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)), C4.67(C23×D5), C23.58(C4×D5), C5⋊5(C22×M4(2)), C8⋊D5⋊17C22, (C10×M4(2))⋊8C2, C10.52(C23×C4), (C23×D5).12C4, (C4×D5).95C23, (C2×C20).881C23, C20.150(C22×C4), D10.55(C22×C4), (C22×C4).373D10, C4.Dic5⋊25C22, (C5×M4(2))⋊24C22, (C22×Dic5).24C4, Dic5.57(C22×C4), (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
Generators and relations for C2×D5×M4(2)
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 >
Subgroups: 862 in 298 conjugacy classes, 159 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, C22×M4(2), C8×D5, C8⋊D5, C2×C5⋊2C8, C4.Dic5, C2×C40, C5×M4(2), C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D5×C2×C8, C2×C8⋊D5, D5×M4(2), C2×C4.Dic5, C10×M4(2), D5×C22×C4, C2×D5×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, C24, D10, C2×M4(2), C23×C4, C4×D5, C22×D5, C22×M4(2), C2×C4×D5, C23×D5, D5×M4(2), D5×C22×C4, C2×D5×M4(2)
►On 80 points
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 78)(26 79)(27 80)(28 73)(29 74)(30 75)(31 76)(32 77)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(1 66 77 34 10)(2 67 78 35 11)(3 68 79 36 12)(4 69 80 37 13)(5 70 73 38 14)(6 71 74 39 15)(7 72 75 40 16)(8 65 76 33 9)(17 41 64 52 25)(18 42 57 53 26)(19 43 58 54 27)(20 44 59 55 28)(21 45 60 56 29)(22 46 61 49 30)(23 47 62 50 31)(24 48 63 51 32)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)
(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)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)
G:=sub<Sym(80)| (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,78)(26,79)(27,80)(28,73)(29,74)(30,75)(31,76)(32,77)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,66,77,34,10)(2,67,78,35,11)(3,68,79,36,12)(4,69,80,37,13)(5,70,73,38,14)(6,71,74,39,15)(7,72,75,40,16)(8,65,76,33,9)(17,41,64,52,25)(18,42,57,53,26)(19,43,58,54,27)(20,44,59,55,28)(21,45,60,56,29)(22,46,61,49,30)(23,47,62,50,31)(24,48,63,51,32), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63), (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)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80)>;
G:=Group( (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,78)(26,79)(27,80)(28,73)(29,74)(30,75)(31,76)(32,77)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,66,77,34,10)(2,67,78,35,11)(3,68,79,36,12)(4,69,80,37,13)(5,70,73,38,14)(6,71,74,39,15)(7,72,75,40,16)(8,65,76,33,9)(17,41,64,52,25)(18,42,57,53,26)(19,43,58,54,27)(20,44,59,55,28)(21,45,60,56,29)(22,46,61,49,30)(23,47,62,50,31)(24,48,63,51,32), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63), (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)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80) );
G=PermutationGroup([[(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,33),(24,34),(25,78),(26,79),(27,80),(28,73),(29,74),(30,75),(31,76),(32,77),(49,72),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71)], [(1,66,77,34,10),(2,67,78,35,11),(3,68,79,36,12),(4,69,80,37,13),(5,70,73,38,14),(6,71,74,39,15),(7,72,75,40,16),(8,65,76,33,9),(17,41,64,52,25),(18,42,57,53,26),(19,43,58,54,27),(20,44,59,55,28),(21,45,60,56,29),(22,46,61,49,30),(23,47,62,50,31),(24,48,63,51,32)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,64),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63)], [(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),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64),(65,69),(67,71),(74,78),(76,80)]])
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 |
Matrix representation of C2×D5×M4(2) ►in 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] >;
C2×D5×M4(2) 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