direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40⋊C4, D10.10SD16, C8⋊8(C2×F5), (C2×C8)⋊7F5, (C2×C40)⋊7C4, C40⋊8(C2×C4), (C8×D5)⋊8C4, C10⋊(C4.Q8), D5⋊(C4.Q8), (C4×D5).82D4, C4.12(C4⋊F5), C20.19(C4⋊C4), (C4×D5).24Q8, D10.29(C2×D4), D5.1(C2×SD16), C4⋊F5.15C22, D10.27(C4⋊C4), C4.35(C22×F5), C20.75(C22×C4), Dic5.13(C2×Q8), (C2×Dic5).33Q8, (C22×D5).97D4, (C4×D5).75C23, (C8×D5).56C22, C22.22(C4⋊F5), Dic5.28(C4⋊C4), C5⋊(C2×C4.Q8), (D5×C2×C8).24C2, (C2×C5⋊2C8)⋊17C4, C2.14(C2×C4⋊F5), C5⋊2C8⋊34(C2×C4), C10.11(C2×C4⋊C4), (C2×C4⋊F5).13C2, (C4×D5).84(C2×C4), (C2×C4).136(C2×F5), (C2×C10).19(C4⋊C4), (C2×C20).126(C2×C4), (C2×C4×D5).394C22, SmallGroup(320,1057)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C40⋊C4
G = < a,b,c | a2=b40=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Subgroups: 538 in 130 conjugacy classes, 60 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C4.Q8, C2×C4⋊C4, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C2×C4.Q8, C8×D5, C2×C5⋊2C8, C2×C40, C4⋊F5, C4⋊F5, C2×C4×D5, C22×F5, C40⋊C4, D5×C2×C8, C2×C4⋊F5, C2×C40⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, F5, C4.Q8, C2×C4⋊C4, C2×SD16, C2×F5, C2×C4.Q8, C4⋊F5, C22×F5, C40⋊C4, C2×C4⋊F5, C2×C40⋊C4
►On 80 points
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)
(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 26 21 6)(2 13 30 9)(3 40 39 12)(4 27 8 15)(5 14 17 18)(7 28 35 24)(10 29 22 33)(11 16 31 36)(19 32 23 20)(25 34 37 38)(41 44 45 72)(42 71 54 75)(43 58 63 78)(46 59 50 47)(48 73 68 53)(49 60 77 56)(51 74 55 62)(52 61 64 65)(57 76 69 80)(66 79 70 67)
G:=sub<Sym(80)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67), (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), (1,26,21,6)(2,13,30,9)(3,40,39,12)(4,27,8,15)(5,14,17,18)(7,28,35,24)(10,29,22,33)(11,16,31,36)(19,32,23,20)(25,34,37,38)(41,44,45,72)(42,71,54,75)(43,58,63,78)(46,59,50,47)(48,73,68,53)(49,60,77,56)(51,74,55,62)(52,61,64,65)(57,76,69,80)(66,79,70,67)>;
G:=Group( (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67), (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), (1,26,21,6)(2,13,30,9)(3,40,39,12)(4,27,8,15)(5,14,17,18)(7,28,35,24)(10,29,22,33)(11,16,31,36)(19,32,23,20)(25,34,37,38)(41,44,45,72)(42,71,54,75)(43,58,63,78)(46,59,50,47)(48,73,68,53)(49,60,77,56)(51,74,55,62)(52,61,64,65)(57,76,69,80)(66,79,70,67) );
G=PermutationGroup([[(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67)], [(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)], [(1,26,21,6),(2,13,30,9),(3,40,39,12),(4,27,8,15),(5,14,17,18),(7,28,35,24),(10,29,22,33),(11,16,31,36),(19,32,23,20),(25,34,37,38),(41,44,45,72),(42,71,54,75),(43,58,63,78),(46,59,50,47),(48,73,68,53),(49,60,77,56),(51,74,55,62),(52,61,64,65),(57,76,69,80),(66,79,70,67)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 10 | 10 | 20 | ··· | 20 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | Q8 | D4 | SD16 | F5 | C2×F5 | C2×F5 | C4⋊F5 | C4⋊F5 | C40⋊C4 |
| kernel | C2×C40⋊C4 | C40⋊C4 | D5×C2×C8 | C2×C4⋊F5 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C2×C40⋊C4 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 26 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 27 | 0 | 34 |
| 0 | 0 | 0 | 0 | 7 | 34 | 34 | 7 |
| 0 | 0 | 0 | 0 | 34 | 0 | 27 | 27 |
| 0 | 0 | 0 | 0 | 14 | 7 | 14 | 0 |
| 17 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 29 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 14 | 14 | 0 | 7 |
| 0 | 0 | 0 | 0 | 27 | 34 | 27 | 0 |
| 0 | 0 | 0 | 0 | 34 | 7 | 7 | 34 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,26,0,0,0,0,0,0,30,30,0,0,0,0,0,0,0,0,35,16,0,0,0,0,0,0,31,6,0,0,0,0,0,0,0,0,27,7,34,14,0,0,0,0,27,34,0,7,0,0,0,0,0,34,27,14,0,0,0,0,34,7,27,0],[17,29,0,0,0,0,0,0,24,24,0,0,0,0,0,0,0,0,28,11,0,0,0,0,0,0,33,13,0,0,0,0,0,0,0,0,7,14,27,34,0,0,0,0,0,14,34,7,0,0,0,0,14,0,27,7,0,0,0,0,14,7,0,34] >;
C2×C40⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{40}\rtimes C_4 % in TeX
G:=Group("C2xC40:C4"); // GroupNames label
G:=SmallGroup(320,1057);
// by ID
G=gap.SmallGroup(320,1057);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,1684,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^2=b^40=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations