non-abelian, soluble, monomial
Aliases: C62.10Dic3, C3⋊(C6.S4), C6.6(C3⋊S4), (C3×C6).16S4, (C2×C6)⋊2Dic9, C6.7(C3.S4), C3.A4⋊2Dic3, (C2×C62).8S3, (C22×C6).4D9, C23.2(C9⋊S3), C32.3(A4⋊C4), C3.2(C6.7S4), C22⋊2(C9⋊Dic3), C2.1(C32.3S4), (C3×C3.A4)⋊3C4, (C6×C3.A4).3C2, (C2×C3.A4).2S3, (C22×C6).5(C3⋊S3), (C2×C6).3(C3⋊Dic3), SmallGroup(432,259)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C3.A4 — C62.10Dic3 |
Generators and relations for C62.10Dic3
G = < a,b,c,d | a6=b6=1, c6=b2, d2=b2c3, ab=ba, cac-1=ab3, dad-1=a2b3, cbc-1=a3b4, dbd-1=a3b2, dcd-1=b4c5 >
Subgroups: 648 in 102 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C6, C2×C4, C23, C9, C32, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C3×C6, C3×C6, C2×Dic3, C22×C6, C22×C6, C3×C9, Dic9, C3.A4, C3⋊Dic3, C62, C62, C6.D4, C3×C18, C2×C3.A4, C2×C3⋊Dic3, C2×C62, C9⋊Dic3, C3×C3.A4, C6.S4, C62⋊5C4, C6×C3.A4, C62.10Dic3
Quotients: C1, C2, C4, S3, Dic3, D9, C3⋊S3, S4, Dic9, C3⋊Dic3, A4⋊C4, C9⋊S3, C3.S4, C3⋊S4, C9⋊Dic3, C6.S4, C6.7S4, C32.3S4, C62.10Dic3
►On 108 points
Generators in S108
(1 78 21)(2 88 22 11 79 31)(3 89 23 12 80 32)(4 81 24)(5 73 25 14 82 34)(6 74 26 15 83 35)(7 84 27)(8 76 28 17 85 19)(9 77 29 18 86 20)(10 87 30)(13 90 33)(16 75 36)(37 101 60 46 92 69)(38 93 61)(39 103 62 48 94 71)(40 104 63 49 95 72)(41 96 64)(42 106 65 51 97 56)(43 107 66 52 98 57)(44 99 67)(45 91 68 54 100 59)(47 102 70)(50 105 55)(53 108 58)
(1 4 7 10 13 16)(2 14 8)(3 6 9 12 15 18)(5 17 11)(19 31 25)(20 23 26 29 32 35)(21 24 27 30 33 36)(22 34 28)(37 40 43 46 49 52)(38 41 44 47 50 53)(39 51 45)(42 54 48)(55 58 61 64 67 70)(56 68 62)(57 60 63 66 69 72)(59 71 65)(73 85 79)(74 77 80 83 86 89)(75 78 81 84 87 90)(76 88 82)(91 103 97)(92 95 98 101 104 107)(93 96 99 102 105 108)(94 106 100)
(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)
(1 51 10 42)(2 50 11 41)(3 49 12 40)(4 48 13 39)(5 47 14 38)(6 46 15 37)(7 45 16 54)(8 44 17 53)(9 43 18 52)(19 108 28 99)(20 107 29 98)(21 106 30 97)(22 105 31 96)(23 104 32 95)(24 103 33 94)(25 102 34 93)(26 101 35 92)(27 100 36 91)(55 88 64 79)(56 87 65 78)(57 86 66 77)(58 85 67 76)(59 84 68 75)(60 83 69 74)(61 82 70 73)(62 81 71 90)(63 80 72 89)
G:=sub<Sym(108)| (1,78,21)(2,88,22,11,79,31)(3,89,23,12,80,32)(4,81,24)(5,73,25,14,82,34)(6,74,26,15,83,35)(7,84,27)(8,76,28,17,85,19)(9,77,29,18,86,20)(10,87,30)(13,90,33)(16,75,36)(37,101,60,46,92,69)(38,93,61)(39,103,62,48,94,71)(40,104,63,49,95,72)(41,96,64)(42,106,65,51,97,56)(43,107,66,52,98,57)(44,99,67)(45,91,68,54,100,59)(47,102,70)(50,105,55)(53,108,58), (1,4,7,10,13,16)(2,14,8)(3,6,9,12,15,18)(5,17,11)(19,31,25)(20,23,26,29,32,35)(21,24,27,30,33,36)(22,34,28)(37,40,43,46,49,52)(38,41,44,47,50,53)(39,51,45)(42,54,48)(55,58,61,64,67,70)(56,68,62)(57,60,63,66,69,72)(59,71,65)(73,85,79)(74,77,80,83,86,89)(75,78,81,84,87,90)(76,88,82)(91,103,97)(92,95,98,101,104,107)(93,96,99,102,105,108)(94,106,100), (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), (1,51,10,42)(2,50,11,41)(3,49,12,40)(4,48,13,39)(5,47,14,38)(6,46,15,37)(7,45,16,54)(8,44,17,53)(9,43,18,52)(19,108,28,99)(20,107,29,98)(21,106,30,97)(22,105,31,96)(23,104,32,95)(24,103,33,94)(25,102,34,93)(26,101,35,92)(27,100,36,91)(55,88,64,79)(56,87,65,78)(57,86,66,77)(58,85,67,76)(59,84,68,75)(60,83,69,74)(61,82,70,73)(62,81,71,90)(63,80,72,89)>;
