metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.7(C4⋊C4), (C2×C12).5Q8, (C2×C4).12D12, C4.Dic3⋊2C4, (C2×C4).1Dic6, C4.32(D6⋊C4), (C2×C12).103D4, (C22×C12).3C4, (C2×C6).18C42, (C22×C4).72D6, C12.8(C22⋊C4), C4.7(Dic3⋊C4), C6.7(C4.D4), C22.9(C4×Dic3), (C22×C4).5Dic3, C3⋊1(C22.C42), C2.2(C12.D4), C6.7(C4.10D4), C22.9(C4⋊Dic3), C23.28(C2×Dic3), C2.2(C12.10D4), C6.6(C2.C42), C2.7(C6.C42), (C22×C12).119C22, C22.27(C6.D4), (C6×C4⋊C4).2C2, (C2×C4⋊C4).3S3, (C2×C4).18(C4×S3), (C2×C6).36(C4⋊C4), (C2×C12).55(C2×C4), (C2×C4.Dic3).6C2, (C2×C4).175(C3⋊D4), (C2×C6).88(C22⋊C4), (C22×C6).125(C2×C4), SmallGroup(192,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.(C4⋊C4)
G = < a,b,c | a12=c4=1, b4=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b3 >
Subgroups: 200 in 98 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C3×C4⋊C4, C22×C12, C22×C12, C22.C42, C2×C4.Dic3, C6×C4⋊C4, C12.(C4⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4.D4, C4.10D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C22.C42, C6.C42, C12.D4, C12.10D4, C12.(C4⋊C4)
►On 96 points
(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 80 34 64 7 74 28 70)(2 79 35 63 8 73 29 69)(3 78 36 62 9 84 30 68)(4 77 25 61 10 83 31 67)(5 76 26 72 11 82 32 66)(6 75 27 71 12 81 33 65)(13 44 91 57 19 38 85 51)(14 43 92 56 20 37 86 50)(15 42 93 55 21 48 87 49)(16 41 94 54 22 47 88 60)(17 40 95 53 23 46 89 59)(18 39 96 52 24 45 90 58)
(1 38 25 54)(2 45 26 49)(3 40 27 56)(4 47 28 51)(5 42 29 58)(6 37 30 53)(7 44 31 60)(8 39 32 55)(9 46 33 50)(10 41 34 57)(11 48 35 52)(12 43 36 59)(13 64 88 83)(14 71 89 78)(15 66 90 73)(16 61 91 80)(17 68 92 75)(18 63 93 82)(19 70 94 77)(20 65 95 84)(21 72 96 79)(22 67 85 74)(23 62 86 81)(24 69 87 76)
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), (1,80,34,64,7,74,28,70)(2,79,35,63,8,73,29,69)(3,78,36,62,9,84,30,68)(4,77,25,61,10,83,31,67)(5,76,26,72,11,82,32,66)(6,75,27,71,12,81,33,65)(13,44,91,57,19,38,85,51)(14,43,92,56,20,37,86,50)(15,42,93,55,21,48,87,49)(16,41,94,54,22,47,88,60)(17,40,95,53,23,46,89,59)(18,39,96,52,24,45,90,58), (1,38,25,54)(2,45,26,49)(3,40,27,56)(4,47,28,51)(5,42,29,58)(6,37,30,53)(7,44,31,60)(8,39,32,55)(9,46,33,50)(10,41,34,57)(11,48,35,52)(12,43,36,59)(13,64,88,83)(14,71,89,78)(15,66,90,73)(16,61,91,80)(17,68,92,75)(18,63,93,82)(19,70,94,77)(20,65,95,84)(21,72,96,79)(22,67,85,74)(23,62,86,81)(24,69,87,76)>;
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), (1,80,34,64,7,74,28,70)(2,79,35,63,8,73,29,69)(3,78,36,62,9,84,30,68)(4,77,25,61,10,83,31,67)(5,76,26,72,11,82,32,66)(6,75,27,71,12,81,33,65)(13,44,91,57,19,38,85,51)(14,43,92,56,20,37,86,50)(15,42,93,55,21,48,87,49)(16,41,94,54,22,47,88,60)(17,40,95,53,23,46,89,59)(18,39,96,52,24,45,90,58), (1,38,25,54)(2,45,26,49)(3,40,27,56)(4,47,28,51)(5,42,29,58)(6,37,30,53)(7,44,31,60)(8,39,32,55)(9,46,33,50)(10,41,34,57)(11,48,35,52)(12,43,36,59)(13,64,88,83)(14,71,89,78)(15,66,90,73)(16,61,91,80)(17,68,92,75)(18,63,93,82)(19,70,94,77)(20,65,95,84)(21,72,96,79)(22,67,85,74)(23,62,86,81)(24,69,87,76) );
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)], [(1,80,34,64,7,74,28,70),(2,79,35,63,8,73,29,69),(3,78,36,62,9,84,30,68),(4,77,25,61,10,83,31,67),(5,76,26,72,11,82,32,66),(6,75,27,71,12,81,33,65),(13,44,91,57,19,38,85,51),(14,43,92,56,20,37,86,50),(15,42,93,55,21,48,87,49),(16,41,94,54,22,47,88,60),(17,40,95,53,23,46,89,59),(18,39,96,52,24,45,90,58)], [(1,38,25,54),(2,45,26,49),(3,40,27,56),(4,47,28,51),(5,42,29,58),(6,37,30,53),(7,44,31,60),(8,39,32,55),(9,46,33,50),(10,41,34,57),(11,48,35,52),(12,43,36,59),(13,64,88,83),(14,71,89,78),(15,66,90,73),(16,61,91,80),(17,68,92,75),(18,63,93,82),(19,70,94,77),(20,65,95,84),(21,72,96,79),(22,67,85,74),(23,62,86,81),(24,69,87,76)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | C4.D4 | C4.10D4 | C12.D4 | C12.10D4 |
| kernel | C12.(C4⋊C4) | C2×C4.Dic3 | C6×C4⋊C4 | C4.Dic3 | C22×C12 | C2×C4⋊C4 | C2×C12 | C2×C12 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 3 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C12.(C4⋊C4) ►in GL6(𝔽73)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 20 | 34 | 0 | 1 |
| 0 | 0 | 39 | 20 | 72 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 11 | 2 | 0 |
| 0 | 0 | 51 | 63 | 0 | 2 |
| 0 | 0 | 72 | 28 | 68 | 62 |
| 0 | 0 | 54 | 71 | 22 | 10 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 8 | 0 | 0 |
| 0 | 0 | 8 | 64 | 0 | 0 |
| 0 | 0 | 72 | 20 | 64 | 65 |
| 0 | 0 | 20 | 27 | 65 | 9 |
G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,64,0,0,0,0,0,0,0,1,20,39,0,0,72,0,34,20,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,5,51,72,54,0,0,11,63,28,71,0,0,2,0,68,22,0,0,0,2,62,10],[27,0,0,0,0,0,0,46,0,0,0,0,0,0,9,8,72,20,0,0,8,64,20,27,0,0,0,0,64,65,0,0,0,0,65,9] >;
C12.(C4⋊C4) in GAP, Magma, Sage, TeX
C_{12}.(C_4\rtimes C_4) % in TeX
G:=Group("C12.(C4:C4)"); // GroupNames label
G:=SmallGroup(192,89);
// by ID
G=gap.SmallGroup(192,89);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,184,1684,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=c^4=1,b^4=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b^3>;
// generators/relations