metabelian, supersoluble, monomial
Aliases: C62.47C23, Dic32⋊15C2, D6.6(C4×S3), D6⋊C4.10S3, (S3×Dic3)⋊3C4, (C4×Dic3)⋊14S3, (C2×C12).194D6, D6⋊Dic3.2C2, C3⋊3(C42⋊2S3), (Dic3×C12)⋊24C2, C6.52(C4○D12), Dic3.12(C4×S3), (C2×Dic3).61D6, (C22×S3).29D6, C2.3(D12⋊5S3), C6.39(D4⋊2S3), (C6×C12).225C22, C6.Dic6⋊15C2, C62.C22⋊13C2, C32⋊7(C42⋊C2), C2.3(D6.3D6), C3⋊1(C23.16D6), (C6×Dic3).57C22, C2.13(C4×S32), (C2×C4).46S32, C6.12(S3×C2×C4), (S3×C6).6(C2×C4), C22.27(C2×S32), (C2×S3×Dic3).6C2, (C3×D6⋊C4).10C2, (S3×C2×C6).10C22, (C3×C6).28(C4○D4), C3⋊Dic3.22(C2×C4), (C3×C6).11(C22×C4), (C2×C6).66(C22×S3), (C3×Dic3).16(C2×C4), (C2×C3⋊Dic3).35C22, SmallGroup(288,525)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.47C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >
Subgroups: 522 in 165 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C42⋊C2, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C4×Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, S3×C2×C4, C22×Dic3, S3×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C42⋊2S3, C23.16D6, Dic32, D6⋊Dic3, C62.C22, Dic3×C12, C3×D6⋊C4, C6.Dic6, C2×S3×Dic3, C62.47C23
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, S32, S3×C2×C4, C4○D12, D4⋊2S3, C2×S32, C42⋊2S3, C23.16D6, D12⋊5S3, C4×S32, D6.3D6, C62.47C23
►On 96 points
Generators in S96
(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)
(1 18 5 16 3 14)(2 13 6 17 4 15)(7 95 11 93 9 91)(8 96 12 94 10 92)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)(43 53 45 49 47 51)(44 54 46 50 48 52)(55 63 57 65 59 61)(56 64 58 66 60 62)(67 74 69 76 71 78)(68 75 70 77 72 73)(79 86 83 90 81 88)(80 87 84 85 82 89)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 41 16 36)(2 40 17 35)(3 39 18 34)(4 38 13 33)(5 37 14 32)(6 42 15 31)(7 78 93 69)(8 77 94 68)(9 76 95 67)(10 75 96 72)(11 74 91 71)(12 73 92 70)(19 53 28 47)(20 52 29 46)(21 51 30 45)(22 50 25 44)(23 49 26 43)(24 54 27 48)(55 88 65 83)(56 87 66 82)(57 86 61 81)(58 85 62 80)(59 90 63 79)(60 89 64 84)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 81 10 84)(8 82 11 79)(9 83 12 80)(13 27 16 30)(14 28 17 25)(15 29 18 26)(31 46 34 43)(32 47 35 44)(33 48 36 45)(37 53 40 50)(38 54 41 51)(39 49 42 52)(55 73 58 76)(56 74 59 77)(57 75 60 78)(61 72 64 69)(62 67 65 70)(63 68 66 71)(85 95 88 92)(86 96 89 93)(87 91 90 94)
G:=sub<Sym(96)| (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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,95,11,93,9,91)(8,96,12,94,10,92)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39)(43,53,45,49,47,51)(44,54,46,50,48,52)(55,63,57,65,59,61)(56,64,58,66,60,62)(67,74,69,76,71,78)(68,75,70,77,72,73)(79,86,83,90,81,88)(80,87,84,85,82,89), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,41,16,36)(2,40,17,35)(3,39,18,34)(4,38,13,33)(5,37,14,32)(6,42,15,31)(7,78,93,69)(8,77,94,68)(9,76,95,67)(10,75,96,72)(11,74,91,71)(12,73,92,70)(19,53,28,47)(20,52,29,46)(21,51,30,45)(22,50,25,44)(23,49,26,43)(24,54,27,48)(55,88,65,83)(56,87,66,82)(57,86,61,81)(58,85,62,80)(59,90,63,79)(60,89,64,84), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,81,10,84)(8,82,11,79)(9,83,12,80)(13,27,16,30)(14,28,17,25)(15,29,18,26)(31,46,34,43)(32,47,35,44)(33,48,36,45)(37,53,40,50)(38,54,41,51)(39,49,42,52)(55,73,58,76)(56,74,59,77)(57,75,60,78)(61,72,64,69)(62,67,65,70)(63,68,66,71)(85,95,88,92)(86,96,89,93)(87,91,90,94)>;
