direct product, metabelian, supersoluble, monomial
Aliases: C2×C6×Dic6, C62⋊11Q8, C62.265C23, C6⋊1(C6×Q8), C6.1(C23×C6), (C2×C12).449D6, C32⋊6(C22×Q8), C23.43(S3×C6), C6.69(S3×C23), (C3×C6).38C24, (C22×C12).43S3, C12.34(C22×C6), (C22×C12).21C6, (C22×C6).174D6, (C6×C12).327C22, (C3×C12).166C23, C12.221(C22×S3), (C22×Dic3).7C6, Dic3.1(C22×C6), (C2×C62).115C22, (C3×Dic3).28C23, (C6×Dic3).162C22, C3⋊1(Q8×C2×C6), C4.32(S3×C2×C6), (C3×C6)⋊5(C2×Q8), (C2×C6)⋊6(C3×Q8), (C2×C6×C12).17C2, C2.3(S3×C22×C6), (C2×C4).86(S3×C6), C22.28(S3×C2×C6), (Dic3×C2×C6).13C2, (C2×C12).111(C2×C6), (C2×C6).67(C22×C6), (C22×C6).68(C2×C6), (C22×C4).12(C3×S3), (C2×C6).345(C22×S3), (C2×Dic3).43(C2×C6), SmallGroup(288,988)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6×Dic6
G = < a,b,c,d | a2=b6=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 570 in 339 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C22×C4, C22×C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22×C6, C22×Q8, C3×Dic3, C3×C12, C62, C2×Dic6, C22×Dic3, C22×C12, C22×C12, C6×Q8, C3×Dic6, C6×Dic3, C6×C12, C2×C62, C22×Dic6, Q8×C2×C6, C6×Dic6, Dic3×C2×C6, C2×C6×C12, C2×C6×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C24, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, C22×Q8, S3×C6, C2×Dic6, C6×Q8, S3×C23, C23×C6, C3×Dic6, S3×C2×C6, C22×Dic6, Q8×C2×C6, C6×Dic6, S3×C22×C6, C2×C6×Dic6
►On 96 points
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 85)(10 86)(11 87)(12 88)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 49)(35 50)(36 51)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)
(1 32 9 28 5 36)(2 33 10 29 6 25)(3 34 11 30 7 26)(4 35 12 31 8 27)(13 71 17 63 21 67)(14 72 18 64 22 68)(15 61 19 65 23 69)(16 62 20 66 24 70)(37 78 41 82 45 74)(38 79 42 83 46 75)(39 80 43 84 47 76)(40 81 44 73 48 77)(49 87 57 95 53 91)(50 88 58 96 54 92)(51 89 59 85 55 93)(52 90 60 86 56 94)
(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 72 7 66)(2 71 8 65)(3 70 9 64)(4 69 10 63)(5 68 11 62)(6 67 12 61)(13 31 19 25)(14 30 20 36)(15 29 21 35)(16 28 22 34)(17 27 23 33)(18 26 24 32)(37 94 43 88)(38 93 44 87)(39 92 45 86)(40 91 46 85)(41 90 47 96)(42 89 48 95)(49 75 55 81)(50 74 56 80)(51 73 57 79)(52 84 58 78)(53 83 59 77)(54 82 60 76)
G:=sub<Sym(96)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (1,32,9,28,5,36)(2,33,10,29,6,25)(3,34,11,30,7,26)(4,35,12,31,8,27)(13,71,17,63,21,67)(14,72,18,64,22,68)(15,61,19,65,23,69)(16,62,20,66,24,70)(37,78,41,82,45,74)(38,79,42,83,46,75)(39,80,43,84,47,76)(40,81,44,73,48,77)(49,87,57,95,53,91)(50,88,58,96,54,92)(51,89,59,85,55,93)(52,90,60,86,56,94), (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), (1,72,7,66)(2,71,8,65)(3,70,9,64)(4,69,10,63)(5,68,11,62)(6,67,12,61)(13,31,19,25)(14,30,20,36)(15,29,21,35)(16,28,22,34)(17,27,23,33)(18,26,24,32)(37,94,43,88)(38,93,44,87)(39,92,45,86)(40,91,46,85)(41,90,47,96)(42,89,48,95)(49,75,55,81)(50,74,56,80)(51,73,57,79)(52,84,58,78)(53,83,59,77)(54,82,60,76)>;
