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, Q8⋊2(C22×D5), (C5×Q8)⋊2C23, C4.6(C23×D5), C10⋊3(C8⋊C22), (C2×C40)⋊13C22, D4⋊D5⋊10C22, (C10×SD16)⋊5C2, D10.84(C2×D4), C8⋊D5⋊8C22, (C5×D4).4C23, D4.4(C22×D5), (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
Subgroups: 1374 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×24], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×2], D4 [×15], Q8 [×2], Q8, C23 [×12], D5 [×6], C10, C10 [×2], C10 [×2], C2×C8, C2×C8, M4(2) [×4], D8 [×8], SD16 [×4], SD16 [×4], C22×C4 [×2], C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×6], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×18], C2×C10, C2×C10 [×4], C2×M4(2), C2×D8 [×2], C2×SD16, C2×SD16, C8⋊C22 [×8], C22×D4, C2×C4○D4, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], D20 [×4], D20 [×6], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×10], C22×C10, C2×C8⋊C22, C8⋊D5 [×4], D40 [×4], C2×C5⋊2C8, D4⋊D5 [×4], Q8⋊D5 [×4], C2×C40, C5×SD16 [×4], C2×C4×D5, C2×C4×D5, C2×D20 [×2], C2×D20, D4×D5 [×4], D4×D5 [×2], Q8⋊2D5 [×4], Q8⋊2D5 [×2], C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C2×C8⋊D5, C2×D40, D40⋊C2 [×8], C2×D4⋊D5, C2×Q8⋊D5, C10×SD16, C2×D4×D5, C2×Q8⋊2D5, C2×D40⋊C2
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C22×D5 [×7], C2×C8⋊C22, D4×D5 [×2], C23×D5, D40⋊C2 [×2], C2×D4×D5, C2×D40⋊C2
Generators and relations
G = < a,b,c,d | a2=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b11, cd=dc >
►On 80 points
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)
(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 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)
(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)(42 52)(43 63)(44 74)(46 56)(47 67)(48 78)(50 60)(51 71)(54 64)(55 75)(58 68)(59 79)(62 72)(66 76)(70 80)
G:=sub<Sym(80)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65), (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)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (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)(42,52)(43,63)(44,74)(46,56)(47,67)(48,78)(50,60)(51,71)(54,64)(55,75)(58,68)(59,79)(62,72)(66,76)(70,80)>;
G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65), (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)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (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)(42,52)(43,63)(44,74)(46,56)(47,67)(48,78)(50,60)(51,71)(54,64)(55,75)(58,68)(59,79)(62,72)(66,76)(70,80) );
G=PermutationGroup([(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65)], [(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),(41,65),(42,64),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74)], [(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),(42,52),(43,63),(44,74),(46,56),(47,67),(48,78),(50,60),(51,71),(54,64),(55,75),(58,68),(59,79),(62,72),(66,76),(70,80)])
Matrix representation ►G ⊆ 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] >;
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 |
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