metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.94D6, C6.512- 1+4, C4⋊C4.271D6, C42⋊2S3⋊2C2, C4.D12⋊12C2, (C2×C6).73C24, C2.9(Q8○D12), C22⋊C4.97D6, C12.6Q8⋊6C2, C42⋊C2⋊13S3, D6.16(C4○D4), C4.95(C4○D12), (C4×C12).24C22, D6⋊C4.96C22, C4.Dic6⋊12C2, C23.9D6.1C2, (C22×C4).210D6, C23.8D6⋊3C2, Dic6⋊C4⋊12C2, C12.197(C4○D4), C12.48D4⋊29C2, (C2×C12).148C23, C4⋊Dic3.34C22, C23.95(C22×S3), Dic3⋊C4.97C22, C22.102(S3×C23), (C22×C6).143C23, (C2×Dic3).26C23, (C4×Dic3).69C22, (C22×S3).166C23, (C22×C12).231C22, C3⋊2(C22.46C24), (C2×Dic6).143C22, C6.D4.96C22, (S3×C4⋊C4)⋊12C2, C6.30(C2×C4○D4), C2.12(S3×C4○D4), (C4×C3⋊D4).5C2, C2.32(C2×C4○D12), (S3×C2×C4).60C22, (C3×C42⋊C2)⋊15C2, (C3×C4⋊C4).309C22, (C2×C4).275(C22×S3), (C2×C3⋊D4).103C22, (C3×C22⋊C4).113C22, SmallGroup(192,1088)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.94D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c5 >
Subgroups: 472 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×C3⋊D4, C22×C12, C22.46C24, C12.6Q8, C42⋊2S3, C23.8D6, C23.9D6, Dic6⋊C4, C4.Dic6, S3×C4⋊C4, C4.D12, C12.48D4, C4×C3⋊D4, C3×C42⋊C2, C42.94D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, S3×C23, C22.46C24, C2×C4○D12, S3×C4○D4, Q8○D12, C42.94D6
►On 96 points
(1 69 88 53)(2 64 89 60)(3 71 90 55)(4 66 91 50)(5 61 92 57)(6 68 93 52)(7 63 94 59)(8 70 95 54)(9 65 96 49)(10 72 85 56)(11 67 86 51)(12 62 87 58)(13 80 30 40)(14 75 31 47)(15 82 32 42)(16 77 33 37)(17 84 34 44)(18 79 35 39)(19 74 36 46)(20 81 25 41)(21 76 26 48)(22 83 27 43)(23 78 28 38)(24 73 29 45)
(1 39 7 45)(2 40 8 46)(3 41 9 47)(4 42 10 48)(5 43 11 37)(6 44 12 38)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 61 22 67)(17 62 23 68)(18 63 24 69)(25 49 31 55)(26 50 32 56)(27 51 33 57)(28 52 34 58)(29 53 35 59)(30 54 36 60)(73 88 79 94)(74 89 80 95)(75 90 81 96)(76 91 82 85)(77 92 83 86)(78 93 84 87)
(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 12 7 6)(2 5 8 11)(3 10 9 4)(13 33 19 27)(14 26 20 32)(15 31 21 25)(16 36 22 30)(17 29 23 35)(18 34 24 28)(37 80 43 74)(38 73 44 79)(39 78 45 84)(40 83 46 77)(41 76 47 82)(42 81 48 75)(49 56 55 50)(51 54 57 60)(52 59 58 53)(61 64 67 70)(62 69 68 63)(65 72 71 66)(85 96 91 90)(86 89 92 95)(87 94 93 88)
G:=sub<Sym(96)| (1,69,88,53)(2,64,89,60)(3,71,90,55)(4,66,91,50)(5,61,92,57)(6,68,93,52)(7,63,94,59)(8,70,95,54)(9,65,96,49)(10,72,85,56)(11,67,86,51)(12,62,87,58)(13,80,30,40)(14,75,31,47)(15,82,32,42)(16,77,33,37)(17,84,34,44)(18,79,35,39)(19,74,36,46)(20,81,25,41)(21,76,26,48)(22,83,27,43)(23,78,28,38)(24,73,29,45), (1,39,7,45)(2,40,8,46)(3,41,9,47)(4,42,10,48)(5,43,11,37)(6,44,12,38)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,49,31,55)(26,50,32,56)(27,51,33,57)(28,52,34,58)(29,53,35,59)(30,54,36,60)(73,88,79,94)(74,89,80,95)(75,90,81,96)(76,91,82,85)(77,92,83,86)(78,93,84,87), (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), (1,12,7,6)(2,5,8,11)(3,10,9,4)(13,33,19,27)(14,26,20,32)(15,31,21,25)(16,36,22,30)(17,29,23,35)(18,34,24,28)(37,80,43,74)(38,73,44,79)(39,78,45,84)(40,83,46,77)(41,76,47,82)(42,81,48,75)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,64,67,70)(62,69,68,63)(65,72,71,66)(85,96,91,90)(86,89,92,95)(87,94,93,88)>;
