metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊4D10, SD16⋊6D10, D40⋊3C22, C40.5C23, C20.24C24, M4(2)⋊12D10, D20.17C23, Dic10.17C23, D40⋊C2⋊3C2, C8⋊D10⋊3C2, D4⋊D5⋊7C22, (C2×Q8)⋊23D10, C4.192(D4×D5), (C8×D5)⋊5C22, Q8⋊D5⋊6C22, C8.C22⋊6D5, Q8.D10⋊1C2, C8.5(C22×D5), Q16⋊D5⋊2C2, C4○D4.30D10, D10.89(C2×D4), (C4×D5).101D4, C20.245(C2×D4), (D4×D5)⋊10C22, C40⋊C2⋊6C22, C8⋊D5⋊6C22, (D5×M4(2))⋊4C2, D4⋊D10⋊10C2, (C5×Q16)⋊2C22, (Q8×D5)⋊12C22, C5⋊Q16⋊5C22, C4.24(C23×D5), C22.49(D4×D5), SD16⋊3D5⋊3C2, (C2×D20)⋊37C22, C5⋊4(D8⋊C22), C5⋊2C8.12C23, (Q8×C10)⋊21C22, (C5×SD16)⋊6C22, (C5×D4).17C23, (C22×D5).52D4, D4.17(C22×D5), (C4×D5).67C23, Q8.17(C22×D5), (C5×Q8).17C23, C20.C23⋊10C2, D4⋊2D5⋊11C22, (C2×C20).115C23, Dic5.102(C2×D4), (C2×Dic5).252D4, Q8⋊2D5⋊11C22, C4○D20.31C22, C10.125(C22×D4), (C5×M4(2))⋊6C22, C4.Dic5⋊15C22, C2.98(C2×D4×D5), (D5×C4○D4)⋊5C2, (C2×C10).70(C2×D4), (C5×C8.C22)⋊2C2, (C2×Q8⋊2D5)⋊17C2, (C2×C4×D5).172C22, (C2×C4).99(C22×D5), (C5×C4○D4).26C22, SmallGroup(320,1449)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1038 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×13], Q8, Q8 [×2], Q8 [×3], C23 [×3], D5 [×5], C10, C10 [×2], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×4], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×2], C22×C4 [×3], C2×D4 [×4], C2×Q8, C2×Q8, C4○D4, C4○D4 [×11], Dic5 [×2], Dic5, C20 [×2], C20 [×3], D10 [×2], D10 [×8], C2×C10, C2×C10, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22, C8.C22 [×3], C2×C4○D4 [×2], C5⋊2C8 [×2], C40 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×7], D20, D20 [×2], D20 [×6], C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×2], D8⋊C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×4], C5⋊Q16 [×2], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, Q8⋊2D5 [×4], Q8⋊2D5 [×2], Q8×C10, C5×C4○D4, D5×M4(2), C8⋊D10, D40⋊C2 [×2], SD16⋊3D5 [×2], Q16⋊D5 [×2], Q8.D10 [×2], C20.C23, D4⋊D10, C5×C8.C22, C2×Q8⋊2D5, D5×C4○D4, D40⋊C22
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D8⋊C22, D4×D5 [×2], C23×D5, C2×D4×D5, D40⋊C22
Generators and relations
G = < a,b,c,d | a40=b2=c2=d2=1, bab=a-1, cac=a21, dad=a29, cbc=a20b, dbd=a8b, cd=dc >
►On 80 points
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)
(1 21)(3 23)(5 25)(7 27)(9 29)(11 31)(13 33)(15 35)(17 37)(19 39)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)(57 77)(59 79)
(1 51)(2 80)(3 69)(4 58)(5 47)(6 76)(7 65)(8 54)(9 43)(10 72)(11 61)(12 50)(13 79)(14 68)(15 57)(16 46)(17 75)(18 64)(19 53)(20 42)(21 71)(22 60)(23 49)(24 78)(25 67)(26 56)(27 45)(28 74)(29 63)(30 52)(31 41)(32 70)(33 59)(34 48)(35 77)(36 66)(37 55)(38 44)(39 73)(40 62)
G:=sub<Sym(80)| (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61), (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,33)(15,35)(17,37)(19,39)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79), (1,51)(2,80)(3,69)(4,58)(5,47)(6,76)(7,65)(8,54)(9,43)(10,72)(11,61)(12,50)(13,79)(14,68)(15,57)(16,46)(17,75)(18,64)(19,53)(20,42)(21,71)(22,60)(23,49)(24,78)(25,67)(26,56)(27,45)(28,74)(29,63)(30,52)(31,41)(32,70)(33,59)(34,48)(35,77)(36,66)(37,55)(38,44)(39,73)(40,62)>;
G:=Group( (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61), (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,33)(15,35)(17,37)(19,39)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79), (1,51)(2,80)(3,69)(4,58)(5,47)(6,76)(7,65)(8,54)(9,43)(10,72)(11,61)(12,50)(13,79)(14,68)(15,57)(16,46)(17,75)(18,64)(19,53)(20,42)(21,71)(22,60)(23,49)(24,78)(25,67)(26,56)(27,45)(28,74)(29,63)(30,52)(31,41)(32,70)(33,59)(34,48)(35,77)(36,66)(37,55)(38,44)(39,73)(40,62) );
G=PermutationGroup([(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61)], [(1,21),(3,23),(5,25),(7,27),(9,29),(11,31),(13,33),(15,35),(17,37),(19,39),(41,61),(43,63),(45,65),(47,67),(49,69),(51,71),(53,73),(55,75),(57,77),(59,79)], [(1,51),(2,80),(3,69),(4,58),(5,47),(6,76),(7,65),(8,54),(9,43),(10,72),(11,61),(12,50),(13,79),(14,68),(15,57),(16,46),(17,75),(18,64),(19,53),(20,42),(21,71),(22,60),(23,49),(24,78),(25,67),(26,56),(27,45),(28,74),(29,63),(30,52),(31,41),(32,70),(33,59),(34,48),(35,77),(36,66),(37,55),(38,44),(39,73),(40,62)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 28 | 13 | 0 | 27 | 0 | 0 | 0 | 0 |
| 28 | 19 | 14 | 0 | 0 | 0 | 0 | 0 |
| 32 | 19 | 22 | 28 | 0 | 0 | 0 | 0 |
| 0 | 32 | 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 25 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 33 | 9 |
| 0 | 0 | 0 | 0 | 1 | 1 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 23 | 0 |
| 28 | 13 | 0 | 27 | 0 | 0 | 0 | 0 |
| 22 | 13 | 27 | 16 | 0 | 0 | 0 | 0 |
| 22 | 9 | 19 | 18 | 0 | 0 | 0 | 0 |
| 9 | 0 | 28 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 38 | 16 | 9 |
| 0 | 0 | 0 | 0 | 39 | 39 | 8 | 32 |
| 0 | 0 | 0 | 0 | 40 | 40 | 5 | 0 |
| 0 | 0 | 0 | 0 | 23 | 32 | 31 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 30 | 0 | 28 | 22 | 0 | 0 | 0 | 0 |
| 36 | 11 | 22 | 23 | 0 | 0 | 0 | 0 |
| 34 | 37 | 5 | 36 | 0 | 0 | 0 | 0 |
| 37 | 20 | 30 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 | 23 | 38 |
| 0 | 0 | 0 | 0 | 23 | 32 | 8 | 39 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 32 |
G:=sub<GL(8,GF(41))| [28,28,32,0,0,0,0,0,13,19,19,32,0,0,0,0,0,14,22,13,0,0,0,0,27,0,28,13,0,0,0,0,0,0,0,0,3,2,1,0,0,0,0,0,3,2,1,9,0,0,0,0,25,33,36,23,0,0,0,0,0,9,0,0],[28,22,22,9,0,0,0,0,13,13,9,0,0,0,0,0,0,27,19,28,0,0,0,0,27,16,18,22,0,0,0,0,0,0,0,0,38,39,40,23,0,0,0,0,38,39,40,32,0,0,0,0,16,8,5,31,0,0,0,0,9,32,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,6,4,1,0,0,0,0,0,0,0,0,1],[30,36,34,37,0,0,0,0,0,11,37,20,0,0,0,0,28,22,5,30,0,0,0,0,22,23,36,36,0,0,0,0,0,0,0,0,9,23,0,0,0,0,0,0,9,32,0,0,0,0,0,0,23,8,9,39,0,0,0,0,38,39,40,32] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
44 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D8⋊C22 | D4×D5 | D4×D5 | D40⋊C22 |
| kernel | D40⋊C22 | D5×M4(2) | C8⋊D10 | D40⋊C2 | SD16⋊3D5 | Q16⋊D5 | Q8.D10 | C20.C23 | D4⋊D10 | C5×C8.C22 | C2×Q8⋊2D5 | D5×C4○D4 | C4×D5 | C2×Dic5 | C22×D5 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{40}\rtimes C_2^2 % in TeX
G:=Group("D40:C2^2"); // GroupNames label
G:=SmallGroup(320,1449);
// by ID
G=gap.SmallGroup(320,1449);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^21,d*a*d=a^29,c*b*c=a^20*b,d*b*d=a^8*b,c*d=d*c>;
// generators/relations