metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1D10, M4(2)⋊16D10, C4≀C2⋊5D5, (D4×D5)⋊4C4, (Q8×D5)⋊4C4, D4⋊2D5⋊4C4, Q8⋊2D5⋊4C4, D4.11(C4×D5), C4.201(D4×D5), Q8.11(C4×D5), D20⋊4C4⋊4C2, D20⋊7C4⋊6C2, (C4×C20)⋊10C22, C4○D4.19D10, (C4×D5).104D4, D20.19(C2×C4), C20.360(C2×D4), C42⋊D5⋊9C2, (D5×M4(2))⋊9C2, C22.28(D4×D5), D4⋊2Dic5⋊2C2, C20.53(C22×C4), (C2×Dic5).45D4, (C4×Dic5)⋊3C22, (C22×D5).31D4, C4.Dic5⋊3C22, (C2×C20).261C23, Dic10.20(C2×C4), C5⋊4(C42⋊C22), C4○D20.10C22, D10.27(C22⋊C4), (C5×M4(2))⋊14C22, Dic5.39(C22⋊C4), (C5×C4≀C2)⋊6C2, C4.18(C2×C4×D5), (D5×C4○D4).2C2, (C4×D5).5(C2×C4), (C5×D4).19(C2×C4), (C2×C10).25(C2×D4), (C5×Q8).20(C2×C4), C2.26(D5×C22⋊C4), (C2×C4×D5).33C22, C10.66(C2×C22⋊C4), (C5×C4○D4).2C22, (C2×C4).368(C22×D5), SmallGroup(320,448)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊D10
G = < a,b,c,d | a4=b4=c10=d2=1, cac-1=ab=ba, dad=ab-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 638 in 154 conjugacy classes, 51 normal (all characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×6], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×12], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], D5 [×3], C10, C10 [×2], C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2) [×2], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×4], C2×C10, C2×C10, C4≀C2, C4≀C2 [×3], C42⋊C2, C2×M4(2), C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C42⋊C22, C8×D5, C8⋊D5, C4.Dic5, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C5×M4(2), C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, D20⋊4C4, D20⋊7C4, D4⋊2Dic5, C5×C4≀C2, C42⋊D5, D5×M4(2), D5×C4○D4, C42⋊D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], C22×D5, C42⋊C22, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, C42⋊D10
►On 80 points
(1 78 63 27)(2 18)(3 80 65 29)(4 20)(5 72 67 21)(6 12)(7 74 69 23)(8 14)(9 76 61 25)(10 16)(11 59 45 34)(13 51 47 36)(15 53 49 38)(17 55 41 40)(19 57 43 32)(22 35)(24 37)(26 39)(28 31)(30 33)(42 64)(44 66)(46 68)(48 70)(50 62)(52 75)(54 77)(56 79)(58 71)(60 73)
(1 40 63 55)(2 56 64 31)(3 32 65 57)(4 58 66 33)(5 34 67 59)(6 60 68 35)(7 36 69 51)(8 52 70 37)(9 38 61 53)(10 54 62 39)(11 21 45 72)(12 73 46 22)(13 23 47 74)(14 75 48 24)(15 25 49 76)(16 77 50 26)(17 27 41 78)(18 79 42 28)(19 29 43 80)(20 71 44 30)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 30)(8 29)(9 28)(10 27)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 50)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(70 80)
G:=sub<Sym(80)| (1,78,63,27)(2,18)(3,80,65,29)(4,20)(5,72,67,21)(6,12)(7,74,69,23)(8,14)(9,76,61,25)(10,16)(11,59,45,34)(13,51,47,36)(15,53,49,38)(17,55,41,40)(19,57,43,32)(22,35)(24,37)(26,39)(28,31)(30,33)(42,64)(44,66)(46,68)(48,70)(50,62)(52,75)(54,77)(56,79)(58,71)(60,73), (1,40,63,55)(2,56,64,31)(3,32,65,57)(4,58,66,33)(5,34,67,59)(6,60,68,35)(7,36,69,51)(8,52,70,37)(9,38,61,53)(10,54,62,39)(11,21,45,72)(12,73,46,22)(13,23,47,74)(14,75,48,24)(15,25,49,76)(16,77,50,26)(17,27,41,78)(18,79,42,28)(19,29,43,80)(20,71,44,30), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,30)(8,29)(9,28)(10,27)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,50)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80)>;
G:=Group( (1,78,63,27)(2,18)(3,80,65,29)(4,20)(5,72,67,21)(6,12)(7,74,69,23)(8,14)(9,76,61,25)(10,16)(11,59,45,34)(13,51,47,36)(15,53,49,38)(17,55,41,40)(19,57,43,32)(22,35)(24,37)(26,39)(28,31)(30,33)(42,64)(44,66)(46,68)(48,70)(50,62)(52,75)(54,77)(56,79)(58,71)(60,73), (1,40,63,55)(2,56,64,31)(3,32,65,57)(4,58,66,33)(5,34,67,59)(6,60,68,35)(7,36,69,51)(8,52,70,37)(9,38,61,53)(10,54,62,39)(11,21,45,72)(12,73,46,22)(13,23,47,74)(14,75,48,24)(15,25,49,76)(16,77,50,26)(17,27,41,78)(18,79,42,28)(19,29,43,80)(20,71,44,30), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,30)(8,29)(9,28)(10,27)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,50)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80) );
G=PermutationGroup([(1,78,63,27),(2,18),(3,80,65,29),(4,20),(5,72,67,21),(6,12),(7,74,69,23),(8,14),(9,76,61,25),(10,16),(11,59,45,34),(13,51,47,36),(15,53,49,38),(17,55,41,40),(19,57,43,32),(22,35),(24,37),(26,39),(28,31),(30,33),(42,64),(44,66),(46,68),(48,70),(50,62),(52,75),(54,77),(56,79),(58,71),(60,73)], [(1,40,63,55),(2,56,64,31),(3,32,65,57),(4,58,66,33),(5,34,67,59),(6,60,68,35),(7,36,69,51),(8,52,70,37),(9,38,61,53),(10,54,62,39),(11,21,45,72),(12,73,46,22),(13,23,47,74),(14,75,48,24),(15,25,49,76),(16,77,50,26),(17,27,41,78),(18,79,42,28),(19,29,43,80),(20,71,44,30)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,30),(8,29),(9,28),(10,27),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,50),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(70,80)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 1 | 1 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4×D5 | C4×D5 | C42⋊C22 | D4×D5 | D4×D5 | C42⋊D10 |
| kernel | C42⋊D10 | D20⋊4C4 | D20⋊7C4 | D4⋊2Dic5 | C5×C4≀C2 | C42⋊D5 | D5×M4(2) | D5×C4○D4 | D4×D5 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C42⋊D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 37 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 21 | 39 | 23 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 25 | 25 | 9 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 | 40 | 37 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 37 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,21,0,0,40,0,9,39,0,0,40,32,0,23,0,0,37,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,25,0,0,0,0,32,25,0,0,0,0,0,9],[35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,0,0,0,1,40,0,0,0,0,0,37,0,1],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,37,1] >;
C42⋊D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_{10} % in TeX
G:=Group("C4^2:D10"); // GroupNames label
G:=SmallGroup(320,448);
// by ID
G=gap.SmallGroup(320,448);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,58,136,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a*b^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations