metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.93D4, M4(2).36D10, C8○D4⋊6D5, C5⋊7(C8○D8), D4⋊D5⋊11C4, Q8⋊D5⋊11C4, D4.7(C4×D5), Q8.7(C4×D5), D4.D5⋊11C4, C5⋊Q16⋊11C4, (C8×Dic5)⋊32C2, C4○D4.34D10, D20.32(C2×C4), C20.447(C2×D4), (C2×C8).278D10, C10.111(C4×D4), C8.22(C5⋊D4), D20⋊7C4⋊15C2, C20.64(C22×C4), D4⋊2Dic5⋊15C2, D20.3C4⋊13C2, C20.53D4⋊15C2, (C2×C40).236C22, (C2×C20).424C23, Dic10.33(C2×C4), D4.8D10.4C2, C4○D20.44C22, C22.3(C4○D20), C4.Dic5.44C22, (C4×Dic5).267C22, (C5×M4(2)).39C22, C4.29(C2×C4×D5), (C5×C8○D4)⋊6C2, C2.26(C4×C5⋊D4), C5⋊2C8.26(C2×C4), (C5×D4).28(C2×C4), C4.138(C2×C5⋊D4), (C5×Q8).29(C2×C4), (C2×C10).9(C4○D4), (C5×C4○D4).39C22, (C2×C4).514(C22×D5), (C2×C5⋊2C8).275C22, SmallGroup(320,771)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.93D4
G = < a,b,c | a40=c2=1, b4=a20, bab-1=cac=a9, cbc=a20b3 >
Subgroups: 326 in 106 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, D5, C10, C10 [×2], C42, C2×C8, C2×C8 [×3], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5 [×3], C20 [×2], C20, D10, C2×C10, C2×C10, C4×C8, C4≀C2 [×2], C8.C4, C8○D4, C8○D4, C4○D8, C5⋊2C8 [×2], C5⋊2C8, C40 [×2], C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8○D8, C8×D5, C8⋊D5, C2×C5⋊2C8, C4.Dic5, C4×Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C4○D20, C5×C4○D4, C8×Dic5, C20.53D4, D20⋊7C4, D4⋊2Dic5, D20.3C4, D4.8D10, C5×C8○D4, C40.93D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C8○D8, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C40.93D4
►On 80 points
(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 16 31 6 21 36 11 26)(2 25 32 15 22 5 12 35)(3 34 33 24 23 14 13 4)(7 30 37 20 27 10 17 40)(8 39 38 29 28 19 18 9)(41 50 51 60 61 70 71 80)(42 59 52 69 62 79 72 49)(43 68 53 78 63 48 73 58)(44 77 54 47 64 57 74 67)(45 46 55 56 65 66 75 76)
(1 48)(2 57)(3 66)(4 75)(5 44)(6 53)(7 62)(8 71)(9 80)(10 49)(11 58)(12 67)(13 76)(14 45)(15 54)(16 63)(17 72)(18 41)(19 50)(20 59)(21 68)(22 77)(23 46)(24 55)(25 64)(26 73)(27 42)(28 51)(29 60)(30 69)(31 78)(32 47)(33 56)(34 65)(35 74)(36 43)(37 52)(38 61)(39 70)(40 79)
G:=sub<Sym(80)| (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), (1,16,31,6,21,36,11,26)(2,25,32,15,22,5,12,35)(3,34,33,24,23,14,13,4)(7,30,37,20,27,10,17,40)(8,39,38,29,28,19,18,9)(41,50,51,60,61,70,71,80)(42,59,52,69,62,79,72,49)(43,68,53,78,63,48,73,58)(44,77,54,47,64,57,74,67)(45,46,55,56,65,66,75,76), (1,48)(2,57)(3,66)(4,75)(5,44)(6,53)(7,62)(8,71)(9,80)(10,49)(11,58)(12,67)(13,76)(14,45)(15,54)(16,63)(17,72)(18,41)(19,50)(20,59)(21,68)(22,77)(23,46)(24,55)(25,64)(26,73)(27,42)(28,51)(29,60)(30,69)(31,78)(32,47)(33,56)(34,65)(35,74)(36,43)(37,52)(38,61)(39,70)(40,79)>;
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), (1,16,31,6,21,36,11,26)(2,25,32,15,22,5,12,35)(3,34,33,24,23,14,13,4)(7,30,37,20,27,10,17,40)(8,39,38,29,28,19,18,9)(41,50,51,60,61,70,71,80)(42,59,52,69,62,79,72,49)(43,68,53,78,63,48,73,58)(44,77,54,47,64,57,74,67)(45,46,55,56,65,66,75,76), (1,48)(2,57)(3,66)(4,75)(5,44)(6,53)(7,62)(8,71)(9,80)(10,49)(11,58)(12,67)(13,76)(14,45)(15,54)(16,63)(17,72)(18,41)(19,50)(20,59)(21,68)(22,77)(23,46)(24,55)(25,64)(26,73)(27,42)(28,51)(29,60)(30,69)(31,78)(32,47)(33,56)(34,65)(35,74)(36,43)(37,52)(38,61)(39,70)(40,79) );
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)], [(1,16,31,6,21,36,11,26),(2,25,32,15,22,5,12,35),(3,34,33,24,23,14,13,4),(7,30,37,20,27,10,17,40),(8,39,38,29,28,19,18,9),(41,50,51,60,61,70,71,80),(42,59,52,69,62,79,72,49),(43,68,53,78,63,48,73,58),(44,77,54,47,64,57,74,67),(45,46,55,56,65,66,75,76)], [(1,48),(2,57),(3,66),(4,75),(5,44),(6,53),(7,62),(8,71),(9,80),(10,49),(11,58),(12,67),(13,76),(14,45),(15,54),(16,63),(17,72),(18,41),(19,50),(20,59),(21,68),(22,77),(23,46),(24,55),(25,64),(26,73),(27,42),(28,51),(29,60),(30,69),(31,78),(32,47),(33,56),(34,65),(35,74),(36,43),(37,52),(38,61),(39,70),(40,79)])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 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 | 4 | 20 | 1 | 1 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C4×D5 | C4×D5 | C8○D8 | C4○D20 | C40.93D4 |
| kernel | C40.93D4 | C8×Dic5 | C20.53D4 | D20⋊7C4 | D4⋊2Dic5 | D20.3C4 | D4.8D10 | C5×C8○D4 | D4⋊D5 | D4.D5 | Q8⋊D5 | C5⋊Q16 | C40 | C8○D4 | C2×C10 | C2×C8 | M4(2) | C4○D4 | C8 | D4 | Q8 | C5 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C40.93D4 ►in GL4(𝔽41) generated by
| 14 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 19 | 22 |
| 0 | 0 | 19 | 9 |
| 3 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 22 | 19 |
| 0 | 0 | 9 | 19 |
| 0 | 14 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 |
| 0 | 0 | 11 | 14 |
G:=sub<GL(4,GF(41))| [14,0,0,0,0,14,0,0,0,0,19,19,0,0,22,9],[3,0,0,0,0,14,0,0,0,0,22,9,0,0,19,19],[0,3,0,0,14,0,0,0,0,0,27,11,0,0,27,14] >;
C40.93D4 in GAP, Magma, Sage, TeX
C_{40}._{93}D_4 % in TeX
G:=Group("C40.93D4"); // GroupNames label
G:=SmallGroup(320,771);
// by ID
G=gap.SmallGroup(320,771);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,58,136,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations