metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.9C23, D40⋊10C22, C20.60C24, C23.21D20, M4(2)⋊19D10, Dic20⋊9C22, D20.23C23, Dic10.23C23, (C2×C8)⋊5D10, (C2×C40)⋊8C22, C4.73(C2×D20), C8⋊D10⋊13C2, C8.9(C22×D5), (C2×C4).157D20, C20.239(C2×D4), (C2×C20).205D4, (C2×M4(2))⋊5D5, D40⋊7C2⋊10C2, C4.57(C23×D5), C8.D10⋊13C2, C4○D20⋊17C22, (C2×D20)⋊53C22, C40⋊C2⋊10C22, C5⋊1(D8⋊C22), (C10×M4(2))⋊5C2, C2.29(C22×D20), C10.27(C22×D4), C22.22(C2×D20), (C2×C20).798C23, (C22×C10).120D4, (C22×C4).267D10, (C2×Dic10)⋊64C22, (C5×M4(2))⋊21C22, (C22×C20).268C22, (C2×C4○D20)⋊27C2, (C2×C10).64(C2×D4), (C2×C4).225(C22×D5), SmallGroup(320,1420)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1054 in 262 conjugacy classes, 107 normal (21 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], D5 [×4], C10, C10 [×3], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], Dic5 [×4], C20 [×2], C20 [×2], D10 [×8], C2×C10, C2×C10 [×2], C2×C10, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C40 [×4], Dic10 [×4], Dic10 [×2], C4×D5 [×8], D20 [×4], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, D8⋊C22, C40⋊C2 [×8], D40 [×4], Dic20 [×4], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20 [×2], C4○D20 [×8], C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, D40⋊7C2 [×4], C8⋊D10 [×4], C8.D10 [×4], C10×M4(2), C2×C4○D20 [×2], C40.9C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], D8⋊C22, C2×D20 [×6], C23×D5, C22×D20, C40.9C23
Generators and relations
G = < a,b,c,d | a40=b2=1, c2=d2=a20, bab=a19, ac=ca, dad-1=a21, bc=cb, bd=db, cd=dc >
►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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 45)(42 64)(44 62)(46 60)(47 79)(48 58)(49 77)(50 56)(51 75)(52 54)(53 73)(55 71)(57 69)(59 67)(61 65)(66 80)(68 78)(70 76)(72 74)
(1 43 21 63)(2 44 22 64)(3 45 23 65)(4 46 24 66)(5 47 25 67)(6 48 26 68)(7 49 27 69)(8 50 28 70)(9 51 29 71)(10 52 30 72)(11 53 31 73)(12 54 32 74)(13 55 33 75)(14 56 34 76)(15 57 35 77)(16 58 36 78)(17 59 37 79)(18 60 38 80)(19 61 39 41)(20 62 40 42)
(1 63 21 43)(2 44 22 64)(3 65 23 45)(4 46 24 66)(5 67 25 47)(6 48 26 68)(7 69 27 49)(8 50 28 70)(9 71 29 51)(10 52 30 72)(11 73 31 53)(12 54 32 74)(13 75 33 55)(14 56 34 76)(15 77 35 57)(16 58 36 78)(17 79 37 59)(18 60 38 80)(19 41 39 61)(20 62 40 42)
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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,45)(42,64)(44,62)(46,60)(47,79)(48,58)(49,77)(50,56)(51,75)(52,54)(53,73)(55,71)(57,69)(59,67)(61,65)(66,80)(68,78)(70,76)(72,74), (1,43,21,63)(2,44,22,64)(3,45,23,65)(4,46,24,66)(5,47,25,67)(6,48,26,68)(7,49,27,69)(8,50,28,70)(9,51,29,71)(10,52,30,72)(11,53,31,73)(12,54,32,74)(13,55,33,75)(14,56,34,76)(15,57,35,77)(16,58,36,78)(17,59,37,79)(18,60,38,80)(19,61,39,41)(20,62,40,42), (1,63,21,43)(2,44,22,64)(3,65,23,45)(4,46,24,66)(5,67,25,47)(6,48,26,68)(7,69,27,49)(8,50,28,70)(9,71,29,51)(10,52,30,72)(11,73,31,53)(12,54,32,74)(13,75,33,55)(14,56,34,76)(15,77,35,57)(16,58,36,78)(17,79,37,59)(18,60,38,80)(19,41,39,61)(20,62,40,42)>;