G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (1,32,9,28,5,36)(2,33,10,29,6,25)(3,34,11,30,7,26)(4,35,12,31,8,27)(13,71,17,63,21,67)(14,72,18,64,22,68)(15,61,19,65,23,69)(16,62,20,66,24,70)(37,78,41,82,45,74)(38,79,42,83,46,75)(39,80,43,84,47,76)(40,81,44,73,48,77)(49,87,57,95,53,91)(50,88,58,96,54,92)(51,89,59,85,55,93)(52,90,60,86,56,94), (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), (1,72,7,66)(2,71,8,65)(3,70,9,64)(4,69,10,63)(5,68,11,62)(6,67,12,61)(13,31,19,25)(14,30,20,36)(15,29,21,35)(16,28,22,34)(17,27,23,33)(18,26,24,32)(37,94,43,88)(38,93,44,87)(39,92,45,86)(40,91,46,85)(41,90,47,96)(42,89,48,95)(49,75,55,81)(50,74,56,80)(51,73,57,79)(52,84,58,78)(53,83,59,77)(54,82,60,76) );
G=PermutationGroup([[(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,85),(10,86),(11,87),(12,88),(13,84),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,49),(35,50),(36,51),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72)], [(1,32,9,28,5,36),(2,33,10,29,6,25),(3,34,11,30,7,26),(4,35,12,31,8,27),(13,71,17,63,21,67),(14,72,18,64,22,68),(15,61,19,65,23,69),(16,62,20,66,24,70),(37,78,41,82,45,74),(38,79,42,83,46,75),(39,80,43,84,47,76),(40,81,44,73,48,77),(49,87,57,95,53,91),(50,88,58,96,54,92),(51,89,59,85,55,93),(52,90,60,86,56,94)], [(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)], [(1,72,7,66),(2,71,8,65),(3,70,9,64),(4,69,10,63),(5,68,11,62),(6,67,12,61),(13,31,19,25),(14,30,20,36),(15,29,21,35),(16,28,22,34),(17,27,23,33),(18,26,24,32),(37,94,43,88),(38,93,44,87),(39,92,45,86),(40,91,46,85),(41,90,47,96),(42,89,48,95),(49,75,55,81),(50,74,56,80),(51,73,57,79),(52,84,58,78),(53,83,59,77),(54,82,60,76)]])
108 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6N | 6O | ··· | 6AI | 12A | ··· | 12AF | 12AG | ··· | 12AV |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | S3×C6 | C3×Dic6 |
| kernel | C2×C6×Dic6 | C6×Dic6 | Dic3×C2×C6 | C2×C6×C12 | C22×Dic6 | C2×Dic6 | C22×Dic3 | C22×C12 | C22×C12 | C62 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 2 | 1 | 2 | 24 | 4 | 2 | 1 | 4 | 6 | 1 | 2 | 8 | 8 | 12 | 2 | 16 |
Matrix representation of C2×C6×Dic6 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 | 8 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[10,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,10,0,0,0,0,0,4,0,0,0,0,0,12,1,0,0,0,11,1],[12,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,5,8,0,0,0,0,8] >;
C2×C6×Dic6 in GAP, Magma, Sage, TeX
C_2\times C_6\times {\rm Dic}_6 % in TeX
G:=Group("C2xC6xDic6"); // GroupNames label
G:=SmallGroup(288,988);
// by ID
G=gap.SmallGroup(288,988);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,336,1571,192,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations