direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Dic6, C12⋊3Q8, C42.3S3, C3⋊1(C4×Q8), C4.9(C4×S3), C6.1(C2×Q8), (C4×C12).5C2, C4○(C4⋊Dic3), (C2×C4).72D6, C12.19(C2×C4), C4○(Dic3⋊C4), C6.1(C4○D4), C6.1(C22×C4), (C2×C6).9C23, C2.1(C2×Dic6), C2.1(C4○D12), Dic3⋊C4.7C2, C4⋊Dic3.13C2, (C4×Dic3).7C2, Dic3.1(C2×C4), (C2×C12).84C22, (C2×Dic6).10C2, C22.8(C22×S3), (C2×Dic3).23C22, C2.4(S3×C2×C4), SmallGroup(96,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Dic6
G = < a,b,c | a4=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 122 in 70 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×Q8, Dic6, C2×Dic3, C2×C12, C4×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C4×Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, Dic6, C4×S3, C22×S3, C4×Q8, C2×Dic6, S3×C2×C4, C4○D12, C4×Dic6
►Regular action on 96 points
(1 83 87 45)(2 84 88 46)(3 73 89 47)(4 74 90 48)(5 75 91 37)(6 76 92 38)(7 77 93 39)(8 78 94 40)(9 79 95 41)(10 80 96 42)(11 81 85 43)(12 82 86 44)(13 56 69 26)(14 57 70 27)(15 58 71 28)(16 59 72 29)(17 60 61 30)(18 49 62 31)(19 50 63 32)(20 51 64 33)(21 52 65 34)(22 53 66 35)(23 54 67 36)(24 55 68 25)
(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 61 7 67)(2 72 8 66)(3 71 9 65)(4 70 10 64)(5 69 11 63)(6 68 12 62)(13 85 19 91)(14 96 20 90)(15 95 21 89)(16 94 22 88)(17 93 23 87)(18 92 24 86)(25 82 31 76)(26 81 32 75)(27 80 33 74)(28 79 34 73)(29 78 35 84)(30 77 36 83)(37 56 43 50)(38 55 44 49)(39 54 45 60)(40 53 46 59)(41 52 47 58)(42 51 48 57)
G:=sub<Sym(96)| (1,83,87,45)(2,84,88,46)(3,73,89,47)(4,74,90,48)(5,75,91,37)(6,76,92,38)(7,77,93,39)(8,78,94,40)(9,79,95,41)(10,80,96,42)(11,81,85,43)(12,82,86,44)(13,56,69,26)(14,57,70,27)(15,58,71,28)(16,59,72,29)(17,60,61,30)(18,49,62,31)(19,50,63,32)(20,51,64,33)(21,52,65,34)(22,53,66,35)(23,54,67,36)(24,55,68,25), (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), (1,61,7,67)(2,72,8,66)(3,71,9,65)(4,70,10,64)(5,69,11,63)(6,68,12,62)(13,85,19,91)(14,96,20,90)(15,95,21,89)(16,94,22,88)(17,93,23,87)(18,92,24,86)(25,82,31,76)(26,81,32,75)(27,80,33,74)(28,79,34,73)(29,78,35,84)(30,77,36,83)(37,56,43,50)(38,55,44,49)(39,54,45,60)(40,53,46,59)(41,52,47,58)(42,51,48,57)>;
G:=Group( (1,83,87,45)(2,84,88,46)(3,73,89,47)(4,74,90,48)(5,75,91,37)(6,76,92,38)(7,77,93,39)(8,78,94,40)(9,79,95,41)(10,80,96,42)(11,81,85,43)(12,82,86,44)(13,56,69,26)(14,57,70,27)(15,58,71,28)(16,59,72,29)(17,60,61,30)(18,49,62,31)(19,50,63,32)(20,51,64,33)(21,52,65,34)(22,53,66,35)(23,54,67,36)(24,55,68,25), (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), (1,61,7,67)(2,72,8,66)(3,71,9,65)(4,70,10,64)(5,69,11,63)(6,68,12,62)(13,85,19,91)(14,96,20,90)(15,95,21,89)(16,94,22,88)(17,93,23,87)(18,92,24,86)(25,82,31,76)(26,81,32,75)(27,80,33,74)(28,79,34,73)(29,78,35,84)(30,77,36,83)(37,56,43,50)(38,55,44,49)(39,54,45,60)(40,53,46,59)(41,52,47,58)(42,51,48,57) );