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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,45)(42,64)(44,62)(46,60)(47,79)(48,58)(49,77)(50,56)(51,75)(52,54)(53,73)(55,71)(57,69)(59,67)(61,65)(66,80)(68,78)(70,76)(72,74), (1,43,21,63)(2,44,22,64)(3,45,23,65)(4,46,24,66)(5,47,25,67)(6,48,26,68)(7,49,27,69)(8,50,28,70)(9,51,29,71)(10,52,30,72)(11,53,31,73)(12,54,32,74)(13,55,33,75)(14,56,34,76)(15,57,35,77)(16,58,36,78)(17,59,37,79)(18,60,38,80)(19,61,39,41)(20,62,40,42), (1,63,21,43)(2,44,22,64)(3,65,23,45)(4,46,24,66)(5,67,25,47)(6,48,26,68)(7,69,27,49)(8,50,28,70)(9,71,29,51)(10,52,30,72)(11,73,31,53)(12,54,32,74)(13,75,33,55)(14,56,34,76)(15,77,35,57)(16,58,36,78)(17,79,37,59)(18,60,38,80)(19,41,39,61)(20,62,40,42) );
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)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,45),(42,64),(44,62),(46,60),(47,79),(48,58),(49,77),(50,56),(51,75),(52,54),(53,73),(55,71),(57,69),(59,67),(61,65),(66,80),(68,78),(70,76),(72,74)], [(1,43,21,63),(2,44,22,64),(3,45,23,65),(4,46,24,66),(5,47,25,67),(6,48,26,68),(7,49,27,69),(8,50,28,70),(9,51,29,71),(10,52,30,72),(11,53,31,73),(12,54,32,74),(13,55,33,75),(14,56,34,76),(15,57,35,77),(16,58,36,78),(17,59,37,79),(18,60,38,80),(19,61,39,41),(20,62,40,42)], [(1,63,21,43),(2,44,22,64),(3,65,23,45),(4,46,24,66),(5,67,25,47),(6,48,26,68),(7,69,27,49),(8,50,28,70),(9,71,29,51),(10,52,30,72),(11,73,31,53),(12,54,32,74),(13,75,33,55),(14,56,34,76),(15,77,35,57),(16,58,36,78),(17,79,37,59),(18,60,38,80),(19,41,39,61),(20,62,40,42)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 21 | 18 | 26 | 18 |
| 4 | 13 | 5 | 8 |
| 27 | 37 | 37 | 21 |
| 16 | 2 | 2 | 11 |
| 34 | 40 | 23 | 27 |
| 7 | 7 | 35 | 27 |
| 0 | 0 | 25 | 12 |
| 0 | 0 | 30 | 16 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 9 | 0 | 0 | 2 |
| 0 | 9 | 18 | 1 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [21,4,27,16,18,13,37,2,26,5,37,2,18,8,21,11],[34,7,0,0,40,7,0,0,23,35,25,30,27,27,12,16],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,9,0,0,0,18,32,0,2,1,0,32] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | D8⋊C22 | C40.9C23 |
| kernel | C40.9C23 | D40⋊7C2 | C8⋊D10 | C8.D10 | C10×M4(2) | C2×C4○D20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 4 | 4 | 4 | 1 | 2 | 3 | 1 | 2 | 4 | 8 | 2 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{40}._9C_2^3 % in TeX
G:=Group("C40.9C2^3"); // GroupNames label
G:=SmallGroup(320,1420);
// by ID
G=gap.SmallGroup(320,1420);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^40=b^2=1,c^2=d^2=a^20,b*a*b=a^19,a*c=c*a,d*a*d^-1=a^21,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations