metabelian, supersoluble, monomial
Aliases: Dic6⋊5Dic3, C62.13C23, C6.3(S3×Q8), C32⋊5(C4×Q8), C3⋊Dic3⋊5Q8, C3⋊2(Q8×Dic3), Dic32.5C2, C12.34(C4×S3), (C3×Dic6)⋊8C4, (C2×C12).127D6, C4⋊Dic3.11S3, (C2×Dic6).8S3, C4.10(S3×Dic3), C6.1(D4⋊2S3), (C6×C12).87C22, (C6×Dic6).10C2, (C2×Dic3).55D6, C12.19(C2×Dic3), C3⋊5(Dic6⋊C4), C2.1(D12⋊S3), C6.8(C22×Dic3), Dic3⋊Dic3.9C2, C6.19(Q8⋊3S3), Dic3.4(C2×Dic3), C2.2(Dic3.D6), (C6×Dic3).52C22, C6.88(S3×C2×C4), (C2×C4).111S32, C22.20(C2×S32), (C3×C6).11(C2×Q8), C2.10(C2×S3×Dic3), (C3×C12).62(C2×C4), (C3×C6).4(C4○D4), (C4×C3⋊Dic3).1C2, (C3×C4⋊Dic3).10C2, (C3×C6).48(C22×C4), (C2×C6).32(C22×S3), (C3×Dic3).9(C2×C4), (C2×C3⋊Dic3).114C22, SmallGroup(288,491)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.13C23
G = < a,b,c,d,e | a6=b6=1, c2=e2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >
Subgroups: 426 in 155 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C2×C3⋊Dic3, C6×C12, Dic6⋊C4, Q8×Dic3, Dic32, Dic3⋊Dic3, C3×C4⋊Dic3, C4×C3⋊Dic3, C6×Dic6, C62.13C23
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, Dic3, D6, C22×C4, C2×Q8, C4○D4, C4×S3, C2×Dic3, C22×S3, C4×Q8, S32, S3×C2×C4, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×Dic3, S3×Dic3, C2×S32, Dic6⋊C4, Q8×Dic3, D12⋊S3, Dic3.D6, C2×S3×Dic3, C62.13C23
►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 64 16 60)(2 65 17 55)(3 66 18 56)(4 61 13 57)(5 62 14 58)(6 63 15 59)(7 51 93 45)(8 52 94 46)(9 53 95 47)(10 54 96 48)(11 49 91 43)(12 50 92 44)(19 76 28 67)(20 77 29 68)(21 78 30 69)(22 73 25 70)(23 74 26 71)(24 75 27 72)(31 90 42 79)(32 85 37 80)(33 86 38 81)(34 87 39 82)(35 88 40 83)(36 89 41 84)
(1 33 4 36)(2 32 5 35)(3 31 6 34)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)(25 53 28 50)(26 52 29 49)(27 51 30 54)(55 80 58 83)(56 79 59 82)(57 84 60 81)(61 89 64 86)(62 88 65 85)(63 87 66 90)(67 92 70 95)(68 91 71 94)(69 96 72 93)
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 86 93 81)(8 87 94 82)(9 88 95 83)(10 89 96 84)(11 90 91 79)(12 85 92 80)(31 43 42 49)(32 44 37 50)(33 45 38 51)(34 46 39 52)(35 47 40 53)(36 48 41 54)(55 67 65 76)(56 68 66 77)(57 69 61 78)(58 70 62 73)(59 71 63 74)(60 72 64 75)
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), (1,64,16,60)(2,65,17,55)(3,66,18,56)(4,61,13,57)(5,62,14,58)(6,63,15,59)(7,51,93,45)(8,52,94,46)(9,53,95,47)(10,54,96,48)(11,49,91,43)(12,50,92,44)(19,76,28,67)(20,77,29,68)(21,78,30,69)(22,73,25,70)(23,74,26,71)(24,75,27,72)(31,90,42,79)(32,85,37,80)(33,86,38,81)(34,87,39,82)(35,88,40,83)(36,89,41,84), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45)(25,53,28,50)(26,52,29,49)(27,51,30,54)(55,80,58,83)(56,79,59,82)(57,84,60,81)(61,89,64,86)(62,88,65,85)(63,87,66,90)(67,92,70,95)(68,91,71,94)(69,96,72,93), (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,86,93,81)(8,87,94,82)(9,88,95,83)(10,89,96,84)(11,90,91,79)(12,85,92,80)(31,43,42,49)(32,44,37,50)(33,45,38,51)(34,46,39,52)(35,47,40,53)(36,48,41,54)(55,67,65,76)(56,68,66,77)(57,69,61,78)(58,70,62,73)(59,71,63,74)(60,72,64,75)>;
