direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.D10, D20⋊7C23, C20.28C24, Dic10⋊6C23, (C2×D4)⋊34D10, C5⋊2C8⋊4C23, (C22×D4)⋊4D5, C10⋊4(C8⋊C22), D4⋊D5⋊17C22, (C2×C20).207D4, C20.249(C2×D4), C4.28(C23×D5), C4○D20⋊19C22, (C2×D20)⋊55C22, (D4×C10)⋊42C22, D4.D5⋊16C22, D4.20(C22×D5), (C5×D4).20C23, (C2×C20).537C23, C10.137(C22×D4), (C22×C10).207D4, (C22×C4).268D10, C23.92(C5⋊D4), C4.Dic5⋊32C22, (C2×Dic10)⋊65C22, (C22×C20).270C22, (D4×C2×C10)⋊3C2, C5⋊5(C2×C8⋊C22), (C2×D4⋊D5)⋊30C2, C4.21(C2×C5⋊D4), (C2×C4○D20)⋊28C2, (C2×D4.D5)⋊30C2, (C2×C5⋊2C8)⋊20C22, (C2×C10).577(C2×D4), (C2×C4).92(C5⋊D4), (C2×C4.Dic5)⋊26C2, C2.10(C22×C5⋊D4), (C2×C4).235(C22×D5), C22.106(C2×C5⋊D4), SmallGroup(320,1465)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 958 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×4], D4 [×13], Q8 [×3], C23, C23 [×11], D5 [×2], C10, C10 [×2], C10 [×6], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×5], C2×Q8, C4○D4 [×6], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×18], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C5⋊2C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×10], C2×C8⋊C22, C2×C5⋊2C8 [×2], C4.Dic5 [×4], D4⋊D5 [×8], D4.D5 [×8], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, D4×C10 [×6], D4×C10 [×3], C23×C10, C2×C4.Dic5, C2×D4⋊D5 [×2], D4.D10 [×8], C2×D4.D5 [×2], C2×C4○D20, D4×C2×C10, C2×D4.D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C8⋊C22, C2×C5⋊D4 [×6], C23×D5, D4.D10 [×2], C22×C5⋊D4, C2×D4.D10
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d9 >
►On 80 points
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 61)(17 62)(18 63)(19 64)(20 65)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 41)(38 42)(39 43)(40 44)
(1 71 11 61)(2 72 12 62)(3 73 13 63)(4 74 14 64)(5 75 15 65)(6 76 16 66)(7 77 17 67)(8 78 18 68)(9 79 19 69)(10 80 20 70)(21 60 31 50)(22 41 32 51)(23 42 33 52)(24 43 34 53)(25 44 35 54)(26 45 36 55)(27 46 37 56)(28 47 38 57)(29 48 39 58)(30 49 40 59)
(1 61)(2 72)(3 63)(4 74)(5 65)(6 76)(7 67)(8 78)(9 69)(10 80)(11 71)(12 62)(13 73)(14 64)(15 75)(16 66)(17 77)(18 68)(19 79)(20 70)(21 31)(23 33)(25 35)(27 37)(29 39)(41 51)(43 53)(45 55)(47 57)(49 59)
(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 24 11 34)(2 33 12 23)(3 22 13 32)(4 31 14 21)(5 40 15 30)(6 29 16 39)(7 38 17 28)(8 27 18 37)(9 36 19 26)(10 25 20 35)(41 73 51 63)(42 62 52 72)(43 71 53 61)(44 80 54 70)(45 69 55 79)(46 78 56 68)(47 67 57 77)(48 76 58 66)(49 65 59 75)(50 74 60 64)
G:=sub<Sym(80)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,61)(17,62)(18,63)(19,64)(20,65)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,41)(38,42)(39,43)(40,44), (1,71,11,61)(2,72,12,62)(3,73,13,63)(4,74,14,64)(5,75,15,65)(6,76,16,66)(7,77,17,67)(8,78,18,68)(9,79,19,69)(10,80,20,70)(21,60,31,50)(22,41,32,51)(23,42,33,52)(24,43,34,53)(25,44,35,54)(26,45,36,55)(27,46,37,56)(28,47,38,57)(29,48,39,58)(30,49,40,59), (1,61)(2,72)(3,63)(4,74)(5,65)(6,76)(7,67)(8,78)(9,69)(10,80)(11,71)(12,62)(13,73)(14,64)(15,75)(16,66)(17,77)(18,68)(19,79)(20,70)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(43,53)(45,55)(47,57)(49,59), (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), (1,24,11,34)(2,33,12,23)(3,22,13,32)(4,31,14,21)(5,40,15,30)(6,29,16,39)(7,38,17,28)(8,27,18,37)(9,36,19,26)(10,25,20,35)(41,73,51,63)(42,62,52,72)(43,71,53,61)(44,80,54,70)(45,69,55,79)(46,78,56,68)(47,67,57,77)(48,76,58,66)(49,65,59,75)(50,74,60,64)>;
