metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.21D20, C40.19D4, D20.21D4, Dic10.21D4, M4(2).10D10, (C2×D40)⋊21C2, C8.C4⋊6D5, C4.57(C2×D20), (C2×C8).71D10, C4.136(D4×D5), C8⋊D10⋊10C2, C20.137(C2×D4), C5⋊2(D4.4D4), D20.3C4⋊6C2, C20.46D4⋊3C2, C10.50(C4⋊D4), C2.23(C4⋊D20), (C2×C20).313C23, (C2×C40).103C22, C4○D20.40C22, (C2×D20).92C22, C22.7(Q8⋊2D5), (C5×M4(2)).7C22, C4.Dic5.38C22, (C5×C8.C4)⋊7C2, (C2×C10).4(C4○D4), (C2×C4).114(C22×D5), SmallGroup(320,524)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.21D20
G = < a,b,c | a8=c2=1, b20=a4, bab-1=cac=a-1, cbc=a4b19 >
Subgroups: 606 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C8 [×3], C2×C4, C2×C4, D4 [×6], Q8, C23 [×2], D5 [×3], C10, C10, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], D8 [×4], SD16 [×2], C2×D4 [×2], C4○D4, Dic5, C20 [×2], D10 [×5], C2×C10, C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], C5⋊2C8, C40 [×2], C40 [×2], Dic10, C4×D5, D20, D20 [×4], C5⋊D4, C2×C20, C22×D5 [×2], D4.4D4, C8×D5, C8⋊D5, C40⋊C2 [×2], D40 [×4], C4.Dic5, C2×C40, C5×M4(2) [×2], C2×D20 [×2], C4○D20, C20.46D4 [×2], C5×C8.C4, D20.3C4, C2×D40, C8⋊D10 [×2], C8.21D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, D4.4D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, C8.21D20
►On 80 points
(1 44 11 54 21 64 31 74)(2 75 32 65 22 55 12 45)(3 46 13 56 23 66 33 76)(4 77 34 67 24 57 14 47)(5 48 15 58 25 68 35 78)(6 79 36 69 26 59 16 49)(7 50 17 60 27 70 37 80)(8 41 38 71 28 61 18 51)(9 52 19 62 29 72 39 42)(10 43 40 73 30 63 20 53)
(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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)
G:=sub<Sym(80)| (1,44,11,54,21,64,31,74)(2,75,32,65,22,55,12,45)(3,46,13,56,23,66,33,76)(4,77,34,67,24,57,14,47)(5,48,15,58,25,68,35,78)(6,79,36,69,26,59,16,49)(7,50,17,60,27,70,37,80)(8,41,38,71,28,61,18,51)(9,52,19,62,29,72,39,42)(10,43,40,73,30,63,20,53), (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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)>;
G:=Group( (1,44,11,54,21,64,31,74)(2,75,32,65,22,55,12,45)(3,46,13,56,23,66,33,76)(4,77,34,67,24,57,14,47)(5,48,15,58,25,68,35,78)(6,79,36,69,26,59,16,49)(7,50,17,60,27,70,37,80)(8,41,38,71,28,61,18,51)(9,52,19,62,29,72,39,42)(10,43,40,73,30,63,20,53), (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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59) );
G=PermutationGroup([(1,44,11,54,21,64,31,74),(2,75,32,65,22,55,12,45),(3,46,13,56,23,66,33,76),(4,77,34,67,24,57,14,47),(5,48,15,58,25,68,35,78),(6,79,36,69,26,59,16,49),(7,50,17,60,27,70,37,80),(8,41,38,71,28,61,18,51),(9,52,19,62,29,72,39,42),(10,43,40,73,30,63,20,53)], [(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)], [(1,58),(2,57),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,50),(10,49),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,80),(20,79),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,70),(30,69),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,62),(38,61),(39,60),(40,59)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 40 | 40 | 2 | 2 | 20 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D20 | D4.4D4 | D4×D5 | Q8⋊2D5 | C8.21D20 |
| kernel | C8.21D20 | C20.46D4 | C5×C8.C4 | D20.3C4 | C2×D40 | C8⋊D10 | C40 | Dic10 | D20 | C8.C4 | C2×C10 | C2×C8 | M4(2) | C8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C8.21D20 ►in GL4(𝔽41) generated by
| 12 | 33 | 0 | 0 |
| 8 | 5 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 33 | 12 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 6 |
| 13 | 2 | 0 | 0 |
| 39 | 25 | 0 | 0 |
| 8 | 5 | 0 | 0 |
| 12 | 33 | 0 | 0 |
| 0 | 0 | 33 | 12 |
| 0 | 0 | 5 | 8 |
G:=sub<GL(4,GF(41))| [12,8,0,0,33,5,0,0,0,0,5,33,0,0,8,12],[0,0,13,39,0,0,2,25,0,40,0,0,1,6,0,0],[8,12,0,0,5,33,0,0,0,0,33,5,0,0,12,8] >;
C8.21D20 in GAP, Magma, Sage, TeX
C_8._{21}D_{20} % in TeX
G:=Group("C8.21D20"); // GroupNames label
G:=SmallGroup(320,524);
// by ID
G=gap.SmallGroup(320,524);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,226,1123,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^20=a^4,b*a*b^-1=c*a*c=a^-1,c*b*c=a^4*b^19>;
// generators/relations