metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10:1C8, C4.19D20, C20.52D4, C10.8M4(2), (C2xC8):1D5, (C2xC40):1C2, C2.5(C8xD5), C5:3(C22:C8), C10.14(C2xC8), (C2xC4).93D10, C2.3(C8:D5), C4.27(C5:D4), (C2xDic5).5C4, (C22xD5).3C4, C22.11(C4xD5), C2.1(D10:C4), C10.17(C22:C4), (C2xC20).107C22, (C2xC4xD5).7C2, (C2xC5:2C8):9C2, (C2xC10).32(C2xC4), SmallGroup(160,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10:1C8
G = < a,b,c | a10=b2=c8=1, bab=a-1, ac=ca, cbc-1=a5b >
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, D5, C22:C4, C2xC8, M4(2), D10, C22:C8, C4xD5, D20, C5:D4, C8xD5, C8:D5, D10:C4, D10:1C8
Smallest permutation representation of D10:1C8
►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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 80)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 50)(40 49)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)
(1 16 50 70 30 80 40 60)(2 17 41 61 21 71 31 51)(3 18 42 62 22 72 32 52)(4 19 43 63 23 73 33 53)(5 20 44 64 24 74 34 54)(6 11 45 65 25 75 35 55)(7 12 46 66 26 76 36 56)(8 13 47 67 27 77 37 57)(9 14 48 68 28 78 38 58)(10 15 49 69 29 79 39 59)
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,80)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,50)(40,49)(51,63)(52,62)(53,61)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64), (1,16,50,70,30,80,40,60)(2,17,41,61,21,71,31,51)(3,18,42,62,22,72,32,52)(4,19,43,63,23,73,33,53)(5,20,44,64,24,74,34,54)(6,11,45,65,25,75,35,55)(7,12,46,66,26,76,36,56)(8,13,47,67,27,77,37,57)(9,14,48,68,28,78,38,58)(10,15,49,69,29,79,39,59)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,80)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,50)(40,49)(51,63)(52,62)(53,61)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64), (1,16,50,70,30,80,40,60)(2,17,41,61,21,71,31,51)(3,18,42,62,22,72,32,52)(4,19,43,63,23,73,33,53)(5,20,44,64,24,74,34,54)(6,11,45,65,25,75,35,55)(7,12,46,66,26,76,36,56)(8,13,47,67,27,77,37,57)(9,14,48,68,28,78,38,58)(10,15,49,69,29,79,39,59) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,30),(11,79),(12,78),(13,77),(14,76),(15,75),(16,74),(17,73),(18,72),(19,71),(20,80),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,50),(40,49),(51,63),(52,62),(53,61),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64)], [(1,16,50,70,30,80,40,60),(2,17,41,61,21,71,31,51),(3,18,42,62,22,72,32,52),(4,19,43,63,23,73,33,53),(5,20,44,64,24,74,34,54),(6,11,45,65,25,75,35,55),(7,12,46,66,26,76,36,56),(8,13,47,67,27,77,37,57),(9,14,48,68,28,78,38,58),(10,15,49,69,29,79,39,59)]])
D10:1C8 is a maximal subgroup of
C42.282D10 C8xD20 C8:6D20 C42.243D10 C42.182D10 C8:9D20 C42.185D10 D5xC22:C8 C5:5(C8xD4) D10:7M4(2) C22:C8:D5 D10:4M4(2) Dic5:2M4(2) C5:2C8:26D4 D4:D20 D10.12D8 D20.8D4 D10.16SD16 C40:6C4:C2 D4:3D20 D4.D20 C40:5C4:C2 D10.11SD16 Q8:2D20 D10:4Q16 D10.7Q16 Q8.D20 D20:4D4 (C2xC8).D10 D10:1C8.C2 C42.200D10 D20:5C8 C42.202D10 D10:5M4(2) C20:6M4(2) C42.31D10 D10.12SD16 D10.17SD16 C4.Q8:D5 C20.(C4oD4) D10.13D8 D10.8Q16 C2.D8:D5 C2.D8:7D5 C8xC5:D4 (C22xC8):D5 C40:32D4 D10:8M4(2) C40:D4 C40:18D4 C4.89(C2xD20) D20:D4 Dic10:D4 D10:6SD16 D10:8SD16 D20:7D4 Dic10.16D4 D10:5Q16 D20.17D4 C60.93D4 D30:4C8 D30:3C8
D10:1C8 is a maximal quotient of
Dic10:3C8 D20:3C8 C5:3(C23:C8) (C2xDic5):C8 D20:4C8 Dic10:4C8 D10:1C16 D20.3C8 C8.25D20 D20.4C8 (C2xC40):15C4 C60.93D4 D30:4C8 D30:3C8
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D5 | M4(2) | D10 | D20 | C5:D4 | C4xD5 | C8xD5 | C8:D5 |
| kernel | D10:1C8 | C2xC5:2C8 | C2xC40 | C2xC4xD5 | C2xDic5 | C22xD5 | D10 | C20 | C2xC8 | C10 | C2xC4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of D10:1C8 ►in GL3(F41) generated by
| 1 | 0 | 0 |
| 0 | 0 | 7 |
| 0 | 35 | 7 |
| 1 | 0 | 0 |
| 0 | 34 | 7 |
| 0 | 40 | 7 |
| 14 | 0 | 0 |
| 0 | 17 | 7 |
| 0 | 35 | 24 |
G:=sub<GL(3,GF(41))| [1,0,0,0,0,35,0,7,7],[1,0,0,0,34,40,0,7,7],[14,0,0,0,17,35,0,7,24] >;
D10:1C8 in GAP, Magma, Sage, TeX
D_{10}\rtimes_1C_8 % in TeX
G:=Group("D10:1C8"); // GroupNames label
G:=SmallGroup(160,27);
// by ID
G=gap.SmallGroup(160,27);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^10=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^5*b>;
// generators/relations
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