G:=Group( (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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,95,11,93,9,91)(8,96,12,94,10,92)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39)(43,53,45,49,47,51)(44,54,46,50,48,52)(55,63,57,65,59,61)(56,64,58,66,60,62)(67,74,69,76,71,78)(68,75,70,77,72,73)(79,86,83,90,81,88)(80,87,84,85,82,89), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,41,16,36)(2,40,17,35)(3,39,18,34)(4,38,13,33)(5,37,14,32)(6,42,15,31)(7,78,93,69)(8,77,94,68)(9,76,95,67)(10,75,96,72)(11,74,91,71)(12,73,92,70)(19,53,28,47)(20,52,29,46)(21,51,30,45)(22,50,25,44)(23,49,26,43)(24,54,27,48)(55,88,65,83)(56,87,66,82)(57,86,61,81)(58,85,62,80)(59,90,63,79)(60,89,64,84), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,81,10,84)(8,82,11,79)(9,83,12,80)(13,27,16,30)(14,28,17,25)(15,29,18,26)(31,46,34,43)(32,47,35,44)(33,48,36,45)(37,53,40,50)(38,54,41,51)(39,49,42,52)(55,73,58,76)(56,74,59,77)(57,75,60,78)(61,72,64,69)(62,67,65,70)(63,68,66,71)(85,95,88,92)(86,96,89,93)(87,91,90,94) );
G=PermutationGroup([[(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)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,95,11,93,9,91),(8,96,12,94,10,92),(19,30,23,28,21,26),(20,25,24,29,22,27),(31,40,33,42,35,38),(32,41,34,37,36,39),(43,53,45,49,47,51),(44,54,46,50,48,52),(55,63,57,65,59,61),(56,64,58,66,60,62),(67,74,69,76,71,78),(68,75,70,77,72,73),(79,86,83,90,81,88),(80,87,84,85,82,89)], [(1,60),(2,55),(3,56),(4,57),(5,58),(6,59),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,41,16,36),(2,40,17,35),(3,39,18,34),(4,38,13,33),(5,37,14,32),(6,42,15,31),(7,78,93,69),(8,77,94,68),(9,76,95,67),(10,75,96,72),(11,74,91,71),(12,73,92,70),(19,53,28,47),(20,52,29,46),(21,51,30,45),(22,50,25,44),(23,49,26,43),(24,54,27,48),(55,88,65,83),(56,87,66,82),(57,86,61,81),(58,85,62,80),(59,90,63,79),(60,89,64,84)], [(1,21,4,24),(2,22,5,19),(3,23,6,20),(7,81,10,84),(8,82,11,79),(9,83,12,80),(13,27,16,30),(14,28,17,25),(15,29,18,26),(31,46,34,43),(32,47,35,44),(33,48,36,45),(37,53,40,50),(38,54,41,51),(39,49,42,52),(55,73,58,76),(56,74,59,77),(57,75,60,78),(61,72,64,69),(62,67,65,70),(63,68,66,71),(85,95,88,92),(86,96,89,93),(87,91,90,94)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12R | 12S | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 4 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D6 | D6 | D6 | C4○D4 | C4×S3 | C4×S3 | C4○D12 | S32 | D4⋊2S3 | C2×S32 | D12⋊5S3 | C4×S32 | D6.3D6 |
| kernel | C62.47C23 | Dic32 | D6⋊Dic3 | C62.C22 | Dic3×C12 | C3×D6⋊C4 | C6.Dic6 | C2×S3×Dic3 | S3×Dic3 | C4×Dic3 | D6⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | Dic3 | D6 | C6 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 3 | 2 | 1 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of C62.47C23 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C62.47C23 in GAP, Magma, Sage, TeX
C_6^2._{47}C_2^3 % in TeX
G:=Group("C6^2.47C2^3"); // GroupNames label
G:=SmallGroup(288,525);
// by ID
G=gap.SmallGroup(288,525);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=b^3,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,d*e=e*d>;
// generators/relations