G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,61)(17,62)(18,63)(19,64)(20,65)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,41)(38,42)(39,43)(40,44), (1,71,11,61)(2,72,12,62)(3,73,13,63)(4,74,14,64)(5,75,15,65)(6,76,16,66)(7,77,17,67)(8,78,18,68)(9,79,19,69)(10,80,20,70)(21,60,31,50)(22,41,32,51)(23,42,33,52)(24,43,34,53)(25,44,35,54)(26,45,36,55)(27,46,37,56)(28,47,38,57)(29,48,39,58)(30,49,40,59), (1,61)(2,72)(3,63)(4,74)(5,65)(6,76)(7,67)(8,78)(9,69)(10,80)(11,71)(12,62)(13,73)(14,64)(15,75)(16,66)(17,77)(18,68)(19,79)(20,70)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(43,53)(45,55)(47,57)(49,59), (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), (1,24,11,34)(2,33,12,23)(3,22,13,32)(4,31,14,21)(5,40,15,30)(6,29,16,39)(7,38,17,28)(8,27,18,37)(9,36,19,26)(10,25,20,35)(41,73,51,63)(42,62,52,72)(43,71,53,61)(44,80,54,70)(45,69,55,79)(46,78,56,68)(47,67,57,77)(48,76,58,66)(49,65,59,75)(50,74,60,64) );
G=PermutationGroup([(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,61),(17,62),(18,63),(19,64),(20,65),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,41),(38,42),(39,43),(40,44)], [(1,71,11,61),(2,72,12,62),(3,73,13,63),(4,74,14,64),(5,75,15,65),(6,76,16,66),(7,77,17,67),(8,78,18,68),(9,79,19,69),(10,80,20,70),(21,60,31,50),(22,41,32,51),(23,42,33,52),(24,43,34,53),(25,44,35,54),(26,45,36,55),(27,46,37,56),(28,47,38,57),(29,48,39,58),(30,49,40,59)], [(1,61),(2,72),(3,63),(4,74),(5,65),(6,76),(7,67),(8,78),(9,69),(10,80),(11,71),(12,62),(13,73),(14,64),(15,75),(16,66),(17,77),(18,68),(19,79),(20,70),(21,31),(23,33),(25,35),(27,37),(29,39),(41,51),(43,53),(45,55),(47,57),(49,59)], [(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)], [(1,24,11,34),(2,33,12,23),(3,22,13,32),(4,31,14,21),(5,40,15,30),(6,29,16,39),(7,38,17,28),(8,27,18,37),(9,36,19,26),(10,25,20,35),(41,73,51,63),(42,62,52,72),(43,71,53,61),(44,80,54,70),(45,69,55,79),(46,78,56,68),(47,67,57,77),(48,76,58,66),(49,65,59,75),(50,74,60,64)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 1 | 2 |
| 0 | 0 | 0 | 35 | 40 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 16 | 32 |
| 0 | 0 | 21 | 6 | 25 | 25 |
| 1 | 30 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 16 | 32 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 23 | 0 | 0 |
| 0 | 0 | 31 | 8 | 27 | 27 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,6,0,0,0,1,0,6,35,0,0,0,0,1,40,0,0,0,0,2,40],[40,11,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,14,21,0,0,23,0,14,6,0,0,0,0,16,25,0,0,0,0,32,25],[1,0,0,0,0,0,30,40,0,0,0,0,0,0,14,0,0,31,0,0,14,0,23,8,0,0,16,16,0,27,0,0,32,0,0,27] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4.D10 |
| kernel | C2×D4.D10 | C2×C4.Dic5 | C2×D4⋊D5 | D4.D10 | C2×D4.D5 | C2×C4○D20 | D4×C2×C10 | C2×C20 | C22×C10 | C22×D4 | C22×C4 | C2×D4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 3 | 1 | 2 | 2 | 12 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4.D_{10} % in TeX
G:=Group("C2xD4.D10"); // GroupNames label
G:=SmallGroup(320,1465);
// by ID
G=gap.SmallGroup(320,1465);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^9>;
// generators/relations