metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D48⋊5C2, Q32⋊3S3, D6.3D8, C16.10D6, Q16.5D6, C48.8C22, C24.22C23, Dic3.14D8, D24.5C22, C3⋊C8.16D4, (S3×C16)⋊3C2, C3⋊4(C4○D16), (C3×Q32)⋊3C2, C2.25(S3×D8), C6.41(C2×D8), C4.10(S3×D4), (C4×S3).23D4, C12.16(C2×D4), C8.6D6⋊4C2, C3⋊C16.8C22, D24⋊C2⋊5C2, C8.28(C22×S3), (S3×C8).14C22, (C3×Q16).6C22, SmallGroup(192,478)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D48⋊5C2
G = < a,b,c | a48=b2=c2=1, bab=a-1, cac=a17, cbc=a40b >
Subgroups: 316 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Dic3, C12, C12, D6, D6, C16, C16, C2×C8, D8, SD16, Q16, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, C3×Q8, C2×C16, D16, SD32, Q32, C4○D8, C3⋊C16, C48, S3×C8, D24, Q8⋊2S3, C3×Q16, Q8⋊3S3, C4○D16, S3×C16, D48, C8.6D6, C3×Q32, D24⋊C2, D48⋊5C2
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, C2×D8, S3×D4, C4○D16, S3×D8, D48⋊5C2
►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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(49 56)(50 55)(51 54)(52 53)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)
(1 65)(2 82)(3 51)(4 68)(5 85)(6 54)(7 71)(8 88)(9 57)(10 74)(11 91)(12 60)(13 77)(14 94)(15 63)(16 80)(17 49)(18 66)(19 83)(20 52)(21 69)(22 86)(23 55)(24 72)(25 89)(26 58)(27 75)(28 92)(29 61)(30 78)(31 95)(32 64)(33 81)(34 50)(35 67)(36 84)(37 53)(38 70)(39 87)(40 56)(41 73)(42 90)(43 59)(44 76)(45 93)(46 62)(47 79)(48 96)
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(49,56)(50,55)(51,54)(52,53)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77), (1,65)(2,82)(3,51)(4,68)(5,85)(6,54)(7,71)(8,88)(9,57)(10,74)(11,91)(12,60)(13,77)(14,94)(15,63)(16,80)(17,49)(18,66)(19,83)(20,52)(21,69)(22,86)(23,55)(24,72)(25,89)(26,58)(27,75)(28,92)(29,61)(30,78)(31,95)(32,64)(33,81)(34,50)(35,67)(36,84)(37,53)(38,70)(39,87)(40,56)(41,73)(42,90)(43,59)(44,76)(45,93)(46,62)(47,79)(48,96)>;
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(49,56)(50,55)(51,54)(52,53)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77), (1,65)(2,82)(3,51)(4,68)(5,85)(6,54)(7,71)(8,88)(9,57)(10,74)(11,91)(12,60)(13,77)(14,94)(15,63)(16,80)(17,49)(18,66)(19,83)(20,52)(21,69)(22,86)(23,55)(24,72)(25,89)(26,58)(27,75)(28,92)(29,61)(30,78)(31,95)(32,64)(33,81)(34,50)(35,67)(36,84)(37,53)(38,70)(39,87)(40,56)(41,73)(42,90)(43,59)(44,76)(45,93)(46,62)(47,79)(48,96) );
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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(49,56),(50,55),(51,54),(52,53),(57,96),(58,95),(59,94),(60,93),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77)], [(1,65),(2,82),(3,51),(4,68),(5,85),(6,54),(7,71),(8,88),(9,57),(10,74),(11,91),(12,60),(13,77),(14,94),(15,63),(16,80),(17,49),(18,66),(19,83),(20,52),(21,69),(22,86),(23,55),(24,72),(25,89),(26,58),(27,75),(28,92),(29,61),(30,78),(31,95),(32,64),(33,81),(34,50),(35,67),(36,84),(37,53),(38,70),(39,87),(40,56),(41,73),(42,90),(43,59),(44,76),(45,93),(46,62),(47,79),(48,96)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 48A | 48B | 48C | 48D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 48 | 48 | 48 | 48 |
| size | 1 | 1 | 6 | 24 | 24 | 2 | 2 | 3 | 3 | 8 | 8 | 2 | 2 | 2 | 6 | 6 | 4 | 16 | 16 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | C4○D16 | S3×D4 | S3×D8 | D48⋊5C2 |
| kernel | D48⋊5C2 | S3×C16 | D48 | C8.6D6 | C3×Q32 | D24⋊C2 | Q32 | C3⋊C8 | C4×S3 | C16 | Q16 | Dic3 | D6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 2 | 4 |
Matrix representation of D48⋊5C2 ►in GL4(𝔽97) generated by
| 26 | 95 | 0 | 0 |
| 2 | 26 | 0 | 0 |
| 0 | 0 | 0 | 96 |
| 0 | 0 | 1 | 96 |
| 26 | 95 | 0 | 0 |
| 95 | 71 | 0 | 0 |
| 0 | 0 | 1 | 96 |
| 0 | 0 | 0 | 96 |
| 0 | 22 | 0 | 0 |
| 75 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(97))| [26,2,0,0,95,26,0,0,0,0,0,1,0,0,96,96],[26,95,0,0,95,71,0,0,0,0,1,0,0,0,96,96],[0,75,0,0,22,0,0,0,0,0,0,1,0,0,1,0] >;
D48⋊5C2 in GAP, Magma, Sage, TeX
D_{48}\rtimes_5C_2 % in TeX
G:=Group("D48:5C2"); // GroupNames label
G:=SmallGroup(192,478);
// by ID
G=gap.SmallGroup(192,478);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,758,135,184,346,185,192,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^48=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^17,c*b*c=a^40*b>;
// generators/relations