direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D40⋊C2, C40⋊4C23, SD16⋊8D10, D20⋊2C23, C20.6C24, D40⋊20C22, (C2×C8)⋊10D10, C4.43(D4×D5), C8⋊4(C22×D5), (C2×D40)⋊26C2, C5⋊2C8⋊2C23, (C2×Q8)⋊21D10, (C2×SD16)⋊4D5, (C4×D5).15D4, C20.81(C2×D4), (D4×D5)⋊6C22, Q8⋊D5⋊8C22, (C5×Q8)⋊2C23, Q8⋊2(C22×D5), C4.6(C23×D5), C10⋊3(C8⋊C22), (C2×C40)⋊13C22, D4⋊D5⋊10C22, (C10×SD16)⋊5C2, D10.84(C2×D4), C8⋊D5⋊8C22, D4.4(C22×D5), (C5×D4).4C23, (C4×D5).3C23, (C2×D4).182D10, (C2×D20)⋊33C22, Dic5.95(C2×D4), (Q8×C10)⋊18C22, Q8⋊2D5⋊5C22, (C5×SD16)⋊8C22, C22.139(D4×D5), (C2×C20).523C23, (C2×Dic5).248D4, (C22×D5).135D4, C10.107(C22×D4), (D4×C10).164C22, (C2×D4×D5)⋊23C2, C5⋊3(C2×C8⋊C22), C2.80(C2×D4×D5), (C2×D4⋊D5)⋊27C2, (C2×C8⋊D5)⋊4C2, (C2×Q8⋊D5)⋊26C2, (C2×C5⋊2C8)⋊15C22, (C2×Q8⋊2D5)⋊14C2, (C2×C10).396(C2×D4), (C2×C4×D5).165C22, (C2×C4).612(C22×D5), SmallGroup(320,1431)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D40⋊C2
G = < a,b,c,d | a2=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b11, cd=dc >
Subgroups: 1374 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×D5, C22×C10, C2×C8⋊C22, C8⋊D5, D40, C2×C5⋊2C8, D4⋊D5, Q8⋊D5, C2×C40, C5×SD16, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, D4×D5, D4×D5, Q8⋊2D5, Q8⋊2D5, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C2×C8⋊D5, C2×D40, D40⋊C2, C2×D4⋊D5, C2×Q8⋊D5, C10×SD16, C2×D4×D5, C2×Q8⋊2D5, C2×D40⋊C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8⋊C22, C22×D4, C22×D5, C2×C8⋊C22, D4×D5, C23×D5, D40⋊C2, C2×D4×D5, C2×D40⋊C2
►On 80 points
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 51)(42 50)(43 49)(44 48)(45 47)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
(1 11)(2 22)(3 33)(5 15)(6 26)(7 37)(9 19)(10 30)(13 23)(14 34)(17 27)(18 38)(21 31)(25 35)(29 39)(41 71)(43 53)(44 64)(45 75)(47 57)(48 68)(49 79)(51 61)(52 72)(55 65)(56 76)(59 69)(60 80)(63 73)(67 77)
G:=sub<Sym(80)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,11)(2,22)(3,33)(5,15)(6,26)(7,37)(9,19)(10,30)(13,23)(14,34)(17,27)(18,38)(21,31)(25,35)(29,39)(41,71)(43,53)(44,64)(45,75)(47,57)(48,68)(49,79)(51,61)(52,72)(55,65)(56,76)(59,69)(60,80)(63,73)(67,77)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,11)(2,22)(3,33)(5,15)(6,26)(7,37)(9,19)(10,30)(13,23)(14,34)(17,27)(18,38)(21,31)(25,35)(29,39)(41,71)(43,53)(44,64)(45,75)(47,57)(48,68)(49,79)(51,61)(52,72)(55,65)(56,76)(59,69)(60,80)(63,73)(67,77) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,51),(42,50),(43,49),(44,48),(45,47),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)], [(1,11),(2,22),(3,33),(5,15),(6,26),(7,37),(9,19),(10,30),(13,23),(14,34),(17,27),(18,38),(21,31),(25,35),(29,39),(41,71),(43,53),(44,64),(45,75),(47,57),(48,68),(49,79),(51,61),(52,72),(55,65),(56,76),(59,69),(60,80),(63,73),(67,77)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | C8⋊C22 | D4×D5 | D4×D5 | D40⋊C2 |
| kernel | C2×D40⋊C2 | C2×C8⋊D5 | C2×D40 | D40⋊C2 | C2×D4⋊D5 | C2×Q8⋊D5 | C10×SD16 | C2×D4×D5 | C2×Q8⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×D40⋊C2 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 7 | 35 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 26 | 26 | 26 |
| 0 | 0 | 15 | 34 | 15 | 34 |
| 0 | 0 | 28 | 28 | 0 | 0 |
| 0 | 0 | 13 | 24 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 39 |
| 0 | 0 | 40 | 0 | 39 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 | 0 | 2 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,7,0,0,0,0,35,0,0,0,0,0,0,0,26,15,28,13,0,0,26,34,28,24,0,0,26,15,0,0,0,0,26,34,0,0],[40,0,0,0,0,0,1,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,39,0,1,0,0,39,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,0,40,0,0,0,0,2,0,40] >;
C2×D40⋊C2 in GAP, Magma, Sage, TeX
C_2\times D_{40}\rtimes C_2 % in TeX
G:=Group("C2xD40:C2"); // GroupNames label
G:=SmallGroup(320,1431);
// by ID
G=gap.SmallGroup(320,1431);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^11,c*d=d*c>;
// generators/relations