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), (1,64,16,60)(2,65,17,55)(3,66,18,56)(4,61,13,57)(5,62,14,58)(6,63,15,59)(7,51,93,45)(8,52,94,46)(9,53,95,47)(10,54,96,48)(11,49,91,43)(12,50,92,44)(19,76,28,67)(20,77,29,68)(21,78,30,69)(22,73,25,70)(23,74,26,71)(24,75,27,72)(31,90,42,79)(32,85,37,80)(33,86,38,81)(34,87,39,82)(35,88,40,83)(36,89,41,84), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45)(25,53,28,50)(26,52,29,49)(27,51,30,54)(55,80,58,83)(56,79,59,82)(57,84,60,81)(61,89,64,86)(62,88,65,85)(63,87,66,90)(67,92,70,95)(68,91,71,94)(69,96,72,93), (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,86,93,81)(8,87,94,82)(9,88,95,83)(10,89,96,84)(11,90,91,79)(12,85,92,80)(31,43,42,49)(32,44,37,50)(33,45,38,51)(34,46,39,52)(35,47,40,53)(36,48,41,54)(55,67,65,76)(56,68,66,77)(57,69,61,78)(58,70,62,73)(59,71,63,74)(60,72,64,75) );
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)], [(1,64,16,60),(2,65,17,55),(3,66,18,56),(4,61,13,57),(5,62,14,58),(6,63,15,59),(7,51,93,45),(8,52,94,46),(9,53,95,47),(10,54,96,48),(11,49,91,43),(12,50,92,44),(19,76,28,67),(20,77,29,68),(21,78,30,69),(22,73,25,70),(23,74,26,71),(24,75,27,72),(31,90,42,79),(32,85,37,80),(33,86,38,81),(34,87,39,82),(35,88,40,83),(36,89,41,84)], [(1,33,4,36),(2,32,5,35),(3,31,6,34),(7,78,10,75),(8,77,11,74),(9,76,12,73),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45),(25,53,28,50),(26,52,29,49),(27,51,30,54),(55,80,58,83),(56,79,59,82),(57,84,60,81),(61,89,64,86),(62,88,65,85),(63,87,66,90),(67,92,70,95),(68,91,71,94),(69,96,72,93)], [(1,27,16,24),(2,28,17,19),(3,29,18,20),(4,30,13,21),(5,25,14,22),(6,26,15,23),(7,86,93,81),(8,87,94,82),(9,88,95,83),(10,89,96,84),(11,90,91,79),(12,85,92,80),(31,43,42,49),(32,44,37,50),(33,45,38,51),(34,46,39,52),(35,47,40,53),(36,48,41,54),(55,67,65,76),(56,68,66,77),(57,69,61,78),(58,70,62,73),(59,71,63,74),(60,72,64,75)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | ··· | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6F | 6G | 6H | 6I | 12A | ··· | 12H | 12I | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 6 | ··· | 6 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | - | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | Q8 | Dic3 | D6 | D6 | C4○D4 | C4×S3 | S32 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 | S3×Dic3 | C2×S32 | D12⋊S3 | Dic3.D6 |
| kernel | C62.13C23 | Dic32 | Dic3⋊Dic3 | C3×C4⋊Dic3 | C4×C3⋊Dic3 | C6×Dic6 | C3×Dic6 | C4⋊Dic3 | C2×Dic6 | C3⋊Dic3 | Dic6 | C2×Dic3 | C2×C12 | C3×C6 | C12 | C2×C4 | C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C62.13C23 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 |
| 7 | 4 | 0 | 0 | 0 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 4 | 0 | 0 | 0 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 | 0 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[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],[7,7,0,0,0,0,4,6,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,7,0,0,0,0,4,6,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[8,0,0,0,0,0,10,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C62.13C23 in GAP, Magma, Sage, TeX
C_6^2._{13}C_2^3 % in TeX
G:=Group("C6^2.13C2^3"); // GroupNames label
G:=SmallGroup(288,491);
// by ID
G=gap.SmallGroup(288,491);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,422,219,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=e^2=b^3,d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
// generators/relations