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), C20.239(C2×D4), (C2×C20).205D4, (C2×C4).157D20, (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, C10.27(C22×D4), C2.29(C22×D20), C22.22(C2×D20), (C2×C20).798C23, (C22×C4).267D10, (C22×C10).120D4, (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
Generators and relations for C40.9C23
G = < a,b,c,d | a40=b2=1, c2=d2=a20, bab=a19, ac=ca, dad-1=a21, bc=cb, bd=db, cd=dc >
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
►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 53)(42 72)(43 51)(44 70)(45 49)(46 68)(48 66)(50 64)(52 62)(54 60)(55 79)(56 58)(57 77)(59 75)(61 73)(63 71)(65 69)(74 80)(76 78)
(1 67 21 47)(2 68 22 48)(3 69 23 49)(4 70 24 50)(5 71 25 51)(6 72 26 52)(7 73 27 53)(8 74 28 54)(9 75 29 55)(10 76 30 56)(11 77 31 57)(12 78 32 58)(13 79 33 59)(14 80 34 60)(15 41 35 61)(16 42 36 62)(17 43 37 63)(18 44 38 64)(19 45 39 65)(20 46 40 66)
(1 47 21 67)(2 68 22 48)(3 49 23 69)(4 70 24 50)(5 51 25 71)(6 72 26 52)(7 53 27 73)(8 74 28 54)(9 55 29 75)(10 76 30 56)(11 57 31 77)(12 78 32 58)(13 59 33 79)(14 80 34 60)(15 61 35 41)(16 42 36 62)(17 63 37 43)(18 44 38 64)(19 65 39 45)(20 46 40 66)
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)(41,53)(42,72)(43,51)(44,70)(45,49)(46,68)(48,66)(50,64)(52,62)(54,60)(55,79)(56,58)(57,77)(59,75)(61,73)(63,71)(65,69)(74,80)(76,78), (1,67,21,47)(2,68,22,48)(3,69,23,49)(4,70,24,50)(5,71,25,51)(6,72,26,52)(7,73,27,53)(8,74,28,54)(9,75,29,55)(10,76,30,56)(11,77,31,57)(12,78,32,58)(13,79,33,59)(14,80,34,60)(15,41,35,61)(16,42,36,62)(17,43,37,63)(18,44,38,64)(19,45,39,65)(20,46,40,66), (1,47,21,67)(2,68,22,48)(3,49,23,69)(4,70,24,50)(5,51,25,71)(6,72,26,52)(7,53,27,73)(8,74,28,54)(9,55,29,75)(10,76,30,56)(11,57,31,77)(12,78,32,58)(13,59,33,79)(14,80,34,60)(15,61,35,41)(16,42,36,62)(17,63,37,43)(18,44,38,64)(19,65,39,45)(20,46,40,66)>;
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)(41,53)(42,72)(43,51)(44,70)(45,49)(46,68)(48,66)(50,64)(52,62)(54,60)(55,79)(56,58)(57,77)(59,75)(61,73)(63,71)(65,69)(74,80)(76,78), (1,67,21,47)(2,68,22,48)(3,69,23,49)(4,70,24,50)(5,71,25,51)(6,72,26,52)(7,73,27,53)(8,74,28,54)(9,75,29,55)(10,76,30,56)(11,77,31,57)(12,78,32,58)(13,79,33,59)(14,80,34,60)(15,41,35,61)(16,42,36,62)(17,43,37,63)(18,44,38,64)(19,45,39,65)(20,46,40,66), (1,47,21,67)(2,68,22,48)(3,49,23,69)(4,70,24,50)(5,51,25,71)(6,72,26,52)(7,53,27,73)(8,74,28,54)(9,55,29,75)(10,76,30,56)(11,57,31,77)(12,78,32,58)(13,59,33,79)(14,80,34,60)(15,61,35,41)(16,42,36,62)(17,63,37,43)(18,44,38,64)(19,65,39,45)(20,46,40,66) );
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),(41,53),(42,72),(43,51),(44,70),(45,49),(46,68),(48,66),(50,64),(52,62),(54,60),(55,79),(56,58),(57,77),(59,75),(61,73),(63,71),(65,69),(74,80),(76,78)], [(1,67,21,47),(2,68,22,48),(3,69,23,49),(4,70,24,50),(5,71,25,51),(6,72,26,52),(7,73,27,53),(8,74,28,54),(9,75,29,55),(10,76,30,56),(11,77,31,57),(12,78,32,58),(13,79,33,59),(14,80,34,60),(15,41,35,61),(16,42,36,62),(17,43,37,63),(18,44,38,64),(19,45,39,65),(20,46,40,66)], [(1,47,21,67),(2,68,22,48),(3,49,23,69),(4,70,24,50),(5,51,25,71),(6,72,26,52),(7,53,27,73),(8,74,28,54),(9,55,29,75),(10,76,30,56),(11,57,31,77),(12,78,32,58),(13,59,33,79),(14,80,34,60),(15,61,35,41),(16,42,36,62),(17,63,37,43),(18,44,38,64),(19,65,39,45),(20,46,40,66)])
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 |
Matrix representation of C40.9C23 ►in 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] >;
C40.9C23 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