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, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, C4≀C2, C8.C4, C8○D4, C8○D4, C4○D8, C5⋊2C8, C5⋊2C8, C40, 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, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C5⋊D4, 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 74 51 44 61 54 71 64)(42 43 52 53 62 63 72 73)(45 70 55 80 65 50 75 60)(46 79 56 49 66 59 76 69)(47 48 57 58 67 68 77 78)
(1 55)(2 64)(3 73)(4 42)(5 51)(6 60)(7 69)(8 78)(9 47)(10 56)(11 65)(12 74)(13 43)(14 52)(15 61)(16 70)(17 79)(18 48)(19 57)(20 66)(21 75)(22 44)(23 53)(24 62)(25 71)(26 80)(27 49)(28 58)(29 67)(30 76)(31 45)(32 54)(33 63)(34 72)(35 41)(36 50)(37 59)(38 68)(39 77)(40 46)
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)(41,74,51,44,61,54,71,64)(42,43,52,53,62,63,72,73)(45,70,55,80,65,50,75,60)(46,79,56,49,66,59,76,69)(47,48,57,58,67,68,77,78), (1,55)(2,64)(3,73)(4,42)(5,51)(6,60)(7,69)(8,78)(9,47)(10,56)(11,65)(12,74)(13,43)(14,52)(15,61)(16,70)(17,79)(18,48)(19,57)(20,66)(21,75)(22,44)(23,53)(24,62)(25,71)(26,80)(27,49)(28,58)(29,67)(30,76)(31,45)(32,54)(33,63)(34,72)(35,41)(36,50)(37,59)(38,68)(39,77)(40,46)>;
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)(41,74,51,44,61,54,71,64)(42,43,52,53,62,63,72,73)(45,70,55,80,65,50,75,60)(46,79,56,49,66,59,76,69)(47,48,57,58,67,68,77,78), (1,55)(2,64)(3,73)(4,42)(5,51)(6,60)(7,69)(8,78)(9,47)(10,56)(11,65)(12,74)(13,43)(14,52)(15,61)(16,70)(17,79)(18,48)(19,57)(20,66)(21,75)(22,44)(23,53)(24,62)(25,71)(26,80)(27,49)(28,58)(29,67)(30,76)(31,45)(32,54)(33,63)(34,72)(35,41)(36,50)(37,59)(38,68)(39,77)(40,46) );
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),(41,74,51,44,61,54,71,64),(42,43,52,53,62,63,72,73),(45,70,55,80,65,50,75,60),(46,79,56,49,66,59,76,69),(47,48,57,58,67,68,77,78)], [(1,55),(2,64),(3,73),(4,42),(5,51),(6,60),(7,69),(8,78),(9,47),(10,56),(11,65),(12,74),(13,43),(14,52),(15,61),(16,70),(17,79),(18,48),(19,57),(20,66),(21,75),(22,44),(23,53),(24,62),(25,71),(26,80),(27,49),(28,58),(29,67),(30,76),(31,45),(32,54),(33,63),(34,72),(35,41),(36,50),(37,59),(38,68),(39,77),(40,46)]])
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