G:=Group( (1,78,21)(2,88,22,11,79,31)(3,89,23,12,80,32)(4,81,24)(5,73,25,14,82,34)(6,74,26,15,83,35)(7,84,27)(8,76,28,17,85,19)(9,77,29,18,86,20)(10,87,30)(13,90,33)(16,75,36)(37,101,60,46,92,69)(38,93,61)(39,103,62,48,94,71)(40,104,63,49,95,72)(41,96,64)(42,106,65,51,97,56)(43,107,66,52,98,57)(44,99,67)(45,91,68,54,100,59)(47,102,70)(50,105,55)(53,108,58), (1,4,7,10,13,16)(2,14,8)(3,6,9,12,15,18)(5,17,11)(19,31,25)(20,23,26,29,32,35)(21,24,27,30,33,36)(22,34,28)(37,40,43,46,49,52)(38,41,44,47,50,53)(39,51,45)(42,54,48)(55,58,61,64,67,70)(56,68,62)(57,60,63,66,69,72)(59,71,65)(73,85,79)(74,77,80,83,86,89)(75,78,81,84,87,90)(76,88,82)(91,103,97)(92,95,98,101,104,107)(93,96,99,102,105,108)(94,106,100), (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), (1,51,10,42)(2,50,11,41)(3,49,12,40)(4,48,13,39)(5,47,14,38)(6,46,15,37)(7,45,16,54)(8,44,17,53)(9,43,18,52)(19,108,28,99)(20,107,29,98)(21,106,30,97)(22,105,31,96)(23,104,32,95)(24,103,33,94)(25,102,34,93)(26,101,35,92)(27,100,36,91)(55,88,64,79)(56,87,65,78)(57,86,66,77)(58,85,67,76)(59,84,68,75)(60,83,69,74)(61,82,70,73)(62,81,71,90)(63,80,72,89) );
G=PermutationGroup([[(1,78,21),(2,88,22,11,79,31),(3,89,23,12,80,32),(4,81,24),(5,73,25,14,82,34),(6,74,26,15,83,35),(7,84,27),(8,76,28,17,85,19),(9,77,29,18,86,20),(10,87,30),(13,90,33),(16,75,36),(37,101,60,46,92,69),(38,93,61),(39,103,62,48,94,71),(40,104,63,49,95,72),(41,96,64),(42,106,65,51,97,56),(43,107,66,52,98,57),(44,99,67),(45,91,68,54,100,59),(47,102,70),(50,105,55),(53,108,58)], [(1,4,7,10,13,16),(2,14,8),(3,6,9,12,15,18),(5,17,11),(19,31,25),(20,23,26,29,32,35),(21,24,27,30,33,36),(22,34,28),(37,40,43,46,49,52),(38,41,44,47,50,53),(39,51,45),(42,54,48),(55,58,61,64,67,70),(56,68,62),(57,60,63,66,69,72),(59,71,65),(73,85,79),(74,77,80,83,86,89),(75,78,81,84,87,90),(76,88,82),(91,103,97),(92,95,98,101,104,107),(93,96,99,102,105,108),(94,106,100)], [(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)], [(1,51,10,42),(2,50,11,41),(3,49,12,40),(4,48,13,39),(5,47,14,38),(6,46,15,37),(7,45,16,54),(8,44,17,53),(9,43,18,52),(19,108,28,99),(20,107,29,98),(21,106,30,97),(22,105,31,96),(23,104,32,95),(24,103,33,94),(25,102,34,93),(26,101,35,92),(27,100,36,91),(55,88,64,79),(56,87,65,78),(57,86,66,77),(58,85,67,76),(59,84,68,75),(60,83,69,74),(61,82,70,73),(62,81,71,90),(63,80,72,89)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 9A | ··· | 9I | 18A | ··· | 18I |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 2 | 54 | 54 | 54 | 54 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | - | + | - | + | + | + | - | - | ||
| image | C1 | C2 | C4 | S3 | S3 | Dic3 | Dic3 | D9 | Dic9 | S4 | A4⋊C4 | C3.S4 | C3⋊S4 | C6.S4 | C6.7S4 |
| kernel | C62.10Dic3 | C6×C3.A4 | C3×C3.A4 | C2×C3.A4 | C2×C62 | C3.A4 | C62 | C22×C6 | C2×C6 | C3×C6 | C32 | C6 | C6 | C3 | C3 |
| # reps | 1 | 1 | 2 | 3 | 1 | 3 | 1 | 9 | 9 | 2 | 2 | 3 | 1 | 3 | 1 |
Matrix representation of C62.10Dic3 ►in GL7(𝔽37)
| 36 | 1 | 0 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 26 | 17 | 0 | 0 | 0 | 0 | 0 |
| 20 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 8 | 28 | 0 | 0 | 0 | 0 | 0 |
| 36 | 29 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 19 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(7,GF(37))| [36,36,0,0,0,0,0,1,0,0,0,0,0,0,0,0,35,1,0,0,0,0,0,34,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,36,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[26,20,0,0,0,0,0,17,6,0,0,0,0,0,0,0,2,36,0,0,0,0,0,3,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0],[8,36,0,0,0,0,0,28,29,0,0,0,0,0,0,0,31,0,0,0,0,0,0,19,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;
C62.10Dic3 in GAP, Magma, Sage, TeX
C_6^2._{10}{\rm Dic}_3 % in TeX
G:=Group("C6^2.10Dic3"); // GroupNames label
G:=SmallGroup(432,259);
// by ID
G=gap.SmallGroup(432,259);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,926,394,675,2524,9077,2287,5298,3989]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^2*b^3,c*b*c^-1=a^3*b^4,d*b*d^-1=a^3*b^2,d*c*d^-1=b^4*c^5>;
// generators/relations