direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4.10C42, C20.63C42, (C2×C8).1C20, (C2×C40).28C4, C4.10(C4×C20), (C2×C20).278D4, C23.1(C5×Q8), (C22×C10).1Q8, (C2×M4(2)).5C10, C20.149(C22⋊C4), (C10×M4(2)).17C2, (C22×C20).387C22, C10.41(C2.C42), (C2×C4).9(C5×D4), C22.2(C5×C4⋊C4), (C2×C4).64(C2×C20), C4.18(C5×C22⋊C4), (C2×C10).47(C4⋊C4), (C2×C20).498(C2×C4), (C22×C4).17(C2×C10), C2.3(C5×C2.C42), SmallGroup(320,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.10C42
G = < a,b,c,d | a5=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 122 in 86 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×M4(2), C40, C2×C20, C22×C10, C4.10C42, C2×C40, C5×M4(2), C22×C20, C10×M4(2), C5×C4.10C42
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2.C42, C2×C20, C5×D4, C5×Q8, C4.10C42, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C2.C42, C5×C4.10C42
►On 80 points
Generators in S80
(1 57 67 17 27)(2 58 68 18 28)(3 59 69 19 29)(4 60 70 20 30)(5 61 71 21 31)(6 62 72 22 32)(7 63 65 23 25)(8 64 66 24 26)(9 39 42 53 79)(10 40 43 54 80)(11 33 44 55 73)(12 34 45 56 74)(13 35 46 49 75)(14 36 47 50 76)(15 37 48 51 77)(16 38 41 52 78)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 79 77 75)(74 80 78 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 33 7 35 5 37 3 39)(2 36 4 34 6 40 8 38)(9 27 11 25 13 31 15 29)(10 26 16 28 14 30 12 32)(17 73 23 75 21 77 19 79)(18 76 20 74 22 80 24 78)(41 58 47 60 45 62 43 64)(42 57 44 63 46 61 48 59)(49 71 51 69 53 67 55 65)(50 70 56 72 54 66 52 68)
G:=sub<Sym(80)| (1,57,67,17,27)(2,58,68,18,28)(3,59,69,19,29)(4,60,70,20,30)(5,61,71,21,31)(6,62,72,22,32)(7,63,65,23,25)(8,64,66,24,26)(9,39,42,53,79)(10,40,43,54,80)(11,33,44,55,73)(12,34,45,56,74)(13,35,46,49,75)(14,36,47,50,76)(15,37,48,51,77)(16,38,41,52,78), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,79,77,75)(74,80,78,76), (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), (1,33,7,35,5,37,3,39)(2,36,4,34,6,40,8,38)(9,27,11,25,13,31,15,29)(10,26,16,28,14,30,12,32)(17,73,23,75,21,77,19,79)(18,76,20,74,22,80,24,78)(41,58,47,60,45,62,43,64)(42,57,44,63,46,61,48,59)(49,71,51,69,53,67,55,65)(50,70,56,72,54,66,52,68)>;
G:=Group( (1,57,67,17,27)(2,58,68,18,28)(3,59,69,19,29)(4,60,70,20,30)(5,61,71,21,31)(6,62,72,22,32)(7,63,65,23,25)(8,64,66,24,26)(9,39,42,53,79)(10,40,43,54,80)(11,33,44,55,73)(12,34,45,56,74)(13,35,46,49,75)(14,36,47,50,76)(15,37,48,51,77)(16,38,41,52,78), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,79,77,75)(74,80,78,76), (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), (1,33,7,35,5,37,3,39)(2,36,4,34,6,40,8,38)(9,27,11,25,13,31,15,29)(10,26,16,28,14,30,12,32)(17,73,23,75,21,77,19,79)(18,76,20,74,22,80,24,78)(41,58,47,60,45,62,43,64)(42,57,44,63,46,61,48,59)(49,71,51,69,53,67,55,65)(50,70,56,72,54,66,52,68) );
G=PermutationGroup([[(1,57,67,17,27),(2,58,68,18,28),(3,59,69,19,29),(4,60,70,20,30),(5,61,71,21,31),(6,62,72,22,32),(7,63,65,23,25),(8,64,66,24,26),(9,39,42,53,79),(10,40,43,54,80),(11,33,44,55,73),(12,34,45,56,74),(13,35,46,49,75),(14,36,47,50,76),(15,37,48,51,77),(16,38,41,52,78)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,79,77,75),(74,80,78,76)], [(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)], [(1,33,7,35,5,37,3,39),(2,36,4,34,6,40,8,38),(9,27,11,25,13,31,15,29),(10,26,16,28,14,30,12,32),(17,73,23,75,21,77,19,79),(18,76,20,74,22,80,24,78),(41,58,47,60,45,62,43,64),(42,57,44,63,46,61,48,59),(49,71,51,69,53,67,55,65),(50,70,56,72,54,66,52,68)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | ··· | 8L | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20T | 40A | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D4 | Q8 | C5×D4 | C5×Q8 | C4.10C42 | C5×C4.10C42 |
| kernel | C5×C4.10C42 | C10×M4(2) | C2×C40 | C4.10C42 | C2×M4(2) | C2×C8 | C2×C20 | C22×C10 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 3 | 12 | 4 | 12 | 48 | 3 | 1 | 12 | 4 | 2 | 8 |
Matrix representation of C5×C4.10C42 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 9 | 8 | 40 | 8 |
| 32 | 9 | 9 | 0 |
| 18 | 0 | 32 | 8 |
| 0 | 23 | 32 | 32 |
| 0 | 1 | 16 | 37 |
| 9 | 0 | 36 | 29 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[9,32,18,0,8,9,0,23,40,9,32,32,8,0,8,32],[0,9,0,0,1,0,0,0,16,36,0,1,37,29,32,0] >;
C5×C4.10C42 in GAP, Magma, Sage, TeX
C_5\times C_4._{10}C_4^2 % in TeX
G:=Group("C5xC4.10C4^2"); // GroupNames label
G:=SmallGroup(320,143);
// by ID
G=gap.SmallGroup(320,143);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,248,3511,172,10085,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations