metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D48:7C2, C4.20D24, C8.13D12, C24.63D4, C16.16D6, C12.38D8, Dic24:7C2, C22.1D24, C24.57C23, C48.18C22, D24.7C22, Dic12.7C22, (C2xC16):6S3, (C2xC48):10C2, C4oD24:1C2, C3:1(C4oD16), C48:C2:7C2, C6.11(C2xD8), (C2xC6).20D8, C2.13(C2xD24), (C2xC4).85D12, C4.38(C2xD12), (C2xC8).313D6, C12.281(C2xD4), (C2xC12).395D4, C8.47(C22xS3), (C2xC24).385C22, SmallGroup(192,463)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D48:7C2
G = < a,b,c | a48=b2=c2=1, bab=a-1, ac=ca, cbc=a24b >
Subgroups: 328 in 84 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, Dic3, C12, D6, C2xC6, C16, C2xC8, D8, SD16, Q16, C4oD4, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC16, D16, SD32, Q32, C4oD8, C48, C24:C2, D24, Dic12, C2xC24, C4oD12, C4oD16, D48, C48:C2, Dic24, C2xC48, C4oD24, D48:7C2
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, D12, C22xS3, C2xD8, D24, C2xD12, C4oD16, C2xD24, D48:7C2
►On 96 points
Generators in S96
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(94 96)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)
G:=sub<Sym(96)| (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(94,96), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)>;
G:=Group( (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(94,96), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75) );
G=PermutationGroup([[(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(49,93),(50,92),(51,91),(52,90),(53,89),(54,88),(55,87),(56,86),(57,85),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(94,96)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,93),(19,94),(20,95),(21,96),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 24 | 24 | 2 | 1 | 1 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | D12 | D12 | D24 | D24 | C4oD16 | D48:7C2 |
| kernel | D48:7C2 | D48 | C48:C2 | Dic24 | C2xC48 | C4oD24 | C2xC16 | C24 | C2xC12 | C16 | C2xC8 | C12 | C2xC6 | C8 | C2xC4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of D48:7C2 ►in GL4(F7) generated by
| 2 | 5 | 5 | 0 |
| 1 | 4 | 0 | 5 |
| 6 | 1 | 4 | 0 |
| 6 | 0 | 2 | 3 |
| 0 | 2 | 4 | 1 |
| 4 | 6 | 3 | 1 |
| 3 | 3 | 0 | 1 |
| 2 | 4 | 1 | 1 |
| 4 | 0 | 5 | 2 |
| 5 | 0 | 0 | 3 |
| 2 | 5 | 4 | 3 |
| 5 | 5 | 1 | 6 |
G:=sub<GL(4,GF(7))| [2,1,6,6,5,4,1,0,5,0,4,2,0,5,0,3],[0,4,3,2,2,6,3,4,4,3,0,1,1,1,1,1],[4,5,2,5,0,0,5,5,5,0,4,1,2,3,3,6] >;
D48:7C2 in GAP, Magma, Sage, TeX
D_{48}\rtimes_7C_2 % in TeX
G:=Group("D48:7C2"); // GroupNames label
G:=SmallGroup(192,463);
// by ID
G=gap.SmallGroup(192,463);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,142,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^48=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^24*b>;
// generators/relations