direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8○D8, D8⋊5C20, Q16⋊5C20, C40.77D4, SD16⋊3C20, C4≀C2⋊7C10, (C4×C40)⋊26C2, (C4×C8)⋊10C10, C8○D4⋊6C10, (C5×D8)⋊17C4, C8.30(C5×D4), C8.11(C2×C20), (C5×Q16)⋊17C4, C4○D8.5C10, D4.3(C2×C20), C4.82(D4×C10), C2.18(D4×C20), Q8.3(C2×C20), C8.C4⋊8C10, C40.108(C2×C4), (C5×SD16)⋊11C4, C20.487(C2×D4), C10.150(C4×D4), C42.73(C2×C10), C4.15(C22×C20), (C4×C20).358C22, (C2×C20).910C23, C20.219(C22×C4), (C2×C40).433C22, M4(2).11(C2×C10), (C5×M4(2)).45C22, (C5×C4≀C2)⋊15C2, (C5×C8○D4)⋊15C2, (C5×C4○D8).10C2, C4○D4.8(C2×C10), (C5×D4).34(C2×C4), (C5×Q8).36(C2×C4), (C5×C8.C4)⋊17C2, C22.1(C5×C4○D4), (C2×C8).101(C2×C10), (C2×C10).49(C4○D4), (C2×C4).85(C22×C10), (C5×C4○D4).53C22, SmallGroup(320,944)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8○D8
G = < a,b,c,d | a5=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >
Subgroups: 154 in 106 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8○D8, C4×C20, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C4×C40, C5×C4≀C2, C5×C8.C4, C5×C8○D4, C5×C4○D8, C5×C8○D8
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C2×C20, C5×D4, C22×C10, C8○D8, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8○D8
►On 80 points
(1 43 25 39 13)(2 44 26 40 14)(3 45 27 33 15)(4 46 28 34 16)(5 47 29 35 9)(6 48 30 36 10)(7 41 31 37 11)(8 42 32 38 12)(17 72 73 52 61)(18 65 74 53 62)(19 66 75 54 63)(20 67 76 55 64)(21 68 77 56 57)(22 69 78 49 58)(23 70 79 50 59)(24 71 80 51 60)
(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 6 3 8 5 2 7 4)(9 14 11 16 13 10 15 12)(17 20 23 18 21 24 19 22)(25 30 27 32 29 26 31 28)(33 38 35 40 37 34 39 36)(41 46 43 48 45 42 47 44)(49 52 55 50 53 56 51 54)(57 60 63 58 61 64 59 62)(65 68 71 66 69 72 67 70)(73 76 79 74 77 80 75 78)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
G:=sub<Sym(80)| (1,43,25,39,13)(2,44,26,40,14)(3,45,27,33,15)(4,46,28,34,16)(5,47,29,35,9)(6,48,30,36,10)(7,41,31,37,11)(8,42,32,38,12)(17,72,73,52,61)(18,65,74,53,62)(19,66,75,54,63)(20,67,76,55,64)(21,68,77,56,57)(22,69,78,49,58)(23,70,79,50,59)(24,71,80,51,60), (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), (1,6,3,8,5,2,7,4)(9,14,11,16,13,10,15,12)(17,20,23,18,21,24,19,22)(25,30,27,32,29,26,31,28)(33,38,35,40,37,34,39,36)(41,46,43,48,45,42,47,44)(49,52,55,50,53,56,51,54)(57,60,63,58,61,64,59,62)(65,68,71,66,69,72,67,70)(73,76,79,74,77,80,75,78), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)>;
G:=Group( (1,43,25,39,13)(2,44,26,40,14)(3,45,27,33,15)(4,46,28,34,16)(5,47,29,35,9)(6,48,30,36,10)(7,41,31,37,11)(8,42,32,38,12)(17,72,73,52,61)(18,65,74,53,62)(19,66,75,54,63)(20,67,76,55,64)(21,68,77,56,57)(22,69,78,49,58)(23,70,79,50,59)(24,71,80,51,60), (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), (1,6,3,8,5,2,7,4)(9,14,11,16,13,10,15,12)(17,20,23,18,21,24,19,22)(25,30,27,32,29,26,31,28)(33,38,35,40,37,34,39,36)(41,46,43,48,45,42,47,44)(49,52,55,50,53,56,51,54)(57,60,63,58,61,64,59,62)(65,68,71,66,69,72,67,70)(73,76,79,74,77,80,75,78), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80) );
G=PermutationGroup([[(1,43,25,39,13),(2,44,26,40,14),(3,45,27,33,15),(4,46,28,34,16),(5,47,29,35,9),(6,48,30,36,10),(7,41,31,37,11),(8,42,32,38,12),(17,72,73,52,61),(18,65,74,53,62),(19,66,75,54,63),(20,67,76,55,64),(21,68,77,56,57),(22,69,78,49,58),(23,70,79,50,59),(24,71,80,51,60)], [(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)], [(1,6,3,8,5,2,7,4),(9,14,11,16,13,10,15,12),(17,20,23,18,21,24,19,22),(25,30,27,32,29,26,31,28),(33,38,35,40,37,34,39,36),(41,46,43,48,45,42,47,44),(49,52,55,50,53,56,51,54),(57,60,63,58,61,64,59,62),(65,68,71,66,69,72,67,70),(73,76,79,74,77,80,75,78)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,54),(10,55),(11,56),(12,49),(13,50),(14,51),(15,52),(16,53),(17,45),(18,46),(19,47),(20,48),(21,41),(22,42),(23,43),(24,44),(25,70),(26,71),(27,72),(28,65),(29,66),(30,67),(31,68),(32,69),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | ··· | 20AJ | 40A | ··· | 40P | 40Q | ··· | 40AN | 40AO | ··· | 40BD |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | D4 | C4○D4 | C5×D4 | C8○D8 | C5×C4○D4 | C5×C8○D8 |
| kernel | C5×C8○D8 | C4×C40 | C5×C4≀C2 | C5×C8.C4 | C5×C8○D4 | C5×C4○D8 | C5×D8 | C5×SD16 | C5×Q16 | C8○D8 | C4×C8 | C4≀C2 | C8.C4 | C8○D4 | C4○D8 | D8 | SD16 | Q16 | C40 | C2×C10 | C8 | C5 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 4 | 8 | 16 | 8 | 2 | 2 | 8 | 8 | 8 | 32 |
Matrix representation of C5×C8○D8 ►in GL2(𝔽41) generated by
| 18 | 0 |
| 0 | 18 |
| 14 | 0 |
| 0 | 14 |
| 27 | 0 |
| 0 | 38 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(41))| [18,0,0,18],[14,0,0,14],[27,0,0,38],[0,1,1,0] >;
C5×C8○D8 in GAP, Magma, Sage, TeX
C_5\times C_8\circ D_8
% in TeX
G:=Group("C5xC8oD8"); // GroupNames label
G:=SmallGroup(320,944);
// by ID
G=gap.SmallGroup(320,944);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,436,7004,3511,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
// generators/relations