metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.33C42, C23.33D20, M4(2)⋊3Dic5, C10.24C4≀C2, C4⋊Dic5⋊18C4, (C4×Dic5)⋊9C4, C20.41(C4⋊C4), (C2×C20).10Q8, C4.3(C4×Dic5), C5⋊6(C42⋊6C4), (C2×C20).492D4, (C5×M4(2))⋊12C4, (C2×C4).25Dic10, (C22×C10).46D4, (C2×M4(2)).6D5, C2.3(D20⋊7C4), (C22×C4).325D10, C22.3(C4⋊Dic5), C4.27(C23.D5), C20.141(C22⋊C4), C4.10(C10.D4), (C10×M4(2)).10C2, (C22×C20).124C22, C23.21D10.8C2, C22.42(D10⋊C4), C10.33(C2.C42), C2.14(C10.10C42), (C2×C4).69(C4×D5), (C2×C4×Dic5).2C2, (C2×C10).35(C4⋊C4), (C2×C20).235(C2×C4), (C2×C4).39(C2×Dic5), (C2×C4).179(C5⋊D4), (C2×C10).118(C22⋊C4), SmallGroup(320,113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.33C42
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, cac-1=a11, cbc-1=a5b >
Subgroups: 358 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×6], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×4], C22⋊C4, C4⋊C4 [×2], C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4, Dic5 [×6], C20 [×4], C2×C10 [×3], C2×C10 [×2], C2×C42, C42⋊C2, C2×M4(2), C40 [×2], C2×Dic5 [×8], C2×C20 [×6], C22×C10, C42⋊6C4, C4×Dic5 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5, C2×C40, C5×M4(2) [×2], C5×M4(2), C22×Dic5, C22×C20, C2×C4×Dic5, C23.21D10, C10×M4(2), C20.33C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, C4≀C2 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C42⋊6C4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, D20⋊7C4 [×2], C10.10C42, C20.33C42
►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 66 21 56)(2 65 22 55)(3 64 23 54)(4 63 24 53)(5 62 25 52)(6 61 26 51)(7 80 27 50)(8 79 28 49)(9 78 29 48)(10 77 30 47)(11 76 31 46)(12 75 32 45)(13 74 33 44)(14 73 34 43)(15 72 35 42)(16 71 36 41)(17 70 37 60)(18 69 38 59)(19 68 39 58)(20 67 40 57)
(1 76 36 51 11 66 26 41)(2 67 37 42 12 77 27 52)(3 78 38 53 13 68 28 43)(4 69 39 44 14 79 29 54)(5 80 40 55 15 70 30 45)(6 71 21 46 16 61 31 56)(7 62 22 57 17 72 32 47)(8 73 23 48 18 63 33 58)(9 64 24 59 19 74 34 49)(10 75 25 50 20 65 35 60)
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), (1,66,21,56)(2,65,22,55)(3,64,23,54)(4,63,24,53)(5,62,25,52)(6,61,26,51)(7,80,27,50)(8,79,28,49)(9,78,29,48)(10,77,30,47)(11,76,31,46)(12,75,32,45)(13,74,33,44)(14,73,34,43)(15,72,35,42)(16,71,36,41)(17,70,37,60)(18,69,38,59)(19,68,39,58)(20,67,40,57), (1,76,36,51,11,66,26,41)(2,67,37,42,12,77,27,52)(3,78,38,53,13,68,28,43)(4,69,39,44,14,79,29,54)(5,80,40,55,15,70,30,45)(6,71,21,46,16,61,31,56)(7,62,22,57,17,72,32,47)(8,73,23,48,18,63,33,58)(9,64,24,59,19,74,34,49)(10,75,25,50,20,65,35,60)>;
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), (1,66,21,56)(2,65,22,55)(3,64,23,54)(4,63,24,53)(5,62,25,52)(6,61,26,51)(7,80,27,50)(8,79,28,49)(9,78,29,48)(10,77,30,47)(11,76,31,46)(12,75,32,45)(13,74,33,44)(14,73,34,43)(15,72,35,42)(16,71,36,41)(17,70,37,60)(18,69,38,59)(19,68,39,58)(20,67,40,57), (1,76,36,51,11,66,26,41)(2,67,37,42,12,77,27,52)(3,78,38,53,13,68,28,43)(4,69,39,44,14,79,29,54)(5,80,40,55,15,70,30,45)(6,71,21,46,16,61,31,56)(7,62,22,57,17,72,32,47)(8,73,23,48,18,63,33,58)(9,64,24,59,19,74,34,49)(10,75,25,50,20,65,35,60) );
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)], [(1,66,21,56),(2,65,22,55),(3,64,23,54),(4,63,24,53),(5,62,25,52),(6,61,26,51),(7,80,27,50),(8,79,28,49),(9,78,29,48),(10,77,30,47),(11,76,31,46),(12,75,32,45),(13,74,33,44),(14,73,34,43),(15,72,35,42),(16,71,36,41),(17,70,37,60),(18,69,38,59),(19,68,39,58),(20,67,40,57)], [(1,76,36,51,11,66,26,41),(2,67,37,42,12,77,27,52),(3,78,38,53,13,68,28,43),(4,69,39,44,14,79,29,54),(5,80,40,55,15,70,30,45),(6,71,21,46,16,61,31,56),(7,62,22,57,17,72,32,47),(8,73,23,48,18,63,33,58),(9,64,24,59,19,74,34,49),(10,75,25,50,20,65,35,60)])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 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 | C4 | C4 | C4 | D4 | Q8 | D4 | D5 | Dic5 | D10 | C4≀C2 | Dic10 | C4×D5 | C5⋊D4 | D20 | D20⋊7C4 |
| kernel | C20.33C42 | C2×C4×Dic5 | C23.21D10 | C10×M4(2) | C4×Dic5 | C4⋊Dic5 | C5×M4(2) | C2×C20 | C2×C20 | C22×C10 | C2×M4(2) | M4(2) | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C20.33C42 ►in GL4(𝔽41) generated by
| 35 | 5 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 40 | 9 |
| 28 | 4 | 0 | 0 |
| 19 | 13 | 0 | 0 |
| 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 39 |
| 0 | 0 | 36 | 9 |
G:=sub<GL(4,GF(41))| [35,1,0,0,5,40,0,0,0,0,32,40,0,0,0,9],[28,19,0,0,4,13,0,0,0,0,1,0,0,0,23,40],[32,0,0,0,0,32,0,0,0,0,32,36,0,0,39,9] >;
C20.33C42 in GAP, Magma, Sage, TeX
C_{20}._{33}C_4^2 % in TeX
G:=Group("C20.33C4^2"); // GroupNames label
G:=SmallGroup(320,113);
// by ID
G=gap.SmallGroup(320,113);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b>;
// generators/relations