direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5×M4(2), C40⋊7C23, C20.68C24, (C2×C8)⋊29D10, C8⋊7(C22×D5), (C2×C40)⋊23C22, C5⋊2C8⋊12C23, (C8×D5)⋊21C22, C10⋊5(C2×M4(2)), C23.58(C4×D5), C4.67(C23×D5), C5⋊5(C22×M4(2)), C8⋊D5⋊17C22, (C10×M4(2))⋊8C2, C10.52(C23×C4), (C23×D5).12C4, (C4×D5).95C23, C20.150(C22×C4), (C2×C20).881C23, D10.55(C22×C4), (C22×C4).373D10, C4.Dic5⋊25C22, (C5×M4(2))⋊24C22, Dic5.57(C22×C4), (C22×Dic5).24C4, (C22×C20).263C22, (D5×C2×C8)⋊28C2, (C2×C4×D5).14C4, C4.122(C2×C4×D5), (C2×C8⋊D5)⋊26C2, (D5×C22×C4).8C2, C2.32(D5×C22×C4), C22.76(C2×C4×D5), (C4×D5).78(C2×C4), (C2×C4).162(C4×D5), (C2×C20).303(C2×C4), (C2×C5⋊2C8)⋊47C22, (C2×C4.Dic5)⋊24C2, (C2×C4×D5).323C22, (C2×C4).604(C22×D5), (C2×C10).125(C22×C4), (C22×C10).145(C2×C4), (C2×Dic5).160(C2×C4), (C22×D5).112(C2×C4), SmallGroup(320,1415)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 862 in 298 conjugacy classes, 159 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×20], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×13], C24, Dic5 [×4], C20 [×2], C20 [×2], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8 [×2], C2×M4(2), C2×M4(2) [×11], C23×C4, C5⋊2C8 [×4], C40 [×4], C4×D5 [×16], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C22×M4(2), C8×D5 [×8], C8⋊D5 [×8], C2×C5⋊2C8 [×2], C4.Dic5 [×4], C2×C40 [×2], C5×M4(2) [×4], C2×C4×D5 [×4], C2×C4×D5 [×8], C22×Dic5, C22×C20, C23×D5, D5×C2×C8 [×2], C2×C8⋊D5 [×2], D5×M4(2) [×8], C2×C4.Dic5, C10×M4(2), D5×C22×C4, C2×D5×M4(2)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, M4(2) [×4], C22×C4 [×14], C24, D10 [×7], C2×M4(2) [×6], C23×C4, C4×D5 [×4], C22×D5 [×7], C22×M4(2), C2×C4×D5 [×6], C23×D5, D5×M4(2) [×2], D5×C22×C4, C2×D5×M4(2)
Generators and relations
G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >
►On 80 points
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(1 66 62 26 15)(2 67 63 27 16)(3 68 64 28 9)(4 69 57 29 10)(5 70 58 30 11)(6 71 59 31 12)(7 72 60 32 13)(8 65 61 25 14)(17 46 79 52 38)(18 47 80 53 39)(19 48 73 54 40)(20 41 74 55 33)(21 42 75 56 34)(22 43 76 49 35)(23 44 77 50 36)(24 45 78 51 37)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 73)
(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 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)
G:=sub<Sym(80)| (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,66,62,26,15)(2,67,63,27,16)(3,68,64,28,9)(4,69,57,29,10)(5,70,58,30,11)(6,71,59,31,12)(7,72,60,32,13)(8,65,61,25,14)(17,46,79,52,38)(18,47,80,53,39)(19,48,73,54,40)(20,41,74,55,33)(21,42,75,56,34)(22,43,76,49,35)(23,44,77,50,36)(24,45,78,51,37), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79)>;
G:=Group( (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,66,62,26,15)(2,67,63,27,16)(3,68,64,28,9)(4,69,57,29,10)(5,70,58,30,11)(6,71,59,31,12)(7,72,60,32,13)(8,65,61,25,14)(17,46,79,52,38)(18,47,80,53,39)(19,48,73,54,40)(20,41,74,55,33)(21,42,75,56,34)(22,43,76,49,35)(23,44,77,50,36)(24,45,78,51,37), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79) );
G=PermutationGroup([(1,78),(2,79),(3,80),(4,73),(5,74),(6,75),(7,76),(8,77),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57),(49,72),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71)], [(1,66,62,26,15),(2,67,63,27,16),(3,68,64,28,9),(4,69,57,29,10),(5,70,58,30,11),(6,71,59,31,12),(7,72,60,32,13),(8,65,61,25,14),(17,46,79,52,38),(18,47,80,53,39),(19,48,73,54,40),(20,41,74,55,33),(21,42,75,56,34),(22,43,76,49,35),(23,44,77,50,36),(24,45,78,51,37)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,73)], [(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 34 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 32 | 36 |
| 0 | 0 | 0 | 18 | 9 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,18,0,0,0,36,9],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,40] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | D10 | D10 | D10 | C4×D5 | C4×D5 | D5×M4(2) |
| kernel | C2×D5×M4(2) | D5×C2×C8 | C2×C8⋊D5 | D5×M4(2) | C2×C4.Dic5 | C10×M4(2) | D5×C22×C4 | C2×C4×D5 | C22×Dic5 | C23×D5 | C2×M4(2) | D10 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 8 | 4 | 8 | 2 | 12 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_5\times M_{4(2)} % in TeX
G:=Group("C2xD5xM4(2)"); // GroupNames label
G:=SmallGroup(320,1415);
// by ID
G=gap.SmallGroup(320,1415);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,80,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations