direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8.26D4, D8⋊4C20, Q16⋊4C20, SD16⋊2C20, C40.105D4, C4≀C2⋊6C10, C8○D4⋊7C10, (C5×D8)⋊16C4, C8.6(C2×C20), C8⋊C4⋊3C10, C8.25(C5×D4), C40.88(C2×C4), (C5×Q16)⋊16C4, C4○D8.3C10, D4.4(C2×C20), C4.83(D4×C10), C2.19(D4×C20), Q8.4(C2×C20), C8.C4⋊4C10, (C5×SD16)⋊10C4, C10.151(C4×D4), C20.488(C2×D4), C42.11(C2×C10), C4.16(C22×C20), (C2×C20).911C23, (C4×C20).252C22, (C2×C40).271C22, C20.220(C22×C4), M4(2).12(C2×C10), (C5×M4(2)).46C22, (C5×C4≀C2)⋊14C2, (C5×C8○D4)⋊16C2, (C5×C4○D8).8C2, (C5×C8⋊C4)⋊12C2, (C2×C8).54(C2×C10), C4○D4.9(C2×C10), (C5×D4).35(C2×C4), (C5×Q8).37(C2×C4), (C5×C8.C4)⋊13C2, C22.2(C5×C4○D4), (C2×C10).50(C4○D4), (C2×C4).86(C22×C10), (C5×C4○D4).54C22, SmallGroup(320,945)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.26D4
G = < a,b,c,d | a5=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >
Subgroups: 154 in 104 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, 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, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C40, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8.26D4, C4×C20, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C8⋊C4, C5×C4≀C2, C5×C8.C4, C5×C8○D4, C5×C4○D8, C5×C8.26D4
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, C8.26D4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8.26D4
►On 80 points
(1 43 10 39 29)(2 44 11 40 30)(3 45 12 33 31)(4 46 13 34 32)(5 47 14 35 25)(6 48 15 36 26)(7 41 16 37 27)(8 42 9 38 28)(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)
(2 6)(4 8)(9 13)(11 15)(17 19 21 23)(18 24 22 20)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)(65 71 69 67)(66 68 70 72)(73 75 77 79)(74 80 78 76)
(1 59 3 61 5 63 7 57)(2 64 4 58 6 60 8 62)(9 65 11 67 13 69 15 71)(10 70 12 72 14 66 16 68)(17 47 19 41 21 43 23 45)(18 44 20 46 22 48 24 42)(25 54 27 56 29 50 31 52)(26 51 28 53 30 55 32 49)(33 73 35 75 37 77 39 79)(34 78 36 80 38 74 40 76)
G:=sub<Sym(80)| (1,43,10,39,29)(2,44,11,40,30)(3,45,12,33,31)(4,46,13,34,32)(5,47,14,35,25)(6,48,15,36,26)(7,41,16,37,27)(8,42,9,38,28)(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), (2,6)(4,8)(9,13)(11,15)(17,19,21,23)(18,24,22,20)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60)(65,71,69,67)(66,68,70,72)(73,75,77,79)(74,80,78,76), (1,59,3,61,5,63,7,57)(2,64,4,58,6,60,8,62)(9,65,11,67,13,69,15,71)(10,70,12,72,14,66,16,68)(17,47,19,41,21,43,23,45)(18,44,20,46,22,48,24,42)(25,54,27,56,29,50,31,52)(26,51,28,53,30,55,32,49)(33,73,35,75,37,77,39,79)(34,78,36,80,38,74,40,76)>;
G:=Group( (1,43,10,39,29)(2,44,11,40,30)(3,45,12,33,31)(4,46,13,34,32)(5,47,14,35,25)(6,48,15,36,26)(7,41,16,37,27)(8,42,9,38,28)(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), (2,6)(4,8)(9,13)(11,15)(17,19,21,23)(18,24,22,20)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60)(65,71,69,67)(66,68,70,72)(73,75,77,79)(74,80,78,76), (1,59,3,61,5,63,7,57)(2,64,4,58,6,60,8,62)(9,65,11,67,13,69,15,71)(10,70,12,72,14,66,16,68)(17,47,19,41,21,43,23,45)(18,44,20,46,22,48,24,42)(25,54,27,56,29,50,31,52)(26,51,28,53,30,55,32,49)(33,73,35,75,37,77,39,79)(34,78,36,80,38,74,40,76) );
G=PermutationGroup([[(1,43,10,39,29),(2,44,11,40,30),(3,45,12,33,31),(4,46,13,34,32),(5,47,14,35,25),(6,48,15,36,26),(7,41,16,37,27),(8,42,9,38,28),(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)], [(2,6),(4,8),(9,13),(11,15),(17,19,21,23),(18,24,22,20),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48),(49,55,53,51),(50,52,54,56),(57,59,61,63),(58,64,62,60),(65,71,69,67),(66,68,70,72),(73,75,77,79),(74,80,78,76)], [(1,59,3,61,5,63,7,57),(2,64,4,58,6,60,8,62),(9,65,11,67,13,69,15,71),(10,70,12,72,14,66,16,68),(17,47,19,41,21,43,23,45),(18,44,20,46,22,48,24,42),(25,54,27,56,29,50,31,52),(26,51,28,53,30,55,32,49),(33,73,35,75,37,77,39,79),(34,78,36,80,38,74,40,76)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20AB | 40A | ··· | 40P | 40Q | ··· | 40AN |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
110 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 | 4 | 4 |
| 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 | C5×C4○D4 | C8.26D4 | C5×C8.26D4 |
| kernel | C5×C8.26D4 | C5×C8⋊C4 | C5×C4≀C2 | C5×C8.C4 | C5×C8○D4 | C5×C4○D8 | C5×D8 | C5×SD16 | C5×Q16 | C8.26D4 | C8⋊C4 | C4≀C2 | C8.C4 | C8○D4 | C4○D8 | D8 | SD16 | Q16 | C40 | C2×C10 | C8 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 4 | 8 | 16 | 8 | 2 | 2 | 8 | 8 | 2 | 8 |
Matrix representation of C5×C8.26D4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 16 | 16 | 12 | 39 |
| 23 | 25 | 25 | 32 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 9 | 0 |
| 1 | 20 | 2 | 5 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 20 | 16 | 11 | 18 |
| 0 | 0 | 0 | 32 |
| 39 | 21 | 21 | 40 |
| 0 | 40 | 0 | 0 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[16,23,0,0,16,25,0,0,12,25,0,9,39,32,1,0],[1,0,0,0,20,40,0,0,2,0,9,0,5,0,0,32],[20,0,39,0,16,0,21,40,11,0,21,0,18,32,40,0] >;
C5×C8.26D4 in GAP, Magma, Sage, TeX
C_5\times C_8._{26}D_4 % in TeX
G:=Group("C5xC8.26D4"); // GroupNames label
G:=SmallGroup(320,945);
// by ID
G=gap.SmallGroup(320,945);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,436,7004,3511,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^2*c^-1>;
// generators/relations