direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40⋊C4, D10.10SD16, C8⋊8(C2×F5), (C2×C8)⋊7F5, C40⋊8(C2×C4), (C2×C40)⋊7C4, (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, (C4×D5).75C23, (C22×D5).97D4, (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
Subgroups: 538 in 130 conjugacy classes, 60 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×4], C10, C10 [×2], C4⋊C4 [×6], C2×C8, C2×C8 [×5], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C4.Q8 [×4], C2×C4⋊C4 [×2], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×8], C22×D5, C2×C4.Q8, C8×D5 [×4], C2×C5⋊2C8, C2×C40, C4⋊F5 [×4], C4⋊F5 [×2], C2×C4×D5, C22×F5 [×2], C40⋊C4 [×4], D5×C2×C8, C2×C4⋊F5 [×2], C2×C40⋊C4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, F5, C4.Q8 [×4], C2×C4⋊C4, C2×SD16 [×2], C2×F5 [×3], C2×C4.Q8, C4⋊F5 [×2], C22×F5, C40⋊C4 [×2], C2×C4⋊F5, C2×C40⋊C4
Generators and relations
G = < a,b,c | a2=b40=c4=1, ab=ba, ac=ca, cbc-1=b3 >
►On 80 points
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)
(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 76 61 56)(42 63 70 59)(43 50 79 62)(44 77 48 65)(45 64 57 68)(46 51 66 71)(47 78 75 74)(49 52 53 80)(54 67 58 55)(60 69 72 73)
G:=sub<Sym(80)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (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)(41,76,61,56)(42,63,70,59)(43,50,79,62)(44,77,48,65)(45,64,57,68)(46,51,66,71)(47,78,75,74)(49,52,53,80)(54,67,58,55)(60,69,72,73)>;
G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (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)(41,76,61,56)(42,63,70,59)(43,50,79,62)(44,77,48,65)(45,64,57,68)(46,51,66,71)(47,78,75,74)(49,52,53,80)(54,67,58,55)(60,69,72,73) );
G=PermutationGroup([(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55)], [(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),(41,76,61,56),(42,63,70,59),(43,50,79,62),(44,77,48,65),(45,64,57,68),(46,51,66,71),(47,78,75,74),(49,52,53,80),(54,67,58,55),(60,69,72,73)])
Matrix representation ►G ⊆ 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] >;
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 |
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