direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊C4, D10.21D8, D10.15SD16, (D4×D5)⋊6C4, (C2×D4)⋊3F5, D4⋊3(C2×F5), D20⋊3(C2×C4), (D4×C10)⋊5C4, (C2×D20)⋊8C4, D5.3(C2×D8), C4⋊F5⋊1C22, D5⋊(D4⋊C4), C10⋊(D4⋊C4), D5⋊C8⋊5C22, (C4×D5).32D4, D10.93(C2×D4), D5.3(C2×SD16), Dic5.3(C2×D4), C4.9(C22⋊F5), C4.13(C22×F5), C20.9(C22⋊C4), C20.13(C22×C4), (D4×D5).12C22, (C4×D5).35C23, (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
Generators and relations for C2×D20⋊C4
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 >
Subgroups: 1066 in 202 conjugacy classes, 60 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, D5, D5, C10, C10, C10, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, F5, D10, D10, D10, C2×C10, C2×C10, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C5⋊C8, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C2×F5, C22×D5, C22×D5, C22×C10, C2×D4⋊C4, D5⋊C8, D5⋊C8, C4⋊F5, C4⋊F5, C2×C5⋊C8, C2×C4×D5, C2×D20, D4×D5, D4×D5, C2×C5⋊D4, D4×C10, C22×F5, C23×D5, D20⋊C4, C2×D5⋊C8, C2×C4⋊F5, C2×D4×D5, C2×D20⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, F5, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C2×F5, C2×D4⋊C4, C22⋊F5, C22×F5, D20⋊C4, C2×C22⋊F5, C2×D20⋊C4
►On 80 points
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(41 80)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)
(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 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
(1 60 33 79)(2 47 22 62)(3 54 31 65)(4 41 40 68)(5 48 29 71)(6 55 38 74)(7 42 27 77)(8 49 36 80)(9 56 25 63)(10 43 34 66)(11 50 23 69)(12 57 32 72)(13 44 21 75)(14 51 30 78)(15 58 39 61)(16 45 28 64)(17 52 37 67)(18 59 26 70)(19 46 35 73)(20 53 24 76)
G:=sub<Sym(80)| (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,60,33,79)(2,47,22,62)(3,54,31,65)(4,41,40,68)(5,48,29,71)(6,55,38,74)(7,42,27,77)(8,49,36,80)(9,56,25,63)(10,43,34,66)(11,50,23,69)(12,57,32,72)(13,44,21,75)(14,51,30,78)(15,58,39,61)(16,45,28,64)(17,52,37,67)(18,59,26,70)(19,46,35,73)(20,53,24,76)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,60,33,79)(2,47,22,62)(3,54,31,65)(4,41,40,68)(5,48,29,71)(6,55,38,74)(7,42,27,77)(8,49,36,80)(9,56,25,63)(10,43,34,66)(11,50,23,69)(12,57,32,72)(13,44,21,75)(14,51,30,78)(15,58,39,61)(16,45,28,64)(17,52,37,67)(18,59,26,70)(19,46,35,73)(20,53,24,76) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(41,80),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79)], [(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),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)], [(1,60,33,79),(2,47,22,62),(3,54,31,65),(4,41,40,68),(5,48,29,71),(6,55,38,74),(7,42,27,77),(8,49,36,80),(9,56,25,63),(10,43,34,66),(11,50,23,69),(12,57,32,72),(13,44,21,75),(14,51,30,78),(15,58,39,61),(16,45,28,64),(17,52,37,67),(18,59,26,70),(19,46,35,73),(20,53,24,76)]])
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 |
Matrix representation of C2×D20⋊C4 ►in 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] >;
C2×D20⋊C4 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