metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.45C42, C42.1Dic5, C20.47M4(2), C5⋊2C16⋊5C4, (C4×C20).3C4, C8.32(C4×D5), C5⋊5(C16⋊C4), C40.77(C2×C4), (C2×C40).25C4, C8⋊C4.4D5, (C2×C8).1Dic5, (C2×C8).150D10, C4.22(C4×Dic5), C20.4C8.6C2, C10.10(C8⋊C4), C4.6(C4.Dic5), (C2×C40).217C22, (C2×C10).38M4(2), C2.3(C42.D5), C22.3(C4.Dic5), (C5×C8⋊C4).3C2, (C2×C20).475(C2×C4), (C2×C4).70(C2×Dic5), SmallGroup(320,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.45C42
G = < a,b,c | a20=c4=1, b4=a5, bab-1=a9, ac=ca, cbc-1=a5b >
Smallest permutation representation of C20.45C42
►On 80 points
Generators in S80
(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 72 31 58 6 77 36 43 11 62 21 48 16 67 26 53)(2 61 32 47 7 66 37 52 12 71 22 57 17 76 27 42)(3 70 33 56 8 75 38 41 13 80 23 46 18 65 28 51)(4 79 34 45 9 64 39 50 14 69 24 55 19 74 29 60)(5 68 35 54 10 73 40 59 15 78 25 44 20 63 30 49)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
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), (1,72,31,58,6,77,36,43,11,62,21,48,16,67,26,53)(2,61,32,47,7,66,37,52,12,71,22,57,17,76,27,42)(3,70,33,56,8,75,38,41,13,80,23,46,18,65,28,51)(4,79,34,45,9,64,39,50,14,69,24,55,19,74,29,60)(5,68,35,54,10,73,40,59,15,78,25,44,20,63,30,49), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)>;
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), (1,72,31,58,6,77,36,43,11,62,21,48,16,67,26,53)(2,61,32,47,7,66,37,52,12,71,22,57,17,76,27,42)(3,70,33,56,8,75,38,41,13,80,23,46,18,65,28,51)(4,79,34,45,9,64,39,50,14,69,24,55,19,74,29,60)(5,68,35,54,10,73,40,59,15,78,25,44,20,63,30,49), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70) );
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)], [(1,72,31,58,6,77,36,43,11,62,21,48,16,67,26,53),(2,61,32,47,7,66,37,52,12,71,22,57,17,76,27,42),(3,70,33,56,8,75,38,41,13,80,23,46,18,65,28,51),(4,79,34,45,9,64,39,50,14,69,24,55,19,74,29,60),(5,68,35,54,10,73,40,59,15,78,25,44,20,63,30,49)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,46,51,56),(42,47,52,57),(43,48,53,58),(44,49,54,59),(45,50,55,60),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | M4(2) | Dic5 | Dic5 | D10 | C4×D5 | C4.Dic5 | C4.Dic5 | C16⋊C4 | C20.45C42 |
| kernel | C20.45C42 | C20.4C8 | C5×C8⋊C4 | C5⋊2C16 | C4×C20 | C2×C40 | C8⋊C4 | C20 | C2×C10 | C42 | C2×C8 | C2×C8 | C8 | C4 | C22 | C5 | C1 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 2 | 8 |
Matrix representation of C20.45C42 ►in GL6(𝔽241)
| 98 | 0 | 0 | 0 | 0 | 0 |
| 37 | 91 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 229 | 113 | 0 | 0 | 0 | 0 |
| 180 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 226 | 37 | 13 | 176 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 156 | 1 | 208 |
| 0 | 0 | 160 | 67 | 58 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 179 | 130 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 164 |
| 0 | 0 | 0 | 0 | 0 | 177 |
G:=sub<GL(6,GF(241))| [98,37,0,0,0,0,0,91,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[229,180,0,0,0,0,113,12,0,0,0,0,0,0,226,0,36,160,0,0,37,0,156,67,0,0,13,0,1,58,0,0,176,1,208,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,40,240,0,0,0,0,179,0,64,0,0,0,130,0,164,177] >;
C20.45C42 in GAP, Magma, Sage, TeX
C_{20}._{45}C_4^2 % in TeX
G:=Group("C20.45C4^2"); // GroupNames label
G:=SmallGroup(320,24);
// by ID
G=gap.SmallGroup(320,24);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,100,1123,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=c^4=1,b^4=a^5,b*a*b^-1=a^9,a*c=c*a,c*b*c^-1=a^5*b>;
// generators/relations
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