metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.43D6, C12⋊C8⋊6C2, C4.82(C2×D12), C4⋊C4.5Dic3, C12.18(C4⋊C4), (C2×C12).23Q8, C12.84(C2×Q8), C6.36(C8○D4), C12.302(C2×D4), (C2×C4).145D12, (C2×C12).143D4, C4.5(C4⋊Dic3), (C2×C4).33Dic6, C4.49(C2×Dic6), (C4×C12).18C22, C42⋊C2.7S3, (C22×C4).354D6, C22⋊C4.2Dic3, (C2×C12).846C23, C2.4(D4.Dic3), C22.5(C4⋊Dic3), C23.20(C2×Dic3), C3⋊2(C42.6C22), (C22×C12).149C22, C22.43(C22×Dic3), (C3×C4⋊C4).8C4, C6.41(C2×C4⋊C4), (C22×C3⋊C8).8C2, C2.9(C2×C4⋊Dic3), (C2×C6).13(C4⋊C4), (C2×C12).90(C2×C4), (C3×C22⋊C4).3C4, (C2×C3⋊C8).313C22, (C22×C6).57(C2×C4), (C2×C4).43(C2×Dic3), (C2×C6).183(C22×C4), (C2×C4).788(C22×S3), (C3×C42⋊C2).8C2, (C2×C4.Dic3).18C2, SmallGroup(192,558)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.43D6
G = < a,b,c,d | a4=b4=c6=1, d2=b, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 200 in 114 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C3⋊C8, C2×C12, C2×C12, C22×C6, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C42.6C22, C12⋊C8, C22×C3⋊C8, C2×C4.Dic3, C3×C42⋊C2, C42.43D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C8○D4, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C42.6C22, C2×C4⋊Dic3, D4.Dic3, C42.43D6
►On 96 points
(1 13 41 26)(2 27 42 14)(3 15 43 28)(4 29 44 16)(5 9 45 30)(6 31 46 10)(7 11 47 32)(8 25 48 12)(17 38 51 71)(18 72 52 39)(19 40 53 65)(20 66 54 33)(21 34 55 67)(22 68 56 35)(23 36 49 69)(24 70 50 37)(57 91 86 74)(58 75 87 92)(59 93 88 76)(60 77 81 94)(61 95 82 78)(62 79 83 96)(63 89 84 80)(64 73 85 90)
(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 22 95)(2 96 23)(3 24 89)(4 90 17)(5 18 91)(6 92 19)(7 20 93)(8 94 21)(9 68 86 13 72 82)(10 83 65 14 87 69)(11 70 88 15 66 84)(12 85 67 16 81 71)(25 64 34 29 60 38)(26 39 61 30 35 57)(27 58 36 31 62 40)(28 33 63 32 37 59)(41 56 78)(42 79 49)(43 50 80)(44 73 51)(45 52 74)(46 75 53)(47 54 76)(48 77 55)
(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)| (1,13,41,26)(2,27,42,14)(3,15,43,28)(4,29,44,16)(5,9,45,30)(6,31,46,10)(7,11,47,32)(8,25,48,12)(17,38,51,71)(18,72,52,39)(19,40,53,65)(20,66,54,33)(21,34,55,67)(22,68,56,35)(23,36,49,69)(24,70,50,37)(57,91,86,74)(58,75,87,92)(59,93,88,76)(60,77,81,94)(61,95,82,78)(62,79,83,96)(63,89,84,80)(64,73,85,90), (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), (1,22,95)(2,96,23)(3,24,89)(4,90,17)(5,18,91)(6,92,19)(7,20,93)(8,94,21)(9,68,86,13,72,82)(10,83,65,14,87,69)(11,70,88,15,66,84)(12,85,67,16,81,71)(25,64,34,29,60,38)(26,39,61,30,35,57)(27,58,36,31,62,40)(28,33,63,32,37,59)(41,56,78)(42,79,49)(43,50,80)(44,73,51)(45,52,74)(46,75,53)(47,54,76)(48,77,55), (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( (1,13,41,26)(2,27,42,14)(3,15,43,28)(4,29,44,16)(5,9,45,30)(6,31,46,10)(7,11,47,32)(8,25,48,12)(17,38,51,71)(18,72,52,39)(19,40,53,65)(20,66,54,33)(21,34,55,67)(22,68,56,35)(23,36,49,69)(24,70,50,37)(57,91,86,74)(58,75,87,92)(59,93,88,76)(60,77,81,94)(61,95,82,78)(62,79,83,96)(63,89,84,80)(64,73,85,90), (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), (1,22,95)(2,96,23)(3,24,89)(4,90,17)(5,18,91)(6,92,19)(7,20,93)(8,94,21)(9,68,86,13,72,82)(10,83,65,14,87,69)(11,70,88,15,66,84)(12,85,67,16,81,71)(25,64,34,29,60,38)(26,39,61,30,35,57)(27,58,36,31,62,40)(28,33,63,32,37,59)(41,56,78)(42,79,49)(43,50,80)(44,73,51)(45,52,74)(46,75,53)(47,54,76)(48,77,55), (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([[(1,13,41,26),(2,27,42,14),(3,15,43,28),(4,29,44,16),(5,9,45,30),(6,31,46,10),(7,11,47,32),(8,25,48,12),(17,38,51,71),(18,72,52,39),(19,40,53,65),(20,66,54,33),(21,34,55,67),(22,68,56,35),(23,36,49,69),(24,70,50,37),(57,91,86,74),(58,75,87,92),(59,93,88,76),(60,77,81,94),(61,95,82,78),(62,79,83,96),(63,89,84,80),(64,73,85,90)], [(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)], [(1,22,95),(2,96,23),(3,24,89),(4,90,17),(5,18,91),(6,92,19),(7,20,93),(8,94,21),(9,68,86,13,72,82),(10,83,65,14,87,69),(11,70,88,15,66,84),(12,85,67,16,81,71),(25,64,34,29,60,38),(26,39,61,30,35,57),(27,58,36,31,62,40),(28,33,63,32,37,59),(41,56,78),(42,79,49),(43,50,80),(44,73,51),(45,52,74),(46,75,53),(47,54,76),(48,77,55)], [(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 | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | D6 | Dic3 | Dic3 | D6 | Dic6 | D12 | C8○D4 | D4.Dic3 |
| kernel | C42.43D6 | C12⋊C8 | C22×C3⋊C8 | C2×C4.Dic3 | C3×C42⋊C2 | C3×C22⋊C4 | C3×C4⋊C4 | C42⋊C2 | C2×C12 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 4 |
Matrix representation of C42.43D6 ►in GL4(𝔽73) generated by
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 21 | 70 |
| 0 | 0 | 50 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 14 | 72 |
| 63 | 41 | 0 | 0 |
| 51 | 10 | 0 | 0 |
| 0 | 0 | 51 | 0 |
| 0 | 0 | 57 | 22 |
G:=sub<GL(4,GF(73))| [66,14,0,0,59,7,0,0,0,0,21,50,0,0,70,52],[1,0,0,0,0,1,0,0,0,0,46,0,0,0,0,46],[72,1,0,0,72,0,0,0,0,0,1,14,0,0,0,72],[63,51,0,0,41,10,0,0,0,0,51,57,0,0,0,22] >;
C42.43D6 in GAP, Magma, Sage, TeX
C_4^2._{43}D_6 % in TeX
G:=Group("C4^2.43D6"); // GroupNames label
G:=SmallGroup(192,558);
// by ID
G=gap.SmallGroup(192,558);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,100,136,6278]);
// 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*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations