metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.48D6, D4⋊S3⋊5C4, D4⋊4(C4×S3), (C4×D4)⋊2S3, (D4×C12)⋊2C2, C6.70(C4×D4), (C4×D12)⋊20C2, D12⋊11(C2×C4), C3⋊4(D8⋊C4), C4⋊C4.242D6, (C2×D4).189D6, (C2×C12).254D4, C6.D8⋊29C2, C4.37(C4○D12), C12.49(C4○D4), D4⋊Dic3⋊11C2, C2.3(D4⋊D6), (C4×C12).85C22, C42.S3⋊5C2, C12.22(C22×C4), C12.Q8⋊32C2, C6.108(C8⋊C22), (C2×C12).336C23, C2.3(D12⋊6C22), (C6×D4).231C22, (C2×D12).237C22, C4⋊Dic3.327C22, C3⋊C8⋊8(C2×C4), C4.22(S3×C2×C4), (C3×D4)⋊9(C2×C4), (C2×D4⋊S3).4C2, C2.16(C4×C3⋊D4), (C2×C6).467(C2×D4), (C2×C3⋊C8).92C22, C22.76(C2×C3⋊D4), (C2×C4).217(C3⋊D4), (C3×C4⋊C4).273C22, (C2×C4).436(C22×S3), SmallGroup(192,573)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.48D6
G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c-1 >
Subgroups: 376 in 132 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C3⋊C8, C3⋊C8, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C4×D4, C2×D8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4⋊S3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×C12, C6×D4, D8⋊C4, C42.S3, C12.Q8, C6.D8, D4⋊Dic3, C4×D12, C2×D4⋊S3, D4×C12, C42.48D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C8⋊C22, S3×C2×C4, C4○D12, C2×C3⋊D4, D8⋊C4, C4×C3⋊D4, D12⋊6C22, D4⋊D6, C42.48D6
►On 96 points
(1 56 93 77)(2 53 94 74)(3 50 95 79)(4 55 96 76)(5 52 89 73)(6 49 90 78)(7 54 91 75)(8 51 92 80)(9 42 30 18)(10 47 31 23)(11 44 32 20)(12 41 25 17)(13 46 26 22)(14 43 27 19)(15 48 28 24)(16 45 29 21)(33 86 71 58)(34 83 72 63)(35 88 65 60)(36 85 66 57)(37 82 67 62)(38 87 68 59)(39 84 69 64)(40 81 70 61)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 79 77 75)(74 80 78 76)(81 87 85 83)(82 88 86 84)(89 95 93 91)(90 96 94 92)
(1 45 65 66 44 2)(3 43 67 72 46 8)(4 7 47 71 68 42)(5 41 69 70 48 6)(9 80 75 14 86 83)(10 82 87 13 76 79)(11 78 77 12 88 81)(15 74 73 16 84 85)(17 39 40 24 90 89)(18 96 91 23 33 38)(19 37 34 22 92 95)(20 94 93 21 35 36)(25 60 61 32 49 56)(26 55 50 31 62 59)(27 58 63 30 51 54)(28 53 52 29 64 57)
(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)| (1,56,93,77)(2,53,94,74)(3,50,95,79)(4,55,96,76)(5,52,89,73)(6,49,90,78)(7,54,91,75)(8,51,92,80)(9,42,30,18)(10,47,31,23)(11,44,32,20)(12,41,25,17)(13,46,26,22)(14,43,27,19)(15,48,28,24)(16,45,29,21)(33,86,71,58)(34,83,72,63)(35,88,65,60)(36,85,66,57)(37,82,67,62)(38,87,68,59)(39,84,69,64)(40,81,70,61), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,95,93,91)(90,96,94,92), (1,45,65,66,44,2)(3,43,67,72,46,8)(4,7,47,71,68,42)(5,41,69,70,48,6)(9,80,75,14,86,83)(10,82,87,13,76,79)(11,78,77,12,88,81)(15,74,73,16,84,85)(17,39,40,24,90,89)(18,96,91,23,33,38)(19,37,34,22,92,95)(20,94,93,21,35,36)(25,60,61,32,49,56)(26,55,50,31,62,59)(27,58,63,30,51,54)(28,53,52,29,64,57), (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( (1,56,93,77)(2,53,94,74)(3,50,95,79