direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C8.Q8, C16⋊1C20, C80⋊11C4, C40.10Q8, C20.44SD16, M5(2).1C10, C8.(C5×Q8), C8.18(C2×C20), C20.86(C4⋊C4), C4.Q8.1C10, C4.9(C5×SD16), C40.127(C2×C4), (C2×C20).282D4, C8.C4.2C10, (C2×C10).26SD16, C10.15(C4.Q8), (C5×M5(2)).3C2, C22.5(C5×SD16), (C2×C40).268C22, C4.6(C5×C4⋊C4), C2.3(C5×C4.Q8), (C2×C4).13(C5×D4), (C5×C4.Q8).6C2, (C2×C8).15(C2×C10), (C5×C8.C4).5C2, SmallGroup(320,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.Q8
G = < a,b,c,d | a5=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >
Smallest permutation representation of C5×C8.Q8
►On 80 points
Generators in S80
(1 55 70 20 44)(2 56 71 21 45)(3 57 72 22 46)(4 58 73 23 47)(5 59 74 24 48)(6 60 75 25 33)(7 61 76 26 34)(8 62 77 27 35)(9 63 78 28 36)(10 64 79 29 37)(11 49 80 30 38)(12 50 65 31 39)(13 51 66 32 40)(14 52 67 17 41)(15 53 68 18 42)(16 54 69 19 43)
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 60 54 64 58 52 62 56)(65 75 69 79 73 67 77 71)(66 68 70 72 74 76 78 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)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 27 25 19)(18 30)(21 23 29 31)(22 26)(24 32)(33 43 41 35)(34 46)(37 39 45 47)(38 42)(40 48)(49 53)(50 56 58 64)(51 59)(52 62 60 54)(57 61)(65 71 73 79)(66 74)(67 77 75 69)(68 80)(72 76)
G:=sub<Sym(80)| (1,55,70,20,44)(2,56,71,21,45)(3,57,72,22,46)(4,58,73,23,47)(5,59,74,24,48)(6,60,75,25,33)(7,61,76,26,34)(8,62,77,27,35)(9,63,78,28,36)(10,64,79,29,37)(11,49,80,30,38)(12,50,65,31,39)(13,51,66,32,40)(14,52,67,17,41)(15,53,68,18,42)(16,54,69,19,43), (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,60,54,64,58,52,62,56)(65,75,69,79,73,67,77,71)(66,68,70,72,74,76,78,80), (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), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,27,25,19)(18,30)(21,23,29,31)(22,26)(24,32)(33,43,41,35)(34,46)(37,39,45,47)(38,42)(40,48)(49,53)(50,56,58,64)(51,59)(52,62,60,54)(57,61)(65,71,73,79)(66,74)(67,77,75,69)(68,80)(72,76)>;
G:=Group( (1,55,70,20,44)(2,56,71,21,45)(3,57,72,22,46)(4,58,73,23,47)(5,59,74,24,48)(6,60,75,25,33)(7,61,76,26,34)(8,62,77,27,35)(9,63,78,28,36)(10,64,79,29,37)(11,49,80,30,38)(12,50,65,31,39)(13,51,66,32,40)(14,52,67,17,41)(15,53,68,18,42)(16,54,69,19,43), (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,60,54,64,58,52,62,56)(65,75,69,79,73,67,77,71)(66,68,70,72,74,76,78,80), (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), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,27,25,19)(18,30)(21,23,29,31)(22,26)(24,32)(33,43,41,35)(34,46)(37,39,45,47)(38,42)(40,48)(49,53)(50,56,58,64)(51,59)(52,62,60,54)(57,61)(65,71,73,79)(66,74)(67,77,75,69)(68,80)(72,76) );
G=PermutationGroup([[(1,55,70,20,44),(2,56,71,21,45),(3,57,72,22,46),(4,58,73,23,47),(5,59,74,24,48),(6,60,75,25,33),(7,61,76,26,34),(8,62,77,27,35),(9,63,78,28,36),(10,64,79,29,37),(11,49,80,30,38),(12,50,65,31,39),(13,51,66,32,40),(14,52,67,17,41),(15,53,68,18,42),(16,54,69,19,43)], [(1,3,5,7,9,11,13,15),(2,12,6,16,10,4,14,8),(17,27,21,31,25,19,29,23),(18,20,22,24,26,28,30,32),(33,43,37,47,41,35,45,39),(34,36,38,40,42,44,46,48),(49,51,53,55,57,59,61,63),(50,60,54,64,58,52,62,56),(65,75,69,79,73,67,77,71),(66,68,70,72,74,76,78,80)], [(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)], [(2,4,10,12),(3,7),(5,13),(6,16,14,8),(11,15),(17,27,25,19),(18,30),(21,23,29,31),(22,26),(24,32),(33,43,41,35),(34,46),(37,39,45,47),(38,42),(40,48),(49,53),(50,56,58,64),(51,59),(52,62,60,54),(57,61),(65,71,73,79),(66,74),(67,77,75,69),(68,80),(72,76)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 16A | 16B | 16C | 16D | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 40M | ··· | 40T | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | Q8 | D4 | SD16 | SD16 | C5×Q8 | C5×D4 | C5×SD16 | C5×SD16 | C8.Q8 | C5×C8.Q8 |
| kernel | C5×C8.Q8 | C5×C4.Q8 | C5×C8.C4 | C5×M5(2) | C80 | C8.Q8 | C4.Q8 | C8.C4 | M5(2) | C16 | C40 | C2×C20 | C20 | C2×C10 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of C5×C8.Q8 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 205 | 0 |
| 0 | 0 | 0 | 205 |
| 222 | 19 | 121 | 9 |
| 222 | 222 | 139 | 111 |
| 0 | 0 | 38 | 19 |
| 0 | 0 | 203 | 0 |
| 120 | 120 | 1 | 181 |
| 102 | 102 | 0 | 171 |
| 203 | 0 | 0 | 120 |
| 38 | 38 | 0 | 19 |
| 1 | 0 | 120 | 111 |
| 0 | 240 | 102 | 231 |
| 0 | 0 | 0 | 19 |
| 0 | 0 | 38 | 0 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[222,222,0,0,19,222,0,0,121,139,38,203,9,111,19,0],[120,102,203,38,120,102,0,38,1,0,0,0,181,171,120,19],[1,0,0,0,0,240,0,0,120,102,0,38,111,231,19,0] >;
C5×C8.Q8 in GAP, Magma, Sage, TeX
C_5\times C_8.Q_8
% in TeX
G:=Group("C5xC8.Q8"); // GroupNames label
G:=SmallGroup(320,170);
// by ID
G=gap.SmallGroup(320,170);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,148,2803,136,3511,10085,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations
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