metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.48D4, C20.48M4(2), (C2×C20).10C8, (C2×C40).26C4, C23.(C5⋊2C8), C5⋊4(C23.C8), (C2×C8).2Dic5, (C2×C8).153D10, C8.33(C5⋊D4), (C22×C10).2C8, C20.4C8⋊11C2, (C22×C20).12C4, (C2×M4(2)).4D5, C4.7(C4.Dic5), (C22×C4).4Dic5, C10.33(C22⋊C8), (C2×C40).222C22, (C10×M4(2)).4C2, C4.26(C23.D5), C20.140(C22⋊C4), C2.7(C20.55D4), (C2×C4).(C5⋊2C8), (C2×C10).60(C2×C8), C22.4(C2×C5⋊2C8), (C2×C20).476(C2×C4), (C2×C4).75(C2×Dic5), SmallGroup(320,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.D4
G = < a,b,c | a40=1, b4=a10, c2=a25, bab-1=a29, cac-1=a9, cbc-1=a15b3 >
Subgroups: 118 in 58 conjugacy classes, 31 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, M5(2), C2×M4(2), C40, C40, C2×C20, C2×C20, C22×C10, C23.C8, C5⋊2C16, C2×C40, C5×M4(2), C22×C20, C20.4C8, C10×M4(2), C40.D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), Dic5, D10, C22⋊C8, C5⋊2C8, C2×Dic5, C5⋊D4, C23.C8, C2×C5⋊2C8, C4.Dic5, C23.D5, C20.55D4, C40.D4
►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 70 26 75 11 80 36 45 21 50 6 55 31 60 16 65)(2 59 27 64 12 69 37 74 22 79 7 44 32 49 17 54)(3 48 28 53 13 58 38 63 23 68 8 73 33 78 18 43)(4 77 29 42 14 47 39 52 24 57 9 62 34 67 19 72)(5 66 30 71 15 76 40 41 25 46 10 51 35 56 20 61)
(1 50 26 75 11 60 36 45 21 70 6 55 31 80 16 65)(2 59 27 44 12 69 37 54 22 79 7 64 32 49 17 74)(3 68 28 53 13 78 38 63 23 48 8 73 33 58 18 43)(4 77 29 62 14 47 39 72 24 57 9 42 34 67 19 52)(5 46 30 71 15 56 40 41 25 66 10 51 35 76 20 61)
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), (1,70,26,75,11,80,36,45,21,50,6,55,31,60,16,65)(2,59,27,64,12,69,37,74,22,79,7,44,32,49,17,54)(3,48,28,53,13,58,38,63,23,68,8,73,33,78,18,43)(4,77,29,42,14,47,39,52,24,57,9,62,34,67,19,72)(5,66,30,71,15,76,40,41,25,46,10,51,35,56,20,61), (1,50,26,75,11,60,36,45,21,70,6,55,31,80,16,65)(2,59,27,44,12,69,37,54,22,79,7,64,32,49,17,74)(3,68,28,53,13,78,38,63,23,48,8,73,33,58,18,43)(4,77,29,62,14,47,39,72,24,57,9,42,34,67,19,52)(5,46,30,71,15,56,40,41,25,66,10,51,35,76,20,61)>;
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), (1,70,26,75,11,80,36,45,21,50,6,55,31,60,16,65)(2,59,27,64,12,69,37,74,22,79,7,44,32,49,17,54)(3,48,28,53,13,58,38,63,23,68,8,73,33,78,18,43)(4,77,29,42,14,47,39,52,24,57,9,62,34,67,19,72)(5,66,30,71,15,76,40,41,25,46,10,51,35,56,20,61), (1,50,26,75,11,60,36,45,21,70,6,55,31,80,16,65)(2,59,27,44,12,69,37,54,22,79,7,64,32,49,17,74)(3,68,28,53,13,78,38,63,23,48,8,73,33,58,18,43)(4,77,29,62,14,47,39,72,24,57,9,42,34,67,19,52)(5,46,30,71,15,56,40,41,25,66,10,51,35,76,20,61) );
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)], [(1,70,26,75,11,80,36,45,21,50,6,55,31,60,16,65),(2,59,27,64,12,69,37,74,22,79,7,44,32,49,17,54),(3,48,28,53,13,58,38,63,23,68,8,73,33,78,18,43),(4,77,29,42,14,47,39,52,24,57,9,62,34,67,19,72),(5,66,30,71,15,76,40,41,25,46,10,51,35,56,20,61)], [(1,50,26,75,11,60,36,45,21,70,6,55,31,80,16,65),(2,59,27,44,12,69,37,54,22,79,7,64,32,49,17,74),(3,68,28,53,13,78,38,63,23,48,8,73,33,58,18,43),(4,77,29,62,14,47,39,72,24,57,9,42,34,67,19,52),(5,46,30,71,15,56,40,41,25,66,10,51,35,76,20,61)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 16A | ··· | 16H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D5 | M4(2) | Dic5 | D10 | Dic5 | C5⋊D4 | C5⋊2C8 | C5⋊2C8 | C4.Dic5 | C23.C8 | C40.D4 |
| kernel | C40.D4 | C20.4C8 | C10×M4(2) | C2×C40 | C22×C20 | C2×C20 | C22×C10 | C40 | C2×M4(2) | C20 | C2×C8 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C5 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 8 | 2 | 8 |
Matrix representation of C40.D4 ►in GL4(𝔽241) generated by
| 0 | 205 | 143 | 64 |
| 106 | 0 | 151 | 75 |
| 0 | 0 | 25 | 210 |
| 0 | 0 | 191 | 216 |
| 87 | 0 | 1 | 124 |
| 134 | 0 | 0 | 109 |
| 177 | 1 | 0 | 114 |
| 128 | 0 | 0 | 154 |
| 154 | 0 | 1 | 118 |
| 107 | 0 | 0 | 132 |
| 64 | 1 | 0 | 27 |
| 113 | 0 | 0 | 87 |
G:=sub<GL(4,GF(241))| [0,106,0,0,205,0,0,0,143,151,25,191,64,75,210,216],[87,134,177,128,0,0,1,0,1,0,0,0,124,109,114,154],[154,107,64,113,0,0,1,0,1,0,0,0,118,132,27,87] >;
C40.D4 in GAP, Magma, Sage, TeX
C_{40}.D_4 % in TeX
G:=Group("C40.D4"); // GroupNames label
G:=SmallGroup(320,111);
// by ID
G=gap.SmallGroup(320,111);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=a^10,c^2=a^25,b*a*b^-1=a^29,c*a*c^-1=a^9,c*b*c^-1=a^15*b^3>;
// generators/relations