direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×D8⋊2C4, D8⋊2C20, Q16⋊2C20, C40.102D4, M5(2)⋊5C10, C20.43SD16, (C5×D8)⋊14C4, C8.1(C2×C20), C4.Q8⋊1C10, C8.22(C5×D4), C40.83(C2×C4), (C5×Q16)⋊14C4, C4○D8.2C10, (C2×C10).25D8, C4.8(C5×SD16), C22.3(C5×D8), (C2×C20).279D4, (C5×M5(2))⋊13C2, (C2×C40).265C22, C10.55(D4⋊C4), C20.119(C22⋊C4), (C5×C4○D8).7C2, (C5×C4.Q8)⋊10C2, (C2×C4).10(C5×D4), C4.4(C5×C22⋊C4), (C2×C8).12(C2×C10), C2.9(C5×D4⋊C4), SmallGroup(320,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D8⋊2C4
G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >
Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C4.Q8, M5(2), C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, D8⋊2C4, C80, C5×C4⋊C4, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C4.Q8, C5×M5(2), C5×C4○D8, C5×D8⋊2C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, D8, SD16, C20, C2×C10, D4⋊C4, C2×C20, C5×D4, D8⋊2C4, C5×C22⋊C4, C5×D8, C5×SD16, C5×D4⋊C4, C5×D8⋊2C4
►On 80 points
Generators in S80
(1 39 31 23 15)(2 40 32 24 16)(3 33 25 17 9)(4 34 26 18 10)(5 35 27 19 11)(6 36 28 20 12)(7 37 29 21 13)(8 38 30 22 14)(41 73 65 57 49)(42 74 66 58 50)(43 75 67 59 51)(44 76 68 60 52)(45 77 69 61 53)(46 78 70 62 54)(47 79 71 63 55)(48 80 72 64 56)
(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)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 56)(16 55)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 64)(24 63)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 72)(32 71)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 80)(40 79)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 42 45 46)(43 48 47 44)(49 50 53 54)(51 56 55 52)(57 58 61 62)(59 64 63 60)(65 66 69 70)(67 72 71 68)(73 74 77 78)(75 80 79 76)
G:=sub<Sym(80)| (1,39,31,23,15)(2,40,32,24,16)(3,33,25,17,9)(4,34,26,18,10)(5,35,27,19,11)(6,36,28,20,12)(7,37,29,21,13)(8,38,30,22,14)(41,73,65,57,49)(42,74,66,58,50)(43,75,67,59,51)(44,76,68,60,52)(45,77,69,61,53)(46,78,70,62,54)(47,79,71,63,55)(48,80,72,64,56), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,56)(16,55)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,64)(24,63)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,72)(32,71)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,80)(40,79), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,42,45,46)(43,48,47,44)(49,50,53,54)(51,56,55,52)(57,58,61,62)(59,64,63,60)(65,66,69,70)(67,72,71,68)(73,74,77,78)(75,80,79,76)>;
G:=Group( (1,39,31,23,15)(2,40,32,24,16)(3,33,25,17,9)(4,34,26,18,10)(5,35,27,19,11)(6,36,28,20,12)(7,37,29,21,13)(8,38,30,22,14)(41,73,65,57,49)(42,74,66,58,50)(43,75,67,59,51)(44,76,68,60,52)(45,77,69,61,53)(46,78,70,62,54)(47,79,71,63,55)(48,80,72,64,56), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,56)(16,55)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,64)(24,63)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,72)(32,71)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,80)(40,79), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,42,45,46)(43,48,47,44)(49,50,53,54)(51,56,55,52)(57,58,61,62)(59,64,63,60)(65,66,69,70)(67,72,71,68)(73,74,77,78)(75,80,79,76) );
G=PermutationGroup([[(1,39,31,23,15),(2,40,32,24,16),(3,33,25,17,9),(4,34,26,18,10),(5,35,27,19,11),(6,36,28,20,12),(7,37,29,21,13),(8,38,30,22,14),(41,73,65,57,49),(42,74,66,58,50),(43,75,67,59,51),(44,76,68,60,52),(45,77,69,61,53),(46,78,70,62,54),(47,79,71,63,55),(48,80,72,64,56)], [(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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,56),(16,55),(17,62),(18,61),(19,60),(20,59),(21,58),(22,57),(23,64),(24,63),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,72),(32,71),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,80),(40,79)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,21),(18,24),(20,22),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(41,42,45,46),(43,48,47,44),(49,50,53,54),(51,56,55,52),(57,58,61,62),(59,64,63,60),(65,66,69,70),(67,72,71,68),(73,74,77,78),(75,80,79,76)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 16A | 16B | 16C | 16D | 20A | ··· | 20H | 20I | ··· | 20T | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | D4 | D4 | SD16 | D8 | C5×D4 | C5×D4 | C5×SD16 | C5×D8 | D8⋊2C4 | C5×D8⋊2C4 |
| kernel | C5×D8⋊2C4 | C5×C4.Q8 | C5×M5(2) | C5×C4○D8 | C5×D8 | C5×Q16 | D8⋊2C4 | C4.Q8 | M5(2) | C4○D8 | D8 | Q16 | C40 | C2×C20 | C20 | C2×C10 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of C5×D8⋊2C4 ►in GL4(𝔽241) generated by
| 87 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 0 | 19 | 43 | 32 |
| 203 | 38 | 166 | 11 |
| 0 | 0 | 222 | 222 |
| 0 | 0 | 19 | 222 |
| 64 | 11 | 158 | 121 |
| 0 | 0 | 19 | 222 |
| 203 | 0 | 166 | 11 |
| 203 | 38 | 166 | 11 |
| 240 | 0 | 11 | 75 |
| 239 | 1 | 199 | 64 |
| 0 | 0 | 222 | 222 |
| 0 | 0 | 222 | 19 |
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[0,203,0,0,19,38,0,0,43,166,222,19,32,11,222,222],[64,0,203,203,11,0,0,38,158,19,166,166,121,222,11,11],[240,239,0,0,0,1,0,0,11,199,222,222,75,64,222,19] >;
C5×D8⋊2C4 in GAP, Magma, Sage, TeX
C_5\times D_8\rtimes_2C_4
% in TeX
G:=Group("C5xD8:2C4"); // GroupNames label
G:=SmallGroup(320,165);
// by ID
G=gap.SmallGroup(320,165);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,3650,136,3511,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations