metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊5D10, SD16⋊7D10, D20.43D4, D40⋊4C22, C40.6C23, C20.25C24, M4(2)⋊13D10, Dic10.43D4, D20.18C23, Dic10.18C23, C5⋊D4.6D4, C8⋊D10⋊4C2, D40⋊C2⋊4C2, (C2×Q8)⋊12D10, (D5×SD16)⋊4C2, C4.117(D4×D5), C5⋊5(D4○SD16), (C8×D5)⋊6C22, C8.C22⋊4D5, D4⋊8D10⋊8C2, Q8.D10⋊2C2, (Q8×D5)⋊4C22, C8.6(C22×D5), D4⋊D5⋊16C22, Q16⋊D5⋊3C2, C4○D4.14D10, D10.57(C2×D4), C20.246(C2×D4), C4○D20⋊9C22, C40⋊C2⋊7C22, C8⋊D5⋊7C22, Q8⋊D5⋊15C22, (C5×Q16)⋊3C22, (D4×D5).4C22, C22.16(D4×D5), C4.25(C23×D5), D20.2C4⋊3C2, D4.8D10⋊5C2, C5⋊2C8.27C23, D4.D5⋊15C22, Dic5.63(C2×D4), (Q8×C10)⋊22C22, Q8⋊2D5⋊4C22, (C5×SD16)⋊7C22, (C5×D4).18C23, (C4×D5).16C23, C5⋊Q16⋊14C22, D4.18(C22×D5), (C5×Q8).18C23, Q8.18(C22×D5), (C2×C20).116C23, Q8.10D10⋊5C2, C10.126(C22×D4), (C5×M4(2))⋊7C22, (C2×D20).188C22, C2.99(C2×D4×D5), (C2×Q8⋊D5)⋊29C2, (C2×C10).71(C2×D4), (C5×C8.C22)⋊3C2, (C2×C5⋊2C8)⋊19C22, (C5×C4○D4).27C22, (C2×C4).100(C22×D5), SmallGroup(320,1450)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1062 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×15], Q8, Q8 [×2], Q8 [×5], C23 [×3], D5 [×5], C10, C10 [×2], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×3], SD16 [×2], SD16 [×8], Q16 [×2], Q16, C2×D4 [×6], C2×Q8, C2×Q8 [×3], C4○D4, C4○D4 [×10], Dic5 [×2], Dic5, C20 [×2], C20 [×3], D10 [×2], D10 [×6], C2×C10, C2×C10, C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22, C8.C22 [×2], 2+ (1+4), 2- (1+4), C5⋊2C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×2], C4×D5 [×7], D20 [×2], D20 [×2], D20 [×5], C5⋊D4 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5 [×3], D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, Q8⋊D5 [×4], C5⋊Q16, C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×D20, C2×D20, C4○D20 [×2], C4○D20 [×3], D4×D5 [×2], D4×D5 [×2], Q8×D5 [×2], Q8×D5, Q8⋊2D5 [×4], Q8⋊2D5, Q8×C10, C5×C4○D4, D20.2C4, C8⋊D10, D5×SD16 [×2], D40⋊C2 [×2], Q16⋊D5 [×2], Q8.D10 [×2], C2×Q8⋊D5, D4.8D10, C5×C8.C22, Q8.10D10, D4⋊8D10, C40.C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○SD16, D4×D5 [×2], C23×D5, C2×D4×D5, C40.C23
Generators and relations
G = < a,b,c,d | a40=b2=1, c2=d2=a20, bab=a29, cac-1=a31, dad-1=a11, bc=cb, dbd-1=a20b, 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)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 69)(42 58)(43 47)(44 76)(45 65)(46 54)(48 72)(49 61)(51 79)(52 68)(53 57)(55 75)(56 64)(59 71)(62 78)(63 67)(66 74)(73 77)
(1 80 21 60)(2 71 22 51)(3 62 23 42)(4 53 24 73)(5 44 25 64)(6 75 26 55)(7 66 27 46)(8 57 28 77)(9 48 29 68)(10 79 30 59)(11 70 31 50)(12 61 32 41)(13 52 33 72)(14 43 34 63)(15 74 35 54)(16 65 36 45)(17 56 37 76)(18 47 38 67)(19 78 39 58)(20 69 40 49)
(1 36 21 16)(2 7 22 27)(3 18 23 38)(4 29 24 9)(5 40 25 20)(6 11 26 31)(8 33 28 13)(10 15 30 35)(12 37 32 17)(14 19 34 39)(41 56 61 76)(42 67 62 47)(43 78 63 58)(44 49 64 69)(45 60 65 80)(46 71 66 51)(48 53 68 73)(50 75 70 55)(52 57 72 77)(54 79 74 59)
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), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,69)(42,58)(43,47)(44,76)(45,65)(46,54)(48,72)(49,61)(51,79)(52,68)(53,57)(55,75)(56,64)(59,71)(62,78)(63,67)(66,74)(73,77), (1,80,21,60)(2,71,22,51)(3,62,23,42)(4,53,24,73)(5,44,25,64)(6,75,26,55)(7,66,27,46)(8,57,28,77)(9,48,29,68)(10,79,30,59)(11,70,31,50)(12,61,32,41)(13,52,33,72)(14,43,34,63)(15,74,35,54)(16,65,36,45)(17,56,37,76)(18,47,38,67)(19,78,39,58)(20,69,40,49), (1,36,21,16)(2,7,22,27)(3,18,23,38)(4,29,24,9)(5,40,25,20)(6,11,26,31)(8,33,28,13)(10,15,30,35)(12,37,32,17)(14,19,34,39)(41,56,61,76)(42,67,62,47)(43,78,63,58)(44,49,64,69)(45,60,65,80)(46,71,66,51)(48,53,68,73)(50,75,70,55)(52,57,72,77)(54,79,74,59)>;
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), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,69)(42,58)(43,47)(44,76)(45,65)(46,54)(48,72)(49,61)(51,79)(52,68)(53,57)(55,75)(56,64)(59,71)(62,78)(63,67)(66,74)(73,77), (1,80,21,60)(2,71,22,51)(3,62,23,42)(4,53,24,73)(5,44,25,64)(6,75,26,55)(7,66,27,46)(8,57,28,77)(9,48,29,68)(10,79,30,59)(11,70,31,50)(12,61,32,41)(13,52,33,72)(14,43,34,63)(15,74,35,54)(16,65,36,45)(17,56,37,76)(18,47,38,67)(19,78,39,58)(20,69,40,49), (1,36,21,16)(2,7,22,27)(3,18,23,38)(4,29,24,9)(5,40,25,20)(6,11,26,31)(8,33,28,13)(10,15,30,35)(12,37,32,17)(14,19,34,39)(41,56,61,76)(42,67,62,47)(43,78,63,58)(44,49,64,69)(45,60,65,80)(46,71,66,51)(48,53,68,73)(50,75,70,55)(52,57,72,77)(54,79,74,59) );
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)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,69),(42,58),(43,47),(44,76),(45,65),(46,54),(48,72),(49,61),(51,79),(52,68),(53,57),(55,75),(56,64),(59,71),(62,78),(63,67),(66,74),(73,77)], [(1,80,21,60),(2,71,22,51),(3,62,23,42),(4,53,24,73),(5,44,25,64),(6,75,26,55),(7,66,27,46),(8,57,28,77),(9,48,29,68),(10,79,30,59),(11,70,31,50),(12,61,32,41),(13,52,33,72),(14,43,34,63),(15,74,35,54),(16,65,36,45),(17,56,37,76),(18,47,38,67),(19,78,39,58),(20,69,40,49)], [(1,36,21,16),(2,7,22,27),(3,18,23,38),(4,29,24,9),(5,40,25,20),(6,11,26,31),(8,33,28,13),(10,15,30,35),(12,37,32,17),(14,19,34,39),(41,56,61,76),(42,67,62,47),(43,78,63,58),(44,49,64,69),(45,60,65,80),(46,71,66,51),(48,53,68,73),(50,75,70,55),(52,57,72,77),(54,79,74,59)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 0 | 0 | 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 0 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 15 |
| 0 | 0 | 0 | 0 | 30 | 0 | 26 | 15 |
| 0 | 0 | 0 | 0 | 30 | 11 | 26 | 15 |
| 34 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 40 | 1 |
| 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 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 26 | 0 | 15 |
| 0 | 0 | 0 | 0 | 30 | 11 | 26 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 26 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 15 | 0 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 26 |
| 0 | 0 | 0 | 0 | 11 | 0 | 15 | 26 |
| 0 | 0 | 0 | 0 | 30 | 11 | 26 | 15 |
G:=sub<GL(8,GF(41))| [0,0,7,7,0,0,0,0,0,0,34,40,0,0,0,0,34,34,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,0,0,30,30,0,0,0,0,15,0,0,11,0,0,0,0,0,15,26,26,0,0,0,0,15,15,15,15],[34,40,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,34,40,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,40,0,0,0,0,0,1,0,0,40],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,26,11,0,0,0,0,0,0,0,26,15,26,0,0,0,0,15,15,26,26],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,0,11,30,0,0,0,0,15,0,0,11,0,0,0,0,0,26,15,26,0,0,0,0,26,26,26,15] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 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 | 5 | 5 | 8 | 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 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 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 | D4○SD16 | D4×D5 | D4×D5 | C40.C23 |
| kernel | C40.C23 | D20.2C4 | C8⋊D10 | D5×SD16 | D40⋊C2 | Q16⋊D5 | Q8.D10 | C2×Q8⋊D5 | D4.8D10 | C5×C8.C22 | Q8.10D10 | D4⋊8D10 | Dic10 | D20 | C5⋊D4 | 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 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_{40}.C_2^3 % in TeX
G:=Group("C40.C2^3"); // GroupNames label
G:=SmallGroup(320,1450);
// by ID
G=gap.SmallGroup(320,1450);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^40=b^2=1,c^2=d^2=a^20,b*a*b=a^29,c*a*c^-1=a^31,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^20*b,c*d=d*c>;
// generators/relations