direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊C4, D10.21D8, D10.15SD16, (D4×D5)⋊6C4, D4⋊3(C2×F5), (C2×D4)⋊3F5, (D4×C10)⋊5C4, (C2×D20)⋊8C4, D20⋊3(C2×C4), D5.3(C2×D8), C4⋊F5⋊1C22, D5⋊(D4⋊C4), C10⋊(D4⋊C4), D5⋊C8⋊5C22, (C4×D5).32D4, D10.93(C2×D4), Dic5.3(C2×D4), D5.3(C2×SD16), C4.9(C22⋊F5), C4.13(C22×F5), C20.9(C22⋊C4), C20.13(C22×C4), (C4×D5).35C23, (D4×D5).12C22, (C2×Dic5).117D4, (C22×D5).145D4, D10.41(C22⋊C4), Dic5.9(C22⋊C4), C22.48(C22⋊F5), C5⋊(C2×D4⋊C4), (C2×C4⋊F5)⋊1C2, (C2×D5⋊C8)⋊1C2, (C5×D4)⋊3(C2×C4), (C2×D4×D5).13C2, (C2×C4).78(C2×F5), (C2×C20).49(C2×C4), (C4×D5).19(C2×C4), C2.14(C2×C22⋊F5), C10.13(C2×C22⋊C4), (C2×C4×D5).196C22, (C2×C10).53(C22⋊C4), SmallGroup(320,1104)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1066 in 202 conjugacy classes, 60 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×22], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×2], D4 [×8], C23 [×11], D5 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C4⋊C4 [×3], C2×C8 [×4], C22×C4 [×2], C2×D4, C2×D4 [×8], C24, Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×4], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C5⋊C8 [×2], C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C5×D4, C2×F5 [×4], C22×D5, C22×D5 [×9], C22×C10, C2×D4⋊C4, D5⋊C8 [×2], D5⋊C8, C4⋊F5 [×2], C4⋊F5, C2×C5⋊C8, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], C2×C5⋊D4, D4×C10, C22×F5, C23×D5, D20⋊C4 [×4], C2×D5⋊C8, C2×C4⋊F5, C2×D4×D5, C2×D20⋊C4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], F5, D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×F5 [×3], C2×D4⋊C4, C22⋊F5 [×2], C22×F5, D20⋊C4 [×2], C2×C22⋊F5, C2×D20⋊C4
Generators and relations
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b17c >
►On 80 points
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)
(1 48 36 73)(2 55 25 76)(3 42 34 79)(4 49 23 62)(5 56 32 65)(6 43 21 68)(7 50 30 71)(8 57 39 74)(9 44 28 77)(10 51 37 80)(11 58 26 63)(12 45 35 66)(13 52 24 69)(14 59 33 72)(15 46 22 75)(16 53 31 78)(17 60 40 61)(18 47 29 64)(19 54 38 67)(20 41 27 70)
G:=sub<Sym(80)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71), (1,48,36,73)(2,55,25,76)(3,42,34,79)(4,49,23,62)(5,56,32,65)(6,43,21,68)(7,50,30,71)(8,57,39,74)(9,44,28,77)(10,51,37,80)(11,58,26,63)(12,45,35,66)(13,52,24,69)(14,59,33,72)(15,46,22,75)(16,53,31,78)(17,60,40,61)(18,47,29,64)(19,54,38,67)(20,41,27,70)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71), (1,48,36,73)(2,55,25,76)(3,42,34,79)(4,49,23,62)(5,56,32,65)(6,43,21,68)(7,50,30,71)(8,57,39,74)(9,44,28,77)(10,51,37,80)(11,58,26,63)(12,45,35,66)(13,52,24,69)(14,59,33,72)(15,46,22,75)(16,53,31,78)(17,60,40,61)(18,47,29,64)(19,54,38,67)(20,41,27,70) );
G=PermutationGroup([(1,36),(2,37),(3,38),(4,39),(5,40),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,61),(57,62),(58,63),(59,64),(60,65)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71)], [(1,48,36,73),(2,55,25,76),(3,42,34,79),(4,49,23,62),(5,56,32,65),(6,43,21,68),(7,50,30,71),(8,57,39,74),(9,44,28,77),(10,51,37,80),(11,58,26,63),(12,45,35,66),(13,52,24,69),(14,59,33,72),(15,46,22,75),(16,53,31,78),(17,60,40,61),(18,47,29,64),(19,54,38,67),(20,41,27,70)])
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 | 5 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 35 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 1 | 36 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 6 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 17 | 19 | 0 | 0 | 0 | 0 |
| 28 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
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],[40,16,0,0,0,0,5,1,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,1,6,0,0,0,0,35,6],[1,0,0,0,0,0,36,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,6,1],[17,28,0,0,0,0,19,24,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 20 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D4 | D8 | SD16 | F5 | C2×F5 | C2×F5 | C22⋊F5 | C22⋊F5 | D20⋊C4 |
| kernel | C2×D20⋊C4 | D20⋊C4 | C2×D5⋊C8 | C2×C4⋊F5 | C2×D4×D5 | C2×D20 | D4×D5 | D4×C10 | C4×D5 | C2×Dic5 | C22×D5 | D10 | D10 | C2×D4 | C2×C4 | D4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes C_4 % in TeX
G:=Group("C2xD20:C4"); // GroupNames label
G:=SmallGroup(320,1104);
// by ID
G=gap.SmallGroup(320,1104);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,1684,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^17*c>;
// generators/relations