metabelian, supersoluble, monomial
Aliases: Dic3⋊2Dic6, C62.10C23, C3⋊2(C12⋊Q8), C6.45(S3×D4), Dic32.3C2, C32⋊4(C4⋊Q8), C6.15(S3×Q8), (C3×Dic3)⋊2Q8, (C2×C12).12D6, C2.7(S3×Dic6), C6.3(C2×Dic6), Dic3⋊C4.2S3, C3⋊Dic3.15D4, (C6×C12).1C22, (C2×Dic3).2D6, C2.6(Dic3⋊D6), Dic3⋊Dic3.7C2, (C6×Dic3).32C22, (C2×C4).12S32, (C3×C6).9(C2×Q8), C22.73(C2×S32), (C3×C6).77(C2×D4), (C3×Dic3⋊C4).4C2, (C2×C6).29(C22×S3), (C2×C32⋊2Q8).2C2, (C2×C32⋊4Q8).3C2, (C2×C3⋊Dic3).12C22, SmallGroup(288,488)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.10C23
G = < a,b,c,d,e | a6=b6=1, c2=b3, d2=a3, e2=a3b3, 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=a3d >
Subgroups: 546 in 155 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4⋊Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C32⋊2Q8, C6×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C12⋊Q8, Dic32, Dic3⋊Dic3, C3×Dic3⋊C4, C2×C32⋊2Q8, C2×C32⋊4Q8, C62.10C23
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, C22×S3, C4⋊Q8, S32, C2×Dic6, S3×D4, S3×Q8, C2×S32, C12⋊Q8, S3×Dic6, Dic3⋊D6, C62.10C23
►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 30 13 24)(2 25 14 19)(3 26 15 20)(4 27 16 21)(5 28 17 22)(6 29 18 23)(7 84 96 86)(8 79 91 87)(9 80 92 88)(10 81 93 89)(11 82 94 90)(12 83 95 85)(31 49 39 46)(32 50 40 47)(33 51 41 48)(34 52 42 43)(35 53 37 44)(36 54 38 45)(55 70 62 76)(56 71 63 77)(57 72 64 78)(58 67 65 73)(59 68 66 74)(60 69 61 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), (1,30,13,24)(2,25,14,19)(3,26,15,20)(4,27,16,21)(5,28,17,22)(6,29,18,23)(7,84,96,86)(8,79,91,87)(9,80,92,88)(10,81,93,89)(11,82,94,90)(12,83,95,85)(31,49,39,46)(32,50,40,47)(33,51,41,48)(34,52,42,43)(35,53,37,44)(36,54,38,45)(55,70,62,76)(56,71,63,77)(57,72,64,78)(58,67,65,73)(59,68,66,74)(60,69,61,75)>;
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), (1,30,13,24)(2,25,14,19)(3,26,15,20)(4,27,16,21)(5,28,17,22)(6,29,18,23)(7,84,96,86)(8,79,91,87)(9,80,92,88)(10,81,93,89)(11,82,94,90)(12,83,95,85)(31,49,39,46)(32,50,40,47)(33,51,41,48)(34,52,42,43)(35,53,37,44)(36,54,38,45)(55,70,62,76)(56,71,63,77)(57,72,64,78)(58,67,65,73)(59,68,66,74)(60,69,61,75) );
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)], [(1,30,13,24),(2,25,14,19),(3,26,15,20),(4,27,16,21),(5,28,17,22),(6,29,18,23),(7,84,96,86),(8,79,91,87),(9,80,92,88),(10,81,93,89),(11,82,94,90),(12,83,95,85),(31,49,39,46),(32,50,40,47),(33,51,41,48),(34,52,42,43),(35,53,37,44),(36,54,38,45),(55,70,62,76),(56,71,63,77),(57,72,64,78),(58,67,65,73),(59,68,66,74),(60,69,61,75)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 18 | 18 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | Dic6 | S32 | S3×D4 | S3×Q8 | C2×S32 | S3×Dic6 | Dic3⋊D6 |
| kernel | C62.10C23 | Dic32 | Dic3⋊Dic3 | C3×Dic3⋊C4 | C2×C32⋊2Q8 | C2×C32⋊4Q8 | Dic3⋊C4 | C3×Dic3 | C3⋊Dic3 | C2×Dic3 | C2×C12 | Dic3 | C2×C4 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 8 | 1 | 2 | 2 | 1 | 4 | 2 |
Matrix representation of C62.10C23 ►in GL6(𝔽13)
| 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 5 | 0 | 0 | 0 | 0 |
| 7 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 |
| 4 | 5 | 0 | 0 | 0 | 0 |
| 7 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 10 | 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 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
G:=sub<GL(6,GF(13))| [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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[4,7,0,0,0,0,5,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,0,8],[4,7,0,0,0,0,5,9,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,10,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10] >;
C62.10C23 in GAP, Magma, Sage, TeX
C_6^2._{10}C_2^3 % in TeX
G:=Group("C6^2.10C2^3"); // GroupNames label
G:=SmallGroup(288,488);
// by ID
G=gap.SmallGroup(288,488);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,422,135,142,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=b^3,d^2=a^3,e^2=a^3*b^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=a^3*d>;
// generators/relations