metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊11D10, Q16⋊10D10, D20.46D4, SD16⋊15D10, C20.17C24, C40.43C23, Dic10.46D4, D20.12C23, Dic10.11C23, C4○D8⋊5D5, C4○D4⋊2D10, (C2×C8)⋊14D10, C5⋊D4.2D4, D8⋊D5⋊6C2, D4⋊D5⋊4C22, (D5×SD16)⋊6C2, C4.144(D4×D5), C5⋊3(D4○SD16), Q8⋊D5⋊3C22, D4⋊8D10⋊6C2, D4⋊D10⋊8C2, (Q8×D5)⋊2C22, C22.9(D4×D5), (C2×C40)⋊17C22, Q16⋊D5⋊6C2, D10.53(C2×D4), C20.350(C2×D4), (C5×D8)⋊16C22, (C8×D5)⋊10C22, C5⋊2C8.8C23, D4.D5⋊3C22, C5⋊Q16⋊2C22, (D4×D5).2C22, C8.17(C22×D5), C4.17(C23×D5), SD16⋊3D5⋊6C2, D4.9D10⋊7C2, D4⋊2D5⋊2C22, C40⋊C2⋊21C22, C8⋊D5⋊16C22, Dic5.59(C2×D4), (C5×Q16)⋊14C22, (C4×D5).10C23, (C5×D4).11C23, D4.11(C22×D5), D4.10D10⋊5C2, D20.3C4⋊10C2, Q8.11(C22×D5), (C5×Q8).11C23, (C2×C20).534C23, C4○D20.55C22, (C5×SD16)⋊16C22, C10.118(C22×D4), C4.Dic5⋊31C22, Q8⋊2D5.2C22, (C2×Dic10)⋊38C22, (C2×D20).187C22, C2.91(C2×D4×D5), (C5×C4○D8)⋊7C2, (C2×C40⋊C2)⋊27C2, (C2×C10).14(C2×D4), (C5×C4○D4)⋊4C22, (C2×C4).233(C22×D5), SmallGroup(320,1442)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1046 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×14], Q8 [×2], Q8 [×6], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, D8 [×2], SD16 [×2], SD16 [×8], Q16, Q16 [×2], C2×D4 [×6], C2×Q8 [×4], C4○D4 [×2], C4○D4 [×9], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10 [×2], C8○D4, C2×SD16 [×3], C4○D8, C4○D8 [×2], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C5⋊2C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×3], C4×D5 [×2], C4×D5 [×4], D20, D20 [×2], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×3], D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×4], C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×Dic10, C2×Dic10, C2×D20, C2×D20, C4○D20, C4○D20 [×2], D4×D5 [×2], D4×D5 [×2], D4⋊2D5 [×2], D4⋊2D5 [×2], Q8×D5 [×2], Q8⋊2D5 [×2], C5×C4○D4 [×2], D20.3C4, C2×C40⋊C2, D8⋊D5 [×2], D5×SD16 [×2], SD16⋊3D5 [×2], Q16⋊D5 [×2], D4⋊D10, D4.9D10, C5×C4○D8, D4⋊8D10, D4.10D10, D8⋊11D10
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, D8⋊11D10
Generators and relations
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >
►On 80 points
(1 41 55 78 63 17 40 27)(2 42 56 79 64 18 31 28)(3 43 57 80 65 19 32 29)(4 44 58 71 66 20 33 30)(5 45 59 72 67 11 34 21)(6 46 60 73 68 12 35 22)(7 47 51 74 69 13 36 23)(8 48 52 75 70 14 37 24)(9 49 53 76 61 15 38 25)(10 50 54 77 62 16 39 26)
(1 27)(2 79)(3 29)(4 71)(5 21)(6 73)(7 23)(8 75)(9 25)(10 77)(11 59)(12 35)(13 51)(14 37)(15 53)(16 39)(17 55)(18 31)(19 57)(20 33)(22 68)(24 70)(26 62)(28 64)(30 66)(32 43)(34 45)(36 47)(38 49)(40 41)(42 56)(44 58)(46 60)(48 52)(50 54)(61 76)(63 78)(65 80)(67 72)(69 74)
(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 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 60)(11 41)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 27)(22 26)(23 25)(28 30)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,41,55,78,63,17,40,27)(2,42,56,79,64,18,31,28)(3,43,57,80,65,19,32,29)(4,44,58,71,66,20,33,30)(5,45,59,72,67,11,34,21)(6,46,60,73,68,12,35,22)(7,47,51,74,69,13,36,23)(8,48,52,75,70,14,37,24)(9,49,53,76,61,15,38,25)(10,50,54,77,62,16,39,26), (1,27)(2,79)(3,29)(4,71)(5,21)(6,73)(7,23)(8,75)(9,25)(10,77)(11,59)(12,35)(13,51)(14,37)(15,53)(16,39)(17,55)(18,31)(19,57)(20,33)(22,68)(24,70)(26,62)(28,64)(30,66)(32,43)(34,45)(36,47)(38,49)(40,41)(42,56)(44,58)(46,60)(48,52)(50,54)(61,76)(63,78)(65,80)(67,72)(69,74), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,60)(11,41)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,27)(22,26)(23,25)(28,30)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,41,55,78,63,17,40,27)(2,42,56,79,64,18,31,28)(3,43,57,80,65,19,32,29)(4,44,58,71,66,20,33,30)(5,45,59,72,67,11,34,21)(6,46,60,73,68,12,35,22)(7,47,51,74,69,13,36,23)(8,48,52,75,70,14,37,24)(9,49,53,76,61,15,38,25)(10,50,54,77,62,16,39,26), (1,27)(2,79)(3,29)(4,71)(5,21)(6,73)(7,23)(8,75)(9,25)(10,77)(11,59)(12,35)(13,51)(14,37)(15,53)(16,39)(17,55)(18,31)(19,57)(20,33)(22,68)(24,70)(26,62)(28,64)(30,66)(32,43)(34,45)(36,47)(38,49)(40,41)(42,56)(44,58)(46,60)(48,52)(50,54)(61,76)(63,78)(65,80)(67,72)(69,74), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,60)(11,41)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,27)(22,26)(23,25)(28,30)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([(1,41,55,78,63,17,40,27),(2,42,56,79,64,18,31,28),(3,43,57,80,65,19,32,29),(4,44,58,71,66,20,33,30),(5,45,59,72,67,11,34,21),(6,46,60,73,68,12,35,22),(7,47,51,74,69,13,36,23),(8,48,52,75,70,14,37,24),(9,49,53,76,61,15,38,25),(10,50,54,77,62,16,39,26)], [(1,27),(2,79),(3,29),(4,71),(5,21),(6,73),(7,23),(8,75),(9,25),(10,77),(11,59),(12,35),(13,51),(14,37),(15,53),(16,39),(17,55),(18,31),(19,57),(20,33),(22,68),(24,70),(26,62),(28,64),(30,66),(32,43),(34,45),(36,47),(38,49),(40,41),(42,56),(44,58),(46,60),(48,52),(50,54),(61,76),(63,78),(65,80),(67,72),(69,74)], [(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)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,60),(11,41),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,27),(22,26),(23,25),(28,30),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,70),(38,69),(39,68),(40,67),(71,79),(72,78),(73,77),(74,76)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 0 | 40 | 29 |
| 0 | 0 | 12 | 1 |
| 2 | 24 | 39 | 17 |
| 17 | 39 | 24 | 2 |
| 39 | 17 | 1 | 12 |
| 24 | 2 | 29 | 40 |
| 39 | 17 | 2 | 24 |
| 24 | 2 | 17 | 39 |
| 14 | 27 | 27 | 14 |
| 14 | 30 | 27 | 11 |
| 28 | 13 | 27 | 14 |
| 28 | 19 | 27 | 11 |
| 1 | 7 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 2 | 14 | 40 | 34 |
| 0 | 39 | 0 | 1 |
G:=sub<GL(4,GF(41))| [0,0,2,17,0,0,24,39,40,12,39,24,29,1,17,2],[39,24,39,24,17,2,17,2,1,29,2,17,12,40,24,39],[14,14,28,28,27,30,13,19,27,27,27,27,14,11,14,11],[1,0,2,0,7,40,14,39,0,0,40,0,0,0,34,1] >;
50 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 | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| 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 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
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 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D4○SD16 | D4×D5 | D4×D5 | D8⋊11D10 |
| kernel | D8⋊11D10 | D20.3C4 | C2×C40⋊C2 | D8⋊D5 | D5×SD16 | SD16⋊3D5 | Q16⋊D5 | D4⋊D10 | D4.9D10 | C5×C4○D8 | D4⋊8D10 | D4.10D10 | Dic10 | D20 | C5⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{11}D_{10} % in TeX
G:=Group("D8:11D10"); // GroupNames label
G:=SmallGroup(320,1442);
// by ID
G=gap.SmallGroup(320,1442);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,570,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations