metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.5Q8, C20.2SD16, C8.1Dic10, C5⋊2C16⋊1C4, C5⋊3(C8.Q8), C8.28(C4×D5), C40.73(C2×C4), (C2×C20).87D4, C4.Q8.1D5, (C2×C8).40D10, C4.6(Q8⋊D5), C20.32(C4⋊C4), C10.7(C4.Q8), C20.4C8.2C2, C40.6C4.4C2, (C2×C40).46C22, (C2×C10).29SD16, C4.1(C10.D4), C22.5(D4.D5), C2.3(C20.Q8), (C5×C4.Q8).1C2, (C2×C4).15(C5⋊D4), SmallGroup(320,45)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.Dic10
G = < a,b,c | a8=b20=1, c2=ab10, bab-1=a3, cac-1=a5, cbc-1=a-1b-1 >
Smallest permutation representation of C8.Dic10
►On 80 points
Generators in S80
(1 29 16 24 9 39 11 34)(2 25 12 30 10 35 17 40)(3 21 18 26 6 31 13 36)(4 27 14 22 7 37 19 32)(5 23 20 28 8 33 15 38)(41 56 77 62 51 46 67 72)(42 63 68 57 52 73 78 47)(43 58 79 64 53 48 69 74)(44 65 70 59 54 75 80 49)(45 60 61 66 55 50 71 76)
(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 71 29 66 16 45 24 50 9 61 39 76 11 55 34 60)(2 75 25 70 12 49 30 54 10 65 35 80 17 59 40 44)(3 79 21 74 18 53 26 58 6 69 31 64 13 43 36 48)(4 63 27 78 14 57 22 42 7 73 37 68 19 47 32 52)(5 67 23 62 20 41 28 46 8 77 33 72 15 51 38 56)
G:=sub<Sym(80)| (1,29,16,24,9,39,11,34)(2,25,12,30,10,35,17,40)(3,21,18,26,6,31,13,36)(4,27,14,22,7,37,19,32)(5,23,20,28,8,33,15,38)(41,56,77,62,51,46,67,72)(42,63,68,57,52,73,78,47)(43,58,79,64,53,48,69,74)(44,65,70,59,54,75,80,49)(45,60,61,66,55,50,71,76), (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), (1,71,29,66,16,45,24,50,9,61,39,76,11,55,34,60)(2,75,25,70,12,49,30,54,10,65,35,80,17,59,40,44)(3,79,21,74,18,53,26,58,6,69,31,64,13,43,36,48)(4,63,27,78,14,57,22,42,7,73,37,68,19,47,32,52)(5,67,23,62,20,41,28,46,8,77,33,72,15,51,38,56)>;
G:=Group( (1,29,16,24,9,39,11,34)(2,25,12,30,10,35,17,40)(3,21,18,26,6,31,13,36)(4,27,14,22,7,37,19,32)(5,23,20,28,8,33,15,38)(41,56,77,62,51,46,67,72)(42,63,68,57,52,73,78,47)(43,58,79,64,53,48,69,74)(44,65,70,59,54,75,80,49)(45,60,61,66,55,50,71,76), (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), (1,71,29,66,16,45,24,50,9,61,39,76,11,55,34,60)(2,75,25,70,12,49,30,54,10,65,35,80,17,59,40,44)(3,79,21,74,18,53,26,58,6,69,31,64,13,43,36,48)(4,63,27,78,14,57,22,42,7,73,37,68,19,47,32,52)(5,67,23,62,20,41,28,46,8,77,33,72,15,51,38,56) );
G=PermutationGroup([[(1,29,16,24,9,39,11,34),(2,25,12,30,10,35,17,40),(3,21,18,26,6,31,13,36),(4,27,14,22,7,37,19,32),(5,23,20,28,8,33,15,38),(41,56,77,62,51,46,67,72),(42,63,68,57,52,73,78,47),(43,58,79,64,53,48,69,74),(44,65,70,59,54,75,80,49),(45,60,61,66,55,50,71,76)], [(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)], [(1,71,29,66,16,45,24,50,9,61,39,76,11,55,34,60),(2,75,25,70,12,49,30,54,10,65,35,80,17,59,40,44),(3,79,21,74,18,53,26,58,6,69,31,64,13,43,36,48),(4,63,27,78,14,57,22,42,7,73,37,68,19,47,32,52),(5,67,23,62,20,41,28,46,8,77,33,72,15,51,38,56)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | ··· | 10F | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | SD16 | SD16 | D10 | Dic10 | C4×D5 | C5⋊D4 | C8.Q8 | Q8⋊D5 | D4.D5 | C8.Dic10 |
| kernel | C8.Dic10 | C20.4C8 | C40.6C4 | C5×C4.Q8 | C5⋊2C16 | C40 | C2×C20 | C4.Q8 | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C8.Dic10 ►in GL4(𝔽241) generated by
| 0 | 218 | 0 | 0 |
| 220 | 38 | 0 | 0 |
| 63 | 221 | 222 | 19 |
| 232 | 22 | 222 | 222 |
| 91 | 0 | 0 | 0 |
| 164 | 150 | 0 | 0 |
| 174 | 207 | 175 | 66 |
| 174 | 109 | 66 | 66 |
| 1 | 0 | 90 | 0 |
| 16 | 36 | 240 | 1 |
| 83 | 222 | 240 | 0 |
| 193 | 169 | 41 | 205 |
G:=sub<GL(4,GF(241))| [0,220,63,232,218,38,221,22,0,0,222,222,0,0,19,222],[91,164,174,174,0,150,207,109,0,0,175,66,0,0,66,66],[1,16,83,193,0,36,222,169,90,240,240,41,0,1,0,205] >;
C8.Dic10 in GAP, Magma, Sage, TeX
C_8.{\rm Dic}_{10} % in TeX
G:=Group("C8.Dic10"); // GroupNames label
G:=SmallGroup(320,45);
// by ID
G=gap.SmallGroup(320,45);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,365,36,758,346,80,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=1,c^2=a*b^10,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^-1*b^-1>;
// generators/relations
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