direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C48⋊C2, C16⋊8D6, C6⋊1SD32, C4.7D24, C48⋊9C22, C8.11D12, C12.32D8, C24.61D4, C24.56C23, D24.6C22, C22.13D24, Dic12⋊6C22, (C2×C16)⋊7S3, C3⋊1(C2×SD32), (C2×C48)⋊11C2, (C2×C6).19D8, C6.10(C2×D8), (C2×D24).5C2, (C2×C4).84D12, C4.37(C2×D12), C2.12(C2×D24), (C2×C8).304D6, (C2×Dic12)⋊8C2, C12.280(C2×D4), (C2×C12).381D4, C8.46(C22×S3), (C2×C24).377C22, SmallGroup(192,462)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C48⋊C2
G = < a,b,c | a2=b48=c2=1, ab=ba, ac=ca, cbc=b23 >
Subgroups: 392 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C16, C2×C8, D8, Q16, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C2×C16, SD32, C2×D8, C2×Q16, C48, D24, D24, Dic12, Dic12, C2×C24, C2×Dic6, C2×D12, C2×SD32, C48⋊C2, C2×C48, C2×D24, C2×Dic12, C2×C48⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, SD32, C2×D8, D24, C2×D12, C2×SD32, C48⋊C2, C2×D24, C2×C48⋊C2
►On 96 points
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)
(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 59)(2 82)(3 57)(4 80)(5 55)(6 78)(7 53)(8 76)(9 51)(10 74)(11 49)(12 72)(13 95)(14 70)(15 93)(16 68)(17 91)(18 66)(19 89)(20 64)(21 87)(22 62)(23 85)(24 60)(25 83)(26 58)(27 81)(28 56)(29 79)(30 54)(31 77)(32 52)(33 75)(34 50)(35 73)(36 96)(37 71)(38 94)(39 69)(40 92)(41 67)(42 90)(43 65)(44 88)(45 63)(46 86)(47 61)(48 84)
G:=sub<Sym(96)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (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), (1,59)(2,82)(3,57)(4,80)(5,55)(6,78)(7,53)(8,76)(9,51)(10,74)(11,49)(12,72)(13,95)(14,70)(15,93)(16,68)(17,91)(18,66)(19,89)(20,64)(21,87)(22,62)(23,85)(24,60)(25,83)(26,58)(27,81)(28,56)(29,79)(30,54)(31,77)(32,52)(33,75)(34,50)(35,73)(36,96)(37,71)(38,94)(39,69)(40,92)(41,67)(42,90)(43,65)(44,88)(45,63)(46,86)(47,61)(48,84)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (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), (1,59)(2,82)(3,57)(4,80)(5,55)(6,78)(7,53)(8,76)(9,51)(10,74)(11,49)(12,72)(13,95)(14,70)(15,93)(16,68)(17,91)(18,66)(19,89)(20,64)(21,87)(22,62)(23,85)(24,60)(25,83)(26,58)(27,81)(28,56)(29,79)(30,54)(31,77)(32,52)(33,75)(34,50)(35,73)(36,96)(37,71)(38,94)(39,69)(40,92)(41,67)(42,90)(43,65)(44,88)(45,63)(46,86)(47,61)(48,84) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58)], [(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)], [(1,59),(2,82),(3,57),(4,80),(5,55),(6,78),(7,53),(8,76),(9,51),(10,74),(11,49),(12,72),(13,95),(14,70),(15,93),(16,68),(17,91),(18,66),(19,89),(20,64),(21,87),(22,62),(23,85),(24,60),(25,83),(26,58),(27,81),(28,56),(29,79),(30,54),(31,77),(32,52),(33,75),(34,50),(35,73),(36,96),(37,71),(38,94),(39,69),(40,92),(41,67),(42,90),(43,65),(44,88),(45,63),(46,86),(47,61),(48,84)]])
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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | D12 | D12 | SD32 | D24 | D24 | C48⋊C2 |
| kernel | C2×C48⋊C2 | C48⋊C2 | C2×C48 | C2×D24 | C2×Dic12 | C2×C16 | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C2×C48⋊C2 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 1 | 0 | 0 |
| 0 | 45 | 43 |
| 0 | 54 | 2 |
| 1 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 1 | 1 |
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[1,0,0,0,45,54,0,43,2],[1,0,0,0,96,1,0,0,1] >;
C2×C48⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{48}\rtimes C_2 % in TeX
G:=Group("C2xC48:C2"); // GroupNames label
G:=SmallGroup(192,462);
// by ID
G=gap.SmallGroup(192,462);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,142,1571,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^48=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^23>;
// generators/relations