G=PermutationGroup([[(1,83,87,45),(2,84,88,46),(3,73,89,47),(4,74,90,48),(5,75,91,37),(6,76,92,38),(7,77,93,39),(8,78,94,40),(9,79,95,41),(10,80,96,42),(11,81,85,43),(12,82,86,44),(13,56,69,26),(14,57,70,27),(15,58,71,28),(16,59,72,29),(17,60,61,30),(18,49,62,31),(19,50,63,32),(20,51,64,33),(21,52,65,34),(22,53,66,35),(23,54,67,36),(24,55,68,25)], [(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)], [(1,61,7,67),(2,72,8,66),(3,71,9,65),(4,70,10,64),(5,69,11,63),(6,68,12,62),(13,85,19,91),(14,96,20,90),(15,95,21,89),(16,94,22,88),(17,93,23,87),(18,92,24,86),(25,82,31,76),(26,81,32,75),(27,80,33,74),(28,79,34,73),(29,78,35,84),(30,77,36,83),(37,56,43,50),(38,55,44,49),(39,54,45,60),(40,53,46,59),(41,52,47,58),(42,51,48,57)]])
C4×Dic6 is a maximal subgroup of
C4.8Dic12 Dic6⋊2C8 C24⋊12Q8 C24⋊Q8 C42.16D6 Dic12⋊C4 C42.27D6 Dic6.3Q8 Dic6⋊C8 C42.198D6 C42.36D6 Dic6⋊8D4 C4⋊Dic12 Dic6⋊3Q8 Dic6⋊4Q8 C42.51D6 C42.59D6 C42.61D6 Dic6.4Q8 Dic6⋊9D4 C12⋊Q16 Dic6⋊5Q8 Dic6⋊6Q8 C42.274D6 C42.277D6 C42.87D6 C42.88D6 C42.89D6 C42.91D6 C42.93D6 C42.96D6 C42.98D6 C42.99D6 C42.102D6 D4⋊5Dic6 C42.105D6 C42.106D6 D4⋊6Dic6 C42.108D6 Dic6⋊23D4 Dic6⋊24D4 C42.229D6 C42.114D6 C42.115D6 Dic6⋊10Q8 C42.122D6 Q8⋊6Dic6 Q8⋊7Dic6 C4×S3×Q8 C42.125D6 C42.232D6 C42.134D6 C42.135D6 C42.136D6 C42.137D6 C42.139D6 Dic6⋊10D4 C42.143D6 Dic6⋊7Q8 D12⋊7Q8 C42.152D6 C42.154D6 C42.159D6 C42.160D6 C42.162D6 C42.164D6 C42.166D6 Dic6⋊11D4 Dic6⋊8Q8 Dic6⋊9Q8 D12⋊8Q8 D12⋊9Q8 C42.177D6 Dic3⋊5Dic6 Dic3⋊6Dic6 Dic5⋊5Dic6 Dic30⋊17C4
C4×Dic6 is a maximal quotient of
(C2×C12)⋊Q8 C6.(C4×Q8) C2.(C4×Dic6) Dic3⋊C4⋊C4 C24⋊12Q8 C24⋊Q8 C12⋊4(C4⋊C4) (C2×Dic6)⋊7C4 (C2×C42).6S3 Dic3⋊5Dic6 Dic3⋊6Dic6 Dic5⋊5Dic6 Dic30⋊17C4
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | 6B | 6C | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D6 | C4○D4 | Dic6 | C4×S3 | C4○D12 |
| kernel | C4×Dic6 | C4×Dic3 | Dic3⋊C4 | C4⋊Dic3 | C4×C12 | C2×Dic6 | Dic6 | C42 | C12 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 3 | 2 | 4 | 4 | 4 |
Matrix representation of C4×Dic6 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 11 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 0 | 1 | 0 |
| 8 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,1,0,0,11,1,0,0,0,0,1,1,0,0,12,0],[8,5,0,0,0,5,0,0,0,0,0,1,0,0,1,0] >;
C4×Dic6 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_6 % in TeX
G:=Group("C4xDic6"); // GroupNames label
G:=SmallGroup(96,75);
// by ID
G=gap.SmallGroup(96,75);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,103,50,2309]);
// Polycyclic
G:=Group<a,b,c|a^4=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations