metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.23D4, D20.25D4, Dic10.25D4, (C10×D8)⋊2C2, (C2×D8)⋊10D5, C4.61(D4×D5), (C2×C8).87D10, C40.6C4⋊3C2, (C2×D4).67D10, C20.169(C2×D4), C20.D4⋊7C2, C5⋊4(D4.4D4), C8.27(C5⋊D4), D4.D10⋊4C2, D20.3C4⋊1C2, (C2×C40).32C22, C2.18(C20⋊2D4), (C2×C20).437C23, C4○D20.46C22, (D4×C10).86C22, C10.111(C4⋊D4), C4.Dic5.19C22, C22.20(D4⋊2D5), C4.81(C2×C5⋊D4), (C2×C4).126(C22×D5), (C2×C10).158(C4○D4), SmallGroup(320,787)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.23D4
G = < a,b,c | a40=c2=1, b4=a20, bab-1=a-1, cac=a9, cbc=a20b3 >
Subgroups: 398 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C8 [×3], C2×C4, C2×C4, D4 [×6], Q8, C23 [×2], D5, C10, C10 [×3], C2×C8, C2×C8, M4(2) [×4], D8 [×4], SD16 [×2], C2×D4 [×2], C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], C5⋊2C8 [×3], C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×4], C22×C10 [×2], D4.4D4, C8×D5, C8⋊D5, C4.Dic5, C4.Dic5 [×2], D4⋊D5 [×2], D4.D5 [×2], C2×C40, C5×D8 [×2], C4○D20, D4×C10 [×2], C40.6C4, C20.D4 [×2], D20.3C4, D4.D10 [×2], C10×D8, C40.23D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4.4D4, D4×D5, D4⋊2D5, C2×C5⋊D4, C20⋊2D4, C40.23D4
(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 51 11 41 21 71 31 61)(2 50 12 80 22 70 32 60)(3 49 13 79 23 69 33 59)(4 48 14 78 24 68 34 58)(5 47 15 77 25 67 35 57)(6 46 16 76 26 66 36 56)(7 45 17 75 27 65 37 55)(8 44 18 74 28 64 38 54)(9 43 19 73 29 63 39 53)(10 42 20 72 30 62 40 52)
(1 76)(2 45)(3 54)(4 63)(5 72)(6 41)(7 50)(8 59)(9 68)(10 77)(11 46)(12 55)(13 64)(14 73)(15 42)(16 51)(17 60)(18 69)(19 78)(20 47)(21 56)(22 65)(23 74)(24 43)(25 52)(26 61)(27 70)(28 79)(29 48)(30 57)(31 66)(32 75)(33 44)(34 53)(35 62)(36 71)(37 80)(38 49)(39 58)(40 67)
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), (1,51,11,41,21,71,31,61)(2,50,12,80,22,70,32,60)(3,49,13,79,23,69,33,59)(4,48,14,78,24,68,34,58)(5,47,15,77,25,67,35,57)(6,46,16,76,26,66,36,56)(7,45,17,75,27,65,37,55)(8,44,18,74,28,64,38,54)(9,43,19,73,29,63,39,53)(10,42,20,72,30,62,40,52), (1,76)(2,45)(3,54)(4,63)(5,72)(6,41)(7,50)(8,59)(9,68)(10,77)(11,46)(12,55)(13,64)(14,73)(15,42)(16,51)(17,60)(18,69)(19,78)(20,47)(21,56)(22,65)(23,74)(24,43)(25,52)(26,61)(27,70)(28,79)(29,48)(30,57)(31,66)(32,75)(33,44)(34,53)(35,62)(36,71)(37,80)(38,49)(39,58)(40,67)>;
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), (1,51,11,41,21,71,31,61)(2,50,12,80,22,70,32,60)(3,49,13,79,23,69,33,59)(4,48,14,78,24,68,34,58)(5,47,15,77,25,67,35,57)(6,46,16,76,26,66,36,56)(7,45,17,75,27,65,37,55)(8,44,18,74,28,64,38,54)(9,43,19,73,29,63,39,53)(10,42,20,72,30,62,40,52), (1,76)(2,45)(3,54)(4,63)(5,72)(6,41)(7,50)(8,59)(9,68)(10,77)(11,46)(12,55)(13,64)(14,73)(15,42)(16,51)(17,60)(18,69)(19,78)(20,47)(21,56)(22,65)(23,74)(24,43)(25,52)(26,61)(27,70)(28,79)(29,48)(30,57)(31,66)(32,75)(33,44)(34,53)(35,62)(36,71)(37,80)(38,49)(39,58)(40,67) );
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)], [(1,51,11,41,21,71,31,61),(2,50,12,80,22,70,32,60),(3,49,13,79,23,69,33,59),(4,48,14,78,24,68,34,58),(5,47,15,77,25,67,35,57),(6,46,16,76,26,66,36,56),(7,45,17,75,27,65,37,55),(8,44,18,74,28,64,38,54),(9,43,19,73,29,63,39,53),(10,42,20,72,30,62,40,52)], [(1,76),(2,45),(3,54),(4,63),(5,72),(6,41),(7,50),(8,59),(9,68),(10,77),(11,46),(12,55),(13,64),(14,73),(15,42),(16,51),(17,60),(18,69),(19,78),(20,47),(21,56),(22,65),(23,74),(24,43),(25,52),(26,61),(27,70),(28,79),(29,48),(30,57),(31,66),(32,75),(33,44),(34,53),(35,62),(36,71),(37,80),(38,49),(39,58),(40,67)])
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
size | 1 | 1 | 2 | 8 | 8 | 20 | 2 | 2 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 40 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D4.4D4 | D4×D5 | D4⋊2D5 | C40.23D4 |
kernel | C40.23D4 | C40.6C4 | C20.D4 | D20.3C4 | D4.D10 | C10×D8 | C40 | Dic10 | D20 | C2×D8 | C2×C10 | C2×C8 | C2×D4 | C8 | C5 | C4 | C22 | C1 |
# reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40.23D4 ►in GL6(𝔽41)
16 | 0 | 0 | 0 | 0 | 0 |
0 | 18 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 1 | 24 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 0 | 1 | 24 |
0 | 15 | 0 | 0 | 0 | 0 |
30 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 17 | 40 | 0 | 0 |
0 | 0 | 1 | 24 | 0 | 0 |
0 | 15 | 0 | 0 | 0 | 0 |
11 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,40,24,0,0,0,0,0,0,0,1,0,0,0,0,40,24],[0,30,0,0,0,0,15,0,0,0,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,40,0,0,0,0,40,0,0,0],[0,11,0,0,0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C40.23D4 in GAP, Magma, Sage, TeX
C_{40}._{23}D_4
% in TeX
G:=Group("C40.23D4");
// GroupNames label
G:=SmallGroup(320,787);
// by ID
G=gap.SmallGroup(320,787);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,1123,297,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations