metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D4).9F5, (D4×C10).6C4, C23.F5⋊3C2, Dic5.5(C2×D4), C23.15(C2×F5), Dic5.D4⋊3C2, (C2×Dic5).120D4, C22.3(C22⋊F5), C22.13(C22×F5), C22.F5.3C22, (C22×Dic5).11C4, Dic5.45(C22⋊C4), (C2×Dic5).173C23, (C2×Dic10).145C22, C5⋊2(M4(2).8C22), (C22×Dic5).189C22, (C2×C4×D5).5C4, (C2×C4).5(C2×F5), (C2×C5⋊D4).9C4, (C2×C20).25(C2×C4), (C2×C22.F5)⋊7C2, C2.22(C2×C22⋊F5), (C2×D4⋊2D5).8C2, C10.21(C2×C22⋊C4), (C22×D5).8(C2×C4), (C2×C10).3(C22⋊C4), (C2×C10).80(C22×C4), (C22×C10).28(C2×C4), (C2×C5⋊D4).89C22, (C2×Dic5).193(C2×C4), SmallGroup(320,1115)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 586 in 150 conjugacy classes, 48 normal (28 characteristic)
C1, C2, C2 [×5], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×4], C2×C4, C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×2], C23, D5, C10, C10 [×4], C2×C8 [×2], M4(2) [×6], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], Dic5, C20, D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×3], C4.D4 [×2], C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, C5⋊C8 [×4], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], M4(2).8C22, C2×C5⋊C8 [×2], C22.F5 [×4], C22.F5 [×2], C2×Dic10, C2×C4×D5, D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Dic5.D4 [×2], C23.F5 [×2], C2×C22.F5 [×2], C2×D4⋊2D5, (C2×D4).9F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×F5 [×3], M4(2).8C22, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×D4).9F5
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d5=1, e4=b2, ebe-1=ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=b-1, bd=db, cd=dc, ece-1=b2c, ede-1=d3 >
►On 80 points
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)
(1 67 5 71)(2 72 6 68)(3 65 7 69)(4 70 8 66)(9 57 13 61)(10 62 14 58)(11 63 15 59)(12 60 16 64)(17 27 21 31)(18 32 22 28)(19 25 23 29)(20 30 24 26)(33 49 37 53)(34 54 38 50)(35 55 39 51)(36 52 40 56)(41 79 45 75)(42 76 46 80)(43 77 47 73)(44 74 48 78)
(1 65)(2 70)(3 67)(4 72)(5 69)(6 66)(7 71)(8 68)(9 39)(10 36)(11 33)(12 38)(13 35)(14 40)(15 37)(16 34)(17 45)(18 42)(19 47)(20 44)(21 41)(22 46)(23 43)(24 48)(25 77)(26 74)(27 79)(28 76)(29 73)(30 78)(31 75)(32 80)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)
(1 33 75 19 57)(2 20 34 58 76)(3 59 21 77 35)(4 78 60 36 22)(5 37 79 23 61)(6 24 38 62 80)(7 63 17 73 39)(8 74 64 40 18)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(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)
G:=sub<Sym(80)| (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,67,5,71)(2,72,6,68)(3,65,7,69)(4,70,8,66)(9,57,13,61)(10,62,14,58)(11,63,15,59)(12,60,16,64)(17,27,21,31)(18,32,22,28)(19,25,23,29)(20,30,24,26)(33,49,37,53)(34,54,38,50)(35,55,39,51)(36,52,40,56)(41,79,45,75)(42,76,46,80)(43,77,47,73)(44,74,48,78), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,39)(10,36)(11,33)(12,38)(13,35)(14,40)(15,37)(16,34)(17,45)(18,42)(19,47)(20,44)(21,41)(22,46)(23,43)(24,48)(25,77)(26,74)(27,79)(28,76)(29,73)(30,78)(31,75)(32,80)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,33,75,19,57)(2,20,34,58,76)(3,59,21,77,35)(4,78,60,36,22)(5,37,79,23,61)(6,24,38,62,80)(7,63,17,73,39)(8,74,64,40,18)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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)>;
G:=Group( (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,67,5,71)(2,72,6,68)(3,65,7,69)(4,70,8,66)(9,57,13,61)(10,62,14,58)(11,63,15,59)(12,60,16,64)(17,27,21,31)(18,32,22,28)(19,25,23,29)(20,30,24,26)(33,49,37,53)(34,54,38,50)(35,55,39,51)(36,52,40,56)(41,79,45,75)(42,76,46,80)(43,77,47,73)(44,74,48,78), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,39)(10,36)(11,33)(12,38)(13,35)(14,40)(15,37)(16,34)(17,45)(18,42)(19,47)(20,44)(21,41)(22,46)(23,43)(24,48)(25,77)(26,74)(27,79)(28,76)(29,73)(30,78)(31,75)(32,80)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,33,75,19,57)(2,20,34,58,76)(3,59,21,77,35)(4,78,60,36,22)(5,37,79,23,61)(6,24,38,62,80)(7,63,17,73,39)(8,74,64,40,18)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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) );
G=PermutationGroup([(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79)], [(1,67,5,71),(2,72,6,68),(3,65,7,69),(4,70,8,66),(9,57,13,61),(10,62,14,58),(11,63,15,59),(12,60,16,64),(17,27,21,31),(18,32,22,28),(19,25,23,29),(20,30,24,26),(33,49,37,53),(34,54,38,50),(35,55,39,51),(36,52,40,56),(41,79,45,75),(42,76,46,80),(43,77,47,73),(44,74,48,78)], [(1,65),(2,70),(3,67),(4,72),(5,69),(6,66),(7,71),(8,68),(9,39),(10,36),(11,33),(12,38),(13,35),(14,40),(15,37),(16,34),(17,45),(18,42),(19,47),(20,44),(21,41),(22,46),(23,43),(24,48),(25,77),(26,74),(27,79),(28,76),(29,73),(30,78),(31,75),(32,80),(49,59),(50,64),(51,61),(52,58),(53,63),(54,60),(55,57),(56,62)], [(1,33,75,19,57),(2,20,34,58,76),(3,59,21,77,35),(4,78,60,36,22),(5,37,79,23,61),(6,24,38,62,80),(7,63,17,73,39),(8,74,64,40,18),(9,71,53,45,29),(10,46,72,30,54),(11,31,47,55,65),(12,56,32,66,48),(13,67,49,41,25),(14,42,68,26,50),(15,27,43,51,69),(16,52,28,70,44)], [(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)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 1 |
| 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 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 22 | 0 | 0 | 9 |
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 31 | 0 | 32 |
| 0 | 0 | 0 | 0 | 9 | 3 | 9 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 39 | 39 | 34 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 15 | 36 | 36 | 0 | 0 | 0 | 0 |
| 33 | 18 | 35 | 5 | 0 | 0 | 0 | 0 |
| 32 | 32 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 29 | 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 29 | 1 |
| 0 | 0 | 0 | 0 | 13 | 1 | 8 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 | 14 | 0 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,4,7,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,28,0,22,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,26,29,19,9,0,0,0,0,5,15,31,3,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0],[40,8,39,0,0,0,0,0,40,7,39,0,0,0,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,33,32,0,0,0,0,0,15,18,32,29,0,0,0,0,36,35,8,15,0,0,0,0,36,5,0,15,0,0,0,0,0,0,0,0,33,0,13,18,0,0,0,0,0,0,1,0,0,0,0,0,37,29,8,14,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 20 | 4 | 5 | 5 | 10 | 10 | 10 | 20 | 4 | 20 | ··· | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | F5 | C2×F5 | C2×F5 | M4(2).8C22 | C22⋊F5 | (C2×D4).9F5 |
| kernel | (C2×D4).9F5 | Dic5.D4 | C23.F5 | C2×C22.F5 | C2×D4⋊2D5 | C2×C4×D5 | C22×Dic5 | C2×C5⋊D4 | D4×C10 | C2×Dic5 | C2×D4 | C2×C4 | C23 | C5 | C22 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 2 |
In GAP, Magma, Sage, TeX
(C_2\times D_4)._9F_5
% in TeX
G:=Group("(C2xD4).9F5"); // GroupNames label
G:=SmallGroup(320,1115);
// by ID
G=gap.SmallGroup(320,1115);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,1684,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=1,e^4=b^2,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^3>;
// generators/relations