direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.47D4, M4(2).32D6, (C2×C4).53D12, C4.67(C2×D12), C4.29(D6⋊C4), (C2×C12).176D4, C12.422(C2×D4), C23.61(C4×S3), C6⋊1(C4.10D4), (C2×Dic6).15C4, (C22×C4).160D6, C12.54(C22⋊C4), (C2×C12).418C23, C22.51(D6⋊C4), (C6×M4(2)).28C2, (C2×M4(2)).17S3, (C22×Dic3).5C4, (C22×Dic6).15C2, C4.Dic3.43C22, (C22×C12).191C22, (C2×Dic6).279C22, (C3×M4(2)).35C22, (C2×C4).55(C4×S3), C2.33(C2×D6⋊C4), C3⋊2(C2×C4.10D4), C22.22(S3×C2×C4), C4.115(C2×C3⋊D4), C6.61(C2×C22⋊C4), (C2×C12).111(C2×C4), (C2×C6).16(C22×C4), (C22×C6).72(C2×C4), (C2×Dic3).6(C2×C4), (C2×C4).257(C3⋊D4), (C2×C6).67(C22⋊C4), (C2×C4).122(C22×S3), (C2×C4.Dic3).26C2, SmallGroup(192,695)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.47D4
G = < a,b,c,d | a2=b12=1, c4=d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c3 >
Subgroups: 344 in 146 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C4.10D4, C2×M4(2), C2×M4(2), C22×Q8, C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C2×C4.10D4, C12.47D4, C2×C4.Dic3, C6×M4(2), C22×Dic6, C2×C12.47D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C4.10D4, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C4.10D4, C12.47D4, C2×D6⋊C4, C2×C12.47D4
►On 96 points
(1 72)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(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 28 69 42 7 34 63 48)(2 27 70 41 8 33 64 47)(3 26 71 40 9 32 65 46)(4 25 72 39 10 31 66 45)(5 36 61 38 11 30 67 44)(6 35 62 37 12 29 68 43)(13 79 93 58 19 73 87 52)(14 78 94 57 20 84 88 51)(15 77 95 56 21 83 89 50)(16 76 96 55 22 82 90 49)(17 75 85 54 23 81 91 60)(18 74 86 53 24 80 92 59)
(1 56 7 50)(2 55 8 49)(3 54 9 60)(4 53 10 59)(5 52 11 58)(6 51 12 57)(13 47 19 41)(14 46 20 40)(15 45 21 39)(16 44 22 38)(17 43 23 37)(18 42 24 48)(25 95 31 89)(26 94 32 88)(27 93 33 87)(28 92 34 86)(29 91 35 85)(30 90 36 96)(61 73 67 79)(62 84 68 78)(63 83 69 77)(64 82 70 76)(65 81 71 75)(66 80 72 74)
G:=sub<Sym(96)| (1,72)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,48)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (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), (1,28,69,42,7,34,63,48)(2,27,70,41,8,33,64,47)(3,26,71,40,9,32,65,46)(4,25,72,39,10,31,66,45)(5,36,61,38,11,30,67,44)(6,35,62,37,12,29,68,43)(13,79,93,58,19,73,87,52)(14,78,94,57,20,84,88,51)(15,77,95,56,21,83,89,50)(16,76,96,55,22,82,90,49)(17,75,85,54,23,81,91,60)(18,74,86,53,24,80,92,59), (1,56,7,50)(2,55,8,49)(3,54,9,60)(4,53,10,59)(5,52,11,58)(6,51,12,57)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,42,24,48)(25,95,31,89)(26,94,32,88)(27,93,33,87)(28,92,34,86)(29,91,35,85)(30,90,36,96)(61,73,67,79)(62,84,68,78)(63,83,69,77)(64,82,70,76)(65,81,71,75)(66,80,72,74)>;
G:=Group( (1,72)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,48)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (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), (1,28,69,42,7,34,63,48)(2,27,70,41,8,33,64,47)(3,26,71,40,9,32,65,46)(4,25,72,39,10,31,66,45)(5,36,61,38,11,30,67,44)(6,35,62,37,12,29,68,43)(13,79,93,58,19,73,87,52)(14,78,94,57,20,84,88,51)(15,77,95,56,21,83,89,50)(16,76,96,55,22,82,90,49)(17,75,85,54,23,81,91,60)(18,74,86,53,24,80,92,59), (1,56,7,50)(2,55,8,49)(3,54,9,60)(4,53,10,59)(5,52,11,58)(6,51,12,57)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,42,24,48)(25,95,31,89)(26,94,32,88)(27,93,33,87)(28,92,34,86)(29,91,35,85)(30,90,36,96)(61,73,67,79)(62,84,68,78)(63,83,69,77)(64,82,70,76)(65,81,71,75)(66,80,72,74) );
G=PermutationGroup([[(1,72),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,85),(21,86),(22,87),(23,88),(24,89),(25,48),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)], [(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)], [(1,28,69,42,7,34,63,48),(2,27,70,41,8,33,64,47),(3,26,71,40,9,32,65,46),(4,25,72,39,10,31,66,45),(5,36,61,38,11,30,67,44),(6,35,62,37,12,29,68,43),(13,79,93,58,19,73,87,52),(14,78,94,57,20,84,88,51),(15,77,95,56,21,83,89,50),(16,76,96,55,22,82,90,49),(17,75,85,54,23,81,91,60),(18,74,86,53,24,80,92,59)], [(1,56,7,50),(2,55,8,49),(3,54,9,60),(4,53,10,59),(5,52,11,58),(6,51,12,57),(13,47,19,41),(14,46,20,40),(15,45,21,39),(16,44,22,38),(17,43,23,37),(18,42,24,48),(25,95,31,89),(26,94,32,88),(27,93,33,87),(28,92,34,86),(29,91,35,85),(30,90,36,96),(61,73,67,79),(62,84,68,78),(63,83,69,77),(64,82,70,76),(65,81,71,75),(66,80,72,74)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C4×S3 | C4.10D4 | C12.47D4 |
| kernel | C2×C12.47D4 | C12.47D4 | C2×C4.Dic3 | C6×M4(2) | C22×Dic6 | C2×Dic6 | C22×Dic3 | C2×M4(2) | C2×C12 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 |
Matrix representation of C2×C12.47D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 48 | 44 | 0 | 0 | 0 | 0 |
| 19 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 64 |
| 0 | 0 | 0 | 0 | 64 | 8 |
| 0 | 0 | 9 | 65 | 0 | 0 |
| 0 | 0 | 65 | 64 | 0 | 0 |
| 25 | 29 | 0 | 0 | 0 | 0 |
| 54 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 64 | 0 | 0 |
| 0 | 0 | 64 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 9 |
| 0 | 0 | 0 | 0 | 9 | 65 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[48,19,0,0,0,0,44,25,0,0,0,0,0,0,0,0,9,65,0,0,0,0,65,64,0,0,65,64,0,0,0,0,64,8,0,0],[25,54,0,0,0,0,29,48,0,0,0,0,0,0,65,64,0,0,0,0,64,8,0,0,0,0,0,0,8,9,0,0,0,0,9,65] >;
C2×C12.47D4 in GAP, Magma, Sage, TeX
C_2\times C_{12}._{47}D_4 % in TeX
G:=Group("C2xC12.47D4"); // GroupNames label
G:=SmallGroup(192,695);
// by ID
G=gap.SmallGroup(192,695);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,422,58,1123,136,438,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=1,c^4=d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^3>;
// generators/relations