G:=Group( (1,69,88,53)(2,64,89,60)(3,71,90,55)(4,66,91,50)(5,61,92,57)(6,68,93,52)(7,63,94,59)(8,70,95,54)(9,65,96,49)(10,72,85,56)(11,67,86,51)(12,62,87,58)(13,80,30,40)(14,75,31,47)(15,82,32,42)(16,77,33,37)(17,84,34,44)(18,79,35,39)(19,74,36,46)(20,81,25,41)(21,76,26,48)(22,83,27,43)(23,78,28,38)(24,73,29,45), (1,39,7,45)(2,40,8,46)(3,41,9,47)(4,42,10,48)(5,43,11,37)(6,44,12,38)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,49,31,55)(26,50,32,56)(27,51,33,57)(28,52,34,58)(29,53,35,59)(30,54,36,60)(73,88,79,94)(74,89,80,95)(75,90,81,96)(76,91,82,85)(77,92,83,86)(78,93,84,87), (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), (1,12,7,6)(2,5,8,11)(3,10,9,4)(13,33,19,27)(14,26,20,32)(15,31,21,25)(16,36,22,30)(17,29,23,35)(18,34,24,28)(37,80,43,74)(38,73,44,79)(39,78,45,84)(40,83,46,77)(41,76,47,82)(42,81,48,75)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,64,67,70)(62,69,68,63)(65,72,71,66)(85,96,91,90)(86,89,92,95)(87,94,93,88) );
G=PermutationGroup([[(1,69,88,53),(2,64,89,60),(3,71,90,55),(4,66,91,50),(5,61,92,57),(6,68,93,52),(7,63,94,59),(8,70,95,54),(9,65,96,49),(10,72,85,56),(11,67,86,51),(12,62,87,58),(13,80,30,40),(14,75,31,47),(15,82,32,42),(16,77,33,37),(17,84,34,44),(18,79,35,39),(19,74,36,46),(20,81,25,41),(21,76,26,48),(22,83,27,43),(23,78,28,38),(24,73,29,45)], [(1,39,7,45),(2,40,8,46),(3,41,9,47),(4,42,10,48),(5,43,11,37),(6,44,12,38),(13,70,19,64),(14,71,20,65),(15,72,21,66),(16,61,22,67),(17,62,23,68),(18,63,24,69),(25,49,31,55),(26,50,32,56),(27,51,33,57),(28,52,34,58),(29,53,35,59),(30,54,36,60),(73,88,79,94),(74,89,80,95),(75,90,81,96),(76,91,82,85),(77,92,83,86),(78,93,84,87)], [(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)], [(1,12,7,6),(2,5,8,11),(3,10,9,4),(13,33,19,27),(14,26,20,32),(15,31,21,25),(16,36,22,30),(17,29,23,35),(18,34,24,28),(37,80,43,74),(38,73,44,79),(39,78,45,84),(40,83,46,77),(41,76,47,82),(42,81,48,75),(49,56,55,50),(51,54,57,60),(52,59,58,53),(61,64,67,70),(62,69,68,63),(65,72,71,66),(85,96,91,90),(86,89,92,95),(87,94,93,88)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- 1+4 | S3×C4○D4 | Q8○D12 |
| kernel | C42.94D6 | C12.6Q8 | C42⋊2S3 | C23.8D6 | C23.9D6 | Dic6⋊C4 | C4.Dic6 | S3×C4⋊C4 | C4.D12 | C12.48D4 | C4×C3⋊D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | D6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.94D6 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 12 |
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 12 | 5 |
| 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 1 | 8 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,1,0,0,0,3,12],[11,4,0,0,9,2,0,0,0,0,5,0,0,0,0,5],[12,1,0,0,12,0,0,0,0,0,8,12,0,0,0,5],[12,0,0,0,12,1,0,0,0,0,5,1,0,0,0,8] >;
C42.94D6 in GAP, Magma, Sage, TeX
C_4^2._{94}D_6 % in TeX
G:=Group("C4^2.94D6"); // GroupNames label
G:=SmallGroup(192,1088);
// by ID
G=gap.SmallGroup(192,1088);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations