metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.35C24, D20.31C23, Dic10.30C23, C4○D4⋊16D10, (C2×D4)⋊42D10, (C2×Q8)⋊31D10, D4⋊D5⋊19C22, (C2×C20).218D4, C20.427(C2×D4), Q8⋊D5⋊18C22, D4⋊D10⋊13C2, C4.35(C23×D5), D4.8D10⋊7C2, C4○D20⋊21C22, (C2×D20)⋊59C22, (D4×C10)⋊46C22, C5⋊5(D8⋊C22), C5⋊2C8.14C23, D4.D5⋊17C22, (Q8×C10)⋊38C22, C5⋊Q16⋊16C22, D4.23(C22×D5), (C5×D4).23C23, D4.9D10⋊13C2, D4.D10⋊13C2, (C5×Q8).23C23, Q8.23(C22×D5), C20.C23⋊13C2, (C2×C20).557C23, (C22×C10).124D4, (C22×C4).282D10, C10.160(C22×D4), C23.34(C5⋊D4), C4.Dic5⋊37C22, (C2×Dic10)⋊69C22, (C22×C20).292C22, (C2×C4○D4)⋊4D5, (C10×C4○D4)⋊4C2, (C2×C4○D20)⋊31C2, C4.121(C2×C5⋊D4), (C2×C5⋊2C8)⋊23C22, (C2×C10).591(C2×D4), (C5×C4○D4)⋊18C22, (C2×C4.Dic5)⋊31C2, C2.33(C22×C5⋊D4), C22.21(C2×C5⋊D4), (C2×C4).203(C5⋊D4), (C2×C4).246(C22×D5), SmallGroup(320,1494)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 830 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×4], C22 [×3], C22 [×9], C5, C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], D5 [×2], C10, C10 [×5], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], Dic5 [×2], C20 [×4], C20 [×2], D10 [×4], C2×C10 [×3], C2×C10 [×5], C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4, C2×C4○D4, C5⋊2C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×D5, C22×C10, C22×C10, D8⋊C22, C2×C5⋊2C8 [×2], C4.Dic5 [×4], D4⋊D5 [×4], D4.D5 [×4], Q8⋊D5 [×4], C5⋊Q16 [×4], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C2×C4.Dic5, D4.D10 [×2], C20.C23 [×2], D4⋊D10 [×2], D4.8D10 [×4], D4.9D10 [×2], C2×C4○D20, C10×C4○D4, C20.C24
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], D8⋊C22, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, C20.C24
Generators and relations
G = < a,b,c,d,e | a20=b2=c2=e2=1, d2=a10, bab=a-1, ac=ca, ad=da, eae=a11, bc=cb, bd=db, ebe=a5b, cd=dc, ece=a10c, de=ed >
►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 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 46 11 56)(2 47 12 57)(3 48 13 58)(4 49 14 59)(5 50 15 60)(6 51 16 41)(7 52 17 42)(8 53 18 43)(9 54 19 44)(10 55 20 45)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 71 36 61)(27 72 37 62)(28 73 38 63)(29 74 39 64)(30 75 40 65)
(1 21)(2 32)(3 23)(4 34)(5 25)(6 36)(7 27)(8 38)(9 29)(10 40)(11 31)(12 22)(13 33)(14 24)(15 35)(16 26)(17 37)(18 28)(19 39)(20 30)(41 71)(42 62)(43 73)(44 64)(45 75)(46 66)(47 77)(48 68)(49 79)(50 70)(51 61)(52 72)(53 63)(54 74)(55 65)(56 76)(57 67)(58 78)(59 69)(60 80)
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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,46,11,56)(2,47,12,57)(3,48,13,58)(4,49,14,59)(5,50,15,60)(6,51,16,41)(7,52,17,42)(8,53,18,43)(9,54,19,44)(10,55,20,45)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,71,36,61)(27,72,37,62)(28,73,38,63)(29,74,39,64)(30,75,40,65), (1,21)(2,32)(3,23)(4,34)(5,25)(6,36)(7,27)(8,38)(9,29)(10,40)(11,31)(12,22)(13,33)(14,24)(15,35)(16,26)(17,37)(18,28)(19,39)(20,30)(41,71)(42,62)(43,73)(44,64)(45,75)(46,66)(47,77)(48,68)(49,79)(50,70)(51,61)(52,72)(53,63)(54,74)(55,65)(56,76)(57,67)(58,78)(59,69)(60,80)>;
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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,46,11,56)(2,47,12,57)(3,48,13,58)(4,49,14,59)(5,50,15,60)(6,51,16,41)(7,52,17,42)(8,53,18,43)(9,54,19,44)(10,55,20,45)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,71,36,61)(27,72,37,62)(28,73,38,63)(29,74,39,64)(30,75,40,65), (1,21)(2,32)(3,23)(4,34)(5,25)(6,36)(7,27)(8,38)(9,29)(10,40)(11,31)(12,22)(13,33)(14,24)(15,35)(16,26)(17,37)(18,28)(19,39)(20,30)(41,71)(42,62)(43,73)(44,64)(45,75)(46,66)(47,77)(48,68)(49,79)(50,70)(51,61)(52,72)(53,63)(54,74)(55,65)(56,76)(57,67)(58,78)(59,69)(60,80) );
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)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,46,11,56),(2,47,12,57),(3,48,13,58),(4,49,14,59),(5,50,15,60),(6,51,16,41),(7,52,17,42),(8,53,18,43),(9,54,19,44),(10,55,20,45),(21,66,31,76),(22,67,32,77),(23,68,33,78),(24,69,34,79),(25,70,35,80),(26,71,36,61),(27,72,37,62),(28,73,38,63),(29,74,39,64),(30,75,40,65)], [(1,21),(2,32),(3,23),(4,34),(5,25),(6,36),(7,27),(8,38),(9,29),(10,40),(11,31),(12,22),(13,33),(14,24),(15,35),(16,26),(17,37),(18,28),(19,39),(20,30),(41,71),(42,62),(43,73),(44,64),(45,75),(46,66),(47,77),(48,68),(49,79),(50,70),(51,61),(52,72),(53,63),(54,74),(55,65),(56,76),(57,67),(58,78),(59,69),(60,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
62 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 | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| 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 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | D8⋊C22 | C20.C24 |
| kernel | C20.C24 | C2×C4.Dic5 | D4.D10 | C20.C23 | D4⋊D10 | D4.8D10 | D4.9D10 | C2×C4○D20 | C10×C4○D4 | C2×C20 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 1 | 3 | 1 | 2 | 2 | 2 | 2 | 8 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{20}.C_2^4 % in TeX
G:=Group("C20.C2^4"); // GroupNames label
G:=SmallGroup(320,1494);
// by ID
G=gap.SmallGroup(320,1494);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=e^2=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^11,b*c=c*b,b*d=d*b,e*b*e=a^5*b,c*d=d*c,e*c*e=a^10*c,d*e=e*d>;
// generators/relations