,56)(8,55,85)(9,83,59)(10,97,104)(11,77,19)(12,28,54)(13,91,107)(14,80,22)(15,31,48)(16,94,101)(17,74,25)(18,34,51)(20,66,99)(21,43,29)(23,69,93)(24,37,32)(26,72,96)(27,40,35)(30,108,68)(33,102,71)(36,105,65)(38,95,50)(39,75,103)(41,98,53)(42,78,106)(44,92,47)(45,81,100)(46,67,79)(49,70,73)(52,64,76)(57,60,63)(84,87,90), (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)>;
G:=Group( (1,54,4,48,7,51)(2,46,5,49,8,52)(3,47,6,50,9,53)(10,29,13,32,16,35)(11,30,14,33,17,36)(12,31,15,34,18,28)(19,56,22,59,25,62)(20,57,23,60,26,63)(21,58,24,61,27,55)(37,76,40,79,43,73)(38,77,41,80,44,74)(39,78,42,81,45,75)(64,91,67,94,70,97)(65,92,68,95,71,98)(66,93,69,96,72,99)(82,107,85,101,88,104)(83,108,86,102,89,105)(84,100,87,103,90,106), (1,42,63,12,90,66)(2,43,55,13,82,67)(3,44,56,14,83,68)(4,45,57,15,84,69)(5,37,58,16,85,70)(6,38,59,17,86,71)(7,39,60,18,87,72)(8,40,61,10,88,64)(9,41,62,11,89,65)(19,30,105,92,53,80)(20,31,106,93,54,81)(21,32,107,94,46,73)(22,33,108,95,47,74)(23,34,100,96,48,75)(24,35,101,97,49,76)(25,36,102,98,50,77)(26,28,103,99,51,78)(27,29,104,91,52,79), (1,4,7)(2,58,88)(3,86,62)(5,61,82)(6,89,56)(8,55,85)(9,83,59)(10,97,104)(11,77,19)(12,28,54)(13,91,107)(14,80,22)(15,31,48)(16,94,101)(17,74,25)(18,34,51)(20,66,99)(21,43,29)(23,69,93)(24,37,32)(26,72,96)(27,40,35)(30,108,68)(33,102,71)(36,105,65)(38,95,50)(39,75,103)(41,98,53)(42,78,106)(44,92,47)(45,81,100)(46,67,79)(49,70,73)(52,64,76)(57,60,63)(84,87,90), (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) );
G=PermutationGroup([[(1,54,4,48,7,51),(2,46,5,49,8,52),(3,47,6,50,9,53),(10,29,13,32,16,35),(11,30,14,33,17,36),(12,31,15,34,18,28),(19,56,22,59,25,62),(20,57,23,60,26,63),(21,58,24,61,27,55),(37,76,40,79,43,73),(38,77,41,80,44,74),(39,78,42,81,45,75),(64,91,67,94,70,97),(65,92,68,95,71,98),(66,93,69,96,72,99),(82,107,85,101,88,104),(83,108,86,102,89,105),(84,100,87,103,90,106)], [(1,42,63,12,90,66),(2,43,55,13,82,67),(3,44,56,14,83,68),(4,45,57,15,84,69),(5,37,58,16,85,70),(6,38,59,17,86,71),(7,39,60,18,87,72),(8,40,61,10,88,64),(9,41,62,11,89,65),(19,30,105,92,53,80),(20,31,106,93,54,81),(21,32,107,94,46,73),(22,33,108,95,47,74),(23,34,100,96,48,75),(24,35,101,97,49,76),(25,36,102,98,50,77),(26,28,103,99,51,78),(27,29,104,91,52,79)], [(1,4,7),(2,58,88),(3,86,62),(5,61,82),(6,89,56),(8,55,85),(9,83,59),(10,97,104),(11,77,19),(12,28,54),(13,91,107),(14,80,22),(15,31,48),(16,94,101),(17,74,25),(18,34,51),(20,66,99),(21,43,29),(23,69,93),(24,37,32),(26,72,96),(27,40,35),(30,108,68),(33,102,71),(36,105,65),(38,95,50),(39,75,103),(41,98,53),(42,78,106),(44,92,47),(45,81,100),(46,67,79),(49,70,73),(52,64,76),(57,60,63),(84,87,90)], [(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)]])
60 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 6A | ··· | 6H | 9A | ··· | 9F | 9G | ··· | 9R | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 3 | 1 | ··· | 1 | 12 | ··· | 12 | 3 | ··· | 3 | 3 | ··· | 3 | 12 | ··· | 12 | 3 | ··· | 3 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C3 | C3 | C3 | C9 | A4 | He3 | 3- 1+2 | C3×A4 | C9×A4 | C9⋊A4 | C32⋊A4 |
| kernel | C62.16C32 | C3×C3.A4 | C6×C18 | C32×A4 | C3×A4 | C3×C9 | C2×C6 | C2×C6 | C32 | C3 | C3 | C3 |
| # reps | 1 | 4 | 2 | 2 | 18 | 1 | 2 | 4 | 2 | 6 | 12 | 6 |
Matrix representation of C62.16C32 ►in GL4(𝔽19) generated by
| 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 12 | 12 | 12 |
| 0 | 7 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 8 | 8 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 8 | 8 | 8 |
| 9 | 0 | 0 | 0 |
| 0 | 7 | 15 | 6 |
| 0 | 9 | 1 | 13 |
| 0 | 10 | 4 | 11 |
G:=sub<GL(4,GF(19))| [11,0,0,0,0,0,12,7,0,0,12,0,0,7,12,0],[1,0,0,0,0,0,11,8,0,11,0,8,0,0,0,8],[1,0,0,0,0,11,0,8,0,0,0,8,0,0,11,8],[9,0,0,0,0,7,9,10,0,15,1,4,0,6,13,11] >;
C62.16C32 in GAP, Magma, Sage, TeX
C_6^2._{16}C_3^2 % in TeX
G:=Group("C6^2.16C3^2"); // GroupNames label
G:=SmallGroup(324,52);
// by ID
G=gap.SmallGroup(324,52);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,361,43,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^3=1,d^3=a^2,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=b^2*c>;
// generators/relations