metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.47D6, C3⋊C8⋊28D4, C3⋊5(C8⋊9D4), (C4×D4).5S3, C6.91(C4×D4), (C6×D4).15C4, (D4×C12).6C2, C12⋊C8⋊21C2, C4.215(S3×D4), (C2×C6)⋊3M4(2), C4⋊C4.7Dic3, C2.8(D4×Dic3), C6.39(C8○D4), C12.374(C2×D4), (C2×D4).5Dic3, (C4×C12).82C22, C42.S3⋊4C2, (C22×C4).326D6, C6.40(C2×M4(2)), C22⋊C4.4Dic3, C12.307(C4○D4), C12.55D4⋊25C2, (C2×C12).849C23, C2.6(D4.Dic3), C22⋊2(C4.Dic3), C4.134(D4⋊2S3), C23.22(C2×Dic3), (C22×C12).99C22, C22.45(C22×Dic3), (C3×C4⋊C4).11C4, (C22×C3⋊C8)⋊19C2, (C3×C22⋊C4).5C4, (C2×C4.Dic3)⋊4C2, (C2×C12).163(C2×C4), C2.8(C2×C4.Dic3), (C2×C3⋊C8).200C22, (C22×C6).60(C2×C4), (C2×C4).34(C2×Dic3), (C2×C4).791(C22×S3), (C2×C6).186(C22×C4), SmallGroup(192,570)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.47D6
G = < a,b,c,d | a4=b4=c6=1, d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 232 in 124 conjugacy classes, 61 normal (55 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C3×D4, C22×C6, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C8⋊9D4, C42.S3, C12⋊C8, C12.55D4, C22×C3⋊C8, C2×C4.Dic3, D4×C12, C42.47D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, M4(2), C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C2×M4(2), C8○D4, C4.Dic3, S3×D4, D4⋊2S3, C22×Dic3, C8⋊9D4, C2×C4.Dic3, D4×Dic3, D4.Dic3, C42.47D6
►On 96 points
(1 65 78 58)(2 63 79 70)(3 67 80 60)(4 57 73 72)(5 69 74 62)(6 59 75 66)(7 71 76 64)(8 61 77 68)(9 47 32 24)(10 21 25 44)(11 41 26 18)(12 23 27 46)(13 43 28 20)(14 17 29 48)(15 45 30 22)(16 19 31 42)(33 89 82 50)(34 55 83 94)(35 91 84 52)(36 49 85 96)(37 93 86 54)(38 51 87 90)(39 95 88 56)(40 53 81 92)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)
(1 36 15 58 49 22)(2 23 50 59 16 37)(3 38 9 60 51 24)(4 17 52 61 10 39)(5 40 11 62 53 18)(6 19 54 63 12 33)(7 34 13 64 55 20)(8 21 56 57 14 35)(25 88 73 48 91 68)(26 69 92 41 74 81)(27 82 75 42 93 70)(28 71 94 43 76 83)(29 84 77 44 95 72)(30 65 96 45 78 85)(31 86 79 46 89 66)(32 67 90 47 80 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)
G:=sub<Sym(96)| (1,65,78,58)(2,63,79,70)(3,67,80,60)(4,57,73,72)(5,69,74,62)(6,59,75,66)(7,71,76,64)(8,61,77,68)(9,47,32,24)(10,21,25,44)(11,41,26,18)(12,23,27,46)(13,43,28,20)(14,17,29,48)(15,45,30,22)(16,19,31,42)(33,89,82,50)(34,55,83,94)(35,91,84,52)(36,49,85,96)(37,93,86,54)(38,51,87,90)(39,95,88,56)(40,53,81,92), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (1,36,15,58,49,22)(2,23,50,59,16,37)(3,38,9,60,51,24)(4,17,52,61,10,39)(5,40,11,62,53,18)(6,19,54,63,12,33)(7,34,13,64,55,20)(8,21,56,57,14,35)(25,88,73,48,91,68)(26,69,92,41,74,81)(27,82,75,42,93,70)(28,71,94,43,76,83)(29,84,77,44,95,72)(30,65,96,45,78,85)(31,86,79,46,89,66)(32,67,90,47,80,87), (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)>;
G:=Group( (1,65,78,58)(2,63,79,70)(3,67,80,60)(4,57,73,72)(5,69,74,62)(6,59,75,66)(7,71,76,64)(8,61,77,68)(9,47,32,24)(10,21,25,44)(11,41,26,18)(12,23,27,46)(13,43,28,20)(14,17,29,48)(15,45,30,22)(16,19,31,42)(33,89,82,50)(34,55,83,94)(35,91,84,52)(36,49,85,96)(37,93,86,54)(38,51,87,90)(39,95,88,56)(40,53,81,92), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (1,36,15,58,49,22)(2,23,50,59,16,37)(3,38,9,60,51,24)(4,17,52,61,10,39)(5,40,11,62,53,18)(6,19,54,63,12,33)(7,34,13,64,55,20)(8,21,56,57,14,35)(25,88,73,48,91,68)(26,69,92,41,74,81)(27,82,75,42,93,70)(28,71,94,43,76,83)(29,84,77,44,95,72)(30,65,96,45,78,85)(31,86,79,46,89,66)(32,67,90,47,80,87), (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) );
G=PermutationGroup([[(1,65,78,58),(2,63,79,70),(3,67,80,60),(4,57,73,72),(5,69,74,62),(6,59,75,66),(7,71,76,64),(8,61,77,68),(9,47,32,24),(10,21,25,44),(11,41,26,18),(12,23,27,46),(13,43,28,20),(14,17,29,48),(15,45,30,22),(16,19,31,42),(33,89,82,50),(34,55,83,94),(35,91,84,52),(36,49,85,96),(37,93,86,54),(38,51,87,90),(39,95,88,56),(40,53,81,92)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96)], [(1,36,15,58,49,22),(2,23,50,59,16,37),(3,38,9,60,51,24),(4,17,52,61,10,39),(5,40,11,62,53,18),(6,19,54,63,12,33),(7,34,13,64,55,20),(8,21,56,57,14,35),(25,88,73,48,91,68),(26,69,92,41,74,81),(27,82,75,42,93,70),(28,71,94,43,76,83),(29,84,77,44,95,72),(30,65,96,45,78,85),(31,86,79,46,89,66),(32,67,90,47,80,87)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | Dic3 | Dic3 | D6 | Dic3 | C4○D4 | M4(2) | C8○D4 | C4.Dic3 | S3×D4 | D4⋊2S3 | D4.Dic3 |
| kernel | C42.47D6 | C42.S3 | C12⋊C8 | C12.55D4 | C22×C3⋊C8 | C2×C4.Dic3 | D4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C6×D4 | C4×D4 | C3⋊C8 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C6 | C6 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C42.47D6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 66 |
| 0 | 0 | 42 | 72 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 66 |
| 0 | 0 | 0 | 72 |
| 0 | 8 | 0 | 0 |
| 24 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 22 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,42,0,0,66,72],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,66,72],[0,24,0,0,8,0,0,0,0,0,22,0,0,0,0,22] >;
C42.47D6 in GAP, Magma, Sage, TeX
C_4^2._{47}D_6 % in TeX
G:=Group("C4^2.47D6"); // GroupNames label
G:=SmallGroup(192,570);
// by ID
G=gap.SmallGroup(192,570);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,219,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations