metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.3D8, D40.4C4, C4.11D40, C40.80D4, M5(2)⋊5D5, C8.4(C4×D5), (C2×C4).9D20, C40.42(C2×C4), (C2×D40).6C2, (C2×C20).99D4, (C2×C8).46D10, C5⋊3(M5(2)⋊C2), C8.37(C5⋊D4), (C5×M5(2))⋊9C2, C40.6C4⋊11C2, (C2×C10).8SD16, (C2×C40).50C22, C2.9(D20⋊5C4), C20.89(C22⋊C4), C22.6(C40⋊C2), C10.32(D4⋊C4), C4.18(D10⋊C4), SmallGroup(320,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40.4C4
G = < a,b,c | a40=b2=1, c4=a10, bab=a-1, cac-1=a21, cbc-1=a15b >
Subgroups: 430 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C8, C2×C4, D4, C23, D5, C10, C10, C16, C2×C8, M4(2), D8, C2×D4, C20, D10, C2×C10, C8.C4, M5(2), C2×D8, C5⋊2C8, C40, D20, C2×C20, C22×D5, M5(2)⋊C2, C80, D40, D40, C4.Dic5, C2×C40, C2×D20, C40.6C4, C5×M5(2), C2×D40, D40.4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, C4×D5, D20, C5⋊D4, M5(2)⋊C2, C40⋊C2, D40, D10⋊C4, D20⋊5C4, D40.4C4
►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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(72 80)(73 79)(74 78)(75 77)
(1 64 26 69 11 74 36 79 21 44 6 49 31 54 16 59)(2 45 27 50 12 55 37 60 22 65 7 70 32 75 17 80)(3 66 28 71 13 76 38 41 23 46 8 51 33 56 18 61)(4 47 29 52 14 57 39 62 24 67 9 72 34 77 19 42)(5 68 30 73 15 78 40 43 25 48 10 53 35 58 20 63)
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(72,80)(73,79)(74,78)(75,77), (1,64,26,69,11,74,36,79,21,44,6,49,31,54,16,59)(2,45,27,50,12,55,37,60,22,65,7,70,32,75,17,80)(3,66,28,71,13,76,38,41,23,46,8,51,33,56,18,61)(4,47,29,52,14,57,39,62,24,67,9,72,34,77,19,42)(5,68,30,73,15,78,40,43,25,48,10,53,35,58,20,63)>;
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(72,80)(73,79)(74,78)(75,77), (1,64,26,69,11,74,36,79,21,44,6,49,31,54,16,59)(2,45,27,50,12,55,37,60,22,65,7,70,32,75,17,80)(3,66,28,71,13,76,38,41,23,46,8,51,33,56,18,61)(4,47,29,52,14,57,39,62,24,67,9,72,34,77,19,42)(5,68,30,73,15,78,40,43,25,48,10,53,35,58,20,63) );
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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(72,80),(73,79),(74,78),(75,77)], [(1,64,26,69,11,74,36,79,21,44,6,49,31,54,16,59),(2,45,27,50,12,55,37,60,22,65,7,70,32,75,17,80),(3,66,28,71,13,76,38,41,23,46,8,51,33,56,18,61),(4,47,29,52,14,57,39,62,24,67,9,72,34,77,19,42),(5,68,30,73,15,78,40,43,25,48,10,53,35,58,20,63)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4×D5 | C5⋊D4 | D20 | D40 | C40⋊C2 | M5(2)⋊C2 | D40.4C4 |
| kernel | D40.4C4 | C40.6C4 | C5×M5(2) | C2×D40 | D40 | C40 | C2×C20 | M5(2) | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of D40.4C4 ►in GL4(𝔽241) generated by
| 221 | 47 | 0 | 0 |
| 194 | 14 | 0 | 0 |
| 221 | 47 | 20 | 194 |
| 194 | 14 | 47 | 227 |
| 20 | 212 | 0 | 0 |
| 47 | 221 | 0 | 0 |
| 131 | 106 | 240 | 0 |
| 170 | 110 | 189 | 1 |
| 1 | 0 | 239 | 0 |
| 0 | 1 | 0 | 239 |
| 142 | 106 | 240 | 0 |
| 135 | 111 | 0 | 240 |
G:=sub<GL(4,GF(241))| [221,194,221,194,47,14,47,14,0,0,20,47,0,0,194,227],[20,47,131,170,212,221,106,110,0,0,240,189,0,0,0,1],[1,0,142,135,0,1,106,111,239,0,240,0,0,239,0,240] >;
D40.4C4 in GAP, Magma, Sage, TeX
D_{40}._4C_4 % in TeX
G:=Group("D40.4C4"); // GroupNames label
G:=SmallGroup(320,74);
// by ID
G=gap.SmallGroup(320,74);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,387,268,570,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=1,c^4=a^10,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^15*b>;
// generators/relations