)(4,55,96,76)(5,52,89,73)(6,49,90,78)(7,54,91,75)(8,51,92,80)(9,42,30,18)(10,47,31,23)(11,44,32,20)(12,41,25,17)(13,46,26,22)(14,43,27,19)(15,48,28,24)(16,45,29,21)(33,86,71,58)(34,83,72,63)(35,88,65,60)(36,85,66,57)(37,82,67,62)(38,87,68,59)(39,84,69,64)(40,81,70,61), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,95,93,91)(90,96,94,92), (1,45,65,66,44,2)(3,43,67,72,46,8)(4,7,47,71,68,42)(5,41,69,70,48,6)(9,80,75,14,86,83)(10,82,87,13,76,79)(11,78,77,12,88,81)(15,74,73,16,84,85)(17,39,40,24,90,89)(18,96,91,23,33,38)(19,37,34,22,92,95)(20,94,93,21,35,36)(25,60,61,32,49,56)(26,55,50,31,62,59)(27,58,63,30,51,54)(28,53,52,29,64,57), (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([[(1,56,93,77),(2,53,94,74),(3,50,95,79),(4,55,96,76),(5,52,89,73),(6,49,90,78),(7,54,91,75),(8,51,92,80),(9,42,30,18),(10,47,31,23),(11,44,32,20),(12,41,25,17),(13,46,26,22),(14,43,27,19),(15,48,28,24),(16,45,29,21),(33,86,71,58),(34,83,72,63),(35,88,65,60),(36,85,66,57),(37,82,67,62),(38,87,68,59),(39,84,69,64),(40,81,70,61)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,79,77,75),(74,80,78,76),(81,87,85,83),(82,88,86,84),(89,95,93,91),(90,96,94,92)], [(1,45,65,66,44,2),(3,43,67,72,46,8),(4,7,47,71,68,42),(5,41,69,70,48,6),(9,80,75,14,86,83),(10,82,87,13,76,79),(11,78,77,12,88,81),(15,74,73,16,84,85),(17,39,40,24,90,89),(18,96,91,23,33,38),(19,37,34,22,92,95),(20,94,93,21,35,36),(25,60,61,32,49,56),(26,55,50,31,62,59),(27,58,63,30,51,54),(28,53,52,29,64,57)], [(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4○D12 | C8⋊C22 | D12⋊6C22 | D4⋊D6 |
| kernel | C42.48D6 | C42.S3 | C12.Q8 | C6.D8 | D4⋊Dic3 | C4×D12 | C2×D4⋊S3 | D4×C12 | D4⋊S3 | C4×D4 | C2×C12 | C42 | C4⋊C4 | C2×D4 | C12 | C2×C4 | D4 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C42.48D6 ►in GL6(𝔽73)
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 60 |
| 0 | 0 | 0 | 0 | 13 | 43 |
| 0 | 0 | 43 | 13 | 0 | 0 |
| 0 | 0 | 60 | 30 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 33 | 33 | 21 |
| 0 | 0 | 40 | 61 | 52 | 12 |
| 0 | 0 | 33 | 21 | 52 | 40 |
| 0 | 0 | 52 | 12 | 33 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 21 | 52 | 40 |
| 0 | 0 | 61 | 40 | 61 | 21 |
| 0 | 0 | 21 | 33 | 33 | 21 |
| 0 | 0 | 12 | 52 | 61 | 40 |
G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,30,13,0,0,0,0,60,43,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,21,40,33,52,0,0,33,61,21,12,0,0,33,52,52,33,0,0,21,12,40,12],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,33,61,21,12,0,0,21,40,33,52,0,0,52,61,33,61,0,0,40,21,21,40] >;
C42.48D6 in GAP, Magma, Sage, TeX
C_4^2._{48}D_6 % in TeX
G:=Group("C4^2.48D6"); // GroupNames label
G:=SmallGroup(192,573);
// by ID
G=gap.SmallGroup(192,573);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,387,58,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations