metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊1C4, C2.2D48, C6.5D16, C24.79D4, C6.3SD32, C12.23SD16, C22.10D24, (C2×C48)⋊3C2, (C2×C16)⋊3S3, C8.19(C4×S3), C24⋊1C4⋊1C2, (C2×C6).16D8, C3⋊2(C2.D16), C24.49(C2×C4), (C2×D24).1C2, (C2×C8).299D6, (C2×C4).74D12, C4.2(C24⋊C2), C4.16(D6⋊C4), (C2×C12).372D4, C8.36(C3⋊D4), C2.3(C48⋊C2), C2.7(C2.D24), C6.15(D4⋊C4), C12.40(C22⋊C4), (C2×C24).372C22, SmallGroup(192,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D48
G = < a,b,c | a2=b48=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Subgroups: 328 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, D6, C2×C6, C16, C4⋊C4, C2×C8, D8, C2×D4, C24, D12, C2×Dic3, C2×C12, C22×S3, C2.D8, C2×C16, C2×D8, C48, D24, D24, C4⋊Dic3, C2×C24, C2×D12, C2.D16, C24⋊1C4, C2×C48, C2×D24, C2.D48
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D16, SD32, C24⋊C2, D24, D6⋊C4, C2.D16, D48, C48⋊C2, C2.D24, C2.D48
►On 96 points
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)
(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 74 75 48)(2 47 76 73)(3 72 77 46)(4 45 78 71)(5 70 79 44)(6 43 80 69)(7 68 81 42)(8 41 82 67)(9 66 83 40)(10 39 84 65)(11 64 85 38)(12 37 86 63)(13 62 87 36)(14 35 88 61)(15 60 89 34)(16 33 90 59)(17 58 91 32)(18 31 92 57)(19 56 93 30)(20 29 94 55)(21 54 95 28)(22 27 96 53)(23 52 49 26)(24 25 50 51)
G:=sub<Sym(96)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74), (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), (1,74,75,48)(2,47,76,73)(3,72,77,46)(4,45,78,71)(5,70,79,44)(6,43,80,69)(7,68,81,42)(8,41,82,67)(9,66,83,40)(10,39,84,65)(11,64,85,38)(12,37,86,63)(13,62,87,36)(14,35,88,61)(15,60,89,34)(16,33,90,59)(17,58,91,32)(18,31,92,57)(19,56,93,30)(20,29,94,55)(21,54,95,28)(22,27,96,53)(23,52,49,26)(24,25,50,51)>;
G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74), (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), (1,74,75,48)(2,47,76,73)(3,72,77,46)(4,45,78,71)(5,70,79,44)(6,43,80,69)(7,68,81,42)(8,41,82,67)(9,66,83,40)(10,39,84,65)(11,64,85,38)(12,37,86,63)(13,62,87,36)(14,35,88,61)(15,60,89,34)(16,33,90,59)(17,58,91,32)(18,31,92,57)(19,56,93,30)(20,29,94,55)(21,54,95,28)(22,27,96,53)(23,52,49,26)(24,25,50,51) );
G=PermutationGroup([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,73),(48,74)], [(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)], [(1,74,75,48),(2,47,76,73),(3,72,77,46),(4,45,78,71),(5,70,79,44),(6,43,80,69),(7,68,81,42),(8,41,82,67),(9,66,83,40),(10,39,84,65),(11,64,85,38),(12,37,86,63),(13,62,87,36),(14,35,88,61),(15,60,89,34),(16,33,90,59),(17,58,91,32),(18,31,92,57),(19,56,93,30),(20,29,94,55),(21,54,95,28),(22,27,96,53),(23,52,49,26),(24,25,50,51)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | 2 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | SD16 | D8 | C4×S3 | C3⋊D4 | D12 | D16 | SD32 | C24⋊C2 | D24 | D48 | C48⋊C2 |
| kernel | C2.D48 | C24⋊1C4 | C2×C48 | C2×D24 | D24 | C2×C16 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C2.D48 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 75 | 0 | 0 |
| 0 | 43 | 45 |
| 0 | 52 | 95 |
| 22 | 0 | 0 |
| 0 | 43 | 45 |
| 0 | 2 | 54 |
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[75,0,0,0,43,52,0,45,95],[22,0,0,0,43,2,0,45,54] >;
C2.D48 in GAP, Magma, Sage, TeX
C_2.D_{48} % in TeX
G:=Group("C2.D48"); // GroupNames label
G:=SmallGroup(192,68);
// by ID
G=gap.SmallGroup(192,68);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,204,422,268,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^48=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations