direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic5, C10⋊4M4(2), C20.41C23, C23.3Dic5, C5⋊6(C2×M4(2)), C20.57(C2×C4), (C2×C20).20C4, (C2×C4).6Dic5, C4.9(C2×Dic5), (C22×C4).4D5, C5⋊2C8⋊12C22, (C2×C4).100D10, (C22×C20).9C2, C4.41(C22×D5), (C22×C10).11C4, C10.34(C22×C4), C2.3(C22×Dic5), (C2×C20).100C22, C22.12(C2×Dic5), (C2×C5⋊2C8)⋊12C2, (C2×C10).52(C2×C4), SmallGroup(160,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic5
G = < a,b,c,d | a2=b4=1, c10=b2, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c9 >
Subgroups: 120 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C5⋊2C8, C2×C20, C2×C20, C22×C10, C2×C5⋊2C8, C4.Dic5, C22×C20, C2×C4.Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, Dic5, D10, C2×M4(2), C2×Dic5, C22×D5, C4.Dic5, C22×Dic5, C2×C4.Dic5
►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 78)(42 79)(43 80)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(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 50 6 55 11 60 16 45)(2 59 7 44 12 49 17 54)(3 48 8 53 13 58 18 43)(4 57 9 42 14 47 19 52)(5 46 10 51 15 56 20 41)(21 72 26 77 31 62 36 67)(22 61 27 66 32 71 37 76)(23 70 28 75 33 80 38 65)(24 79 29 64 34 69 39 74)(25 68 30 73 35 78 40 63)
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)(41,78)(42,79)(43,80)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (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), (1,50,6,55,11,60,16,45)(2,59,7,44,12,49,17,54)(3,48,8,53,13,58,18,43)(4,57,9,42,14,47,19,52)(5,46,10,51,15,56,20,41)(21,72,26,77,31,62,36,67)(22,61,27,66,32,71,37,76)(23,70,28,75,33,80,38,65)(24,79,29,64,34,69,39,74)(25,68,30,73,35,78,40,63)>;
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)(41,78)(42,79)(43,80)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (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), (1,50,6,55,11,60,16,45)(2,59,7,44,12,49,17,54)(3,48,8,53,13,58,18,43)(4,57,9,42,14,47,19,52)(5,46,10,51,15,56,20,41)(21,72,26,77,31,62,36,67)(22,61,27,66,32,71,37,76)(23,70,28,75,33,80,38,65)(24,79,29,64,34,69,39,74)(25,68,30,73,35,78,40,63) );
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),(41,78),(42,79),(43,80),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,26,31,36),(22,27,32,37),(23,28,33,38),(24,29,34,39),(25,30,35,40),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70)], [(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)], [(1,50,6,55,11,60,16,45),(2,59,7,44,12,49,17,54),(3,48,8,53,13,58,18,43),(4,57,9,42,14,47,19,52),(5,46,10,51,15,56,20,41),(21,72,26,77,31,62,36,67),(22,61,27,66,32,71,37,76),(23,70,28,75,33,80,38,65),(24,79,29,64,34,69,39,74),(25,68,30,73,35,78,40,63)]])
C2×C4.Dic5 is a maximal subgroup of
C42⋊6Dic5 (C2×C20).Q8 C42⋊1Dic5 C20.32C42 C20.60(C4⋊C4) C20.40C42 M4(2)⋊Dic5 (C2×C40)⋊C4 C20.34C42 M4(2)⋊4Dic5 C20.51C42 C20.29M4(2) D10⋊4M4(2) Dic5⋊2M4(2) C20⋊13M4(2) C20.47(C4⋊C4) C4○D20⋊9C4 C20.64(C4⋊C4) C20.35C42 C42.43D10 C4⋊C4⋊36D10 C4.(C2×D20) C42⋊4D10 C42.47D10 C20⋊7M4(2) C4⋊D4⋊D5 C4.(D4×D5) C22⋊Q8⋊D5 C5⋊(C8.D4) C20.42C42 C20.65(C4⋊C4) (C22×C8)⋊D5 M4(2)×Dic5 Dic5⋊5M4(2) C23.Dic10 M4(2).Dic5 D10⋊8M4(2) M4(2).31D10 C24.4Dic5 (D4×C10)⋊18C4 (Q8×C10)⋊16C4 C4○D4⋊Dic5 (D4×C10).24C4 (D4×C10)⋊21C4 (D4×C10).29C4 C2×D5×M4(2) C40.47C23 C20.76C24 C20.C24
C2×C4.Dic5 is a maximal quotient of
C20⋊13M4(2) C42.6Dic5 C42.7Dic5 C42.47D10 C20⋊7M4(2) C42.210D10 C24.4Dic5
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D5 | M4(2) | Dic5 | D10 | Dic5 | C4.Dic5 |
| kernel | C2×C4.Dic5 | C2×C5⋊2C8 | C4.Dic5 | C22×C20 | C2×C20 | C22×C10 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 2 | 4 | 6 | 6 | 2 | 16 |
Matrix representation of C2×C4.Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 33 |
| 0 | 0 | 0 | 32 |
| 7 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 5 | 15 |
| 0 | 0 | 0 | 8 |
| 9 | 0 | 0 | 0 |
| 19 | 32 | 0 | 0 |
| 0 | 0 | 33 | 19 |
| 0 | 0 | 23 | 8 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,33,32],[7,40,0,0,1,0,0,0,0,0,5,0,0,0,15,8],[9,19,0,0,0,32,0,0,0,0,33,23,0,0,19,8] >;
C2×C4.Dic5 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_5 % in TeX
G:=Group("C2xC4.Dic5"); // GroupNames label
G:=SmallGroup(160,142);
// by ID
G=gap.SmallGroup(160,142);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^10=b^2,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations