metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊2D20, C20⋊1D4, D10⋊2D4, C4⋊C4⋊3D5, (C2×D20)⋊4C2, C5⋊2(C4⋊D4), C10.7(C2×D4), C2.13(D4×D5), C2.9(C2×D20), (C2×C4).12D10, D10⋊C4⋊8C2, (C2×C20).5C22, C10.34(C4○D4), (C2×C10).36C23, C2.6(Q8⋊2D5), (C22×D5).7C22, C22.50(C22×D5), (C2×Dic5).34C22, (C2×C4×D5)⋊1C2, (C5×C4⋊C4)⋊6C2, SmallGroup(160,116)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊2D20
G = < a,b,c | a4=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 384 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C5, C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], D5 [×4], C10 [×3], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20 [×2], C4⋊2D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, C2×D20, D4×D5, Q8⋊2D5, C4⋊2D20
►On 80 points
(1 59 69 37)(2 38 70 60)(3 41 71 39)(4 40 72 42)(5 43 73 21)(6 22 74 44)(7 45 75 23)(8 24 76 46)(9 47 77 25)(10 26 78 48)(11 49 79 27)(12 28 80 50)(13 51 61 29)(14 30 62 52)(15 53 63 31)(16 32 64 54)(17 55 65 33)(18 34 66 56)(19 57 67 35)(20 36 68 58)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
G:=sub<Sym(80)| (1,59,69,37)(2,38,70,60)(3,41,71,39)(4,40,72,42)(5,43,73,21)(6,22,74,44)(7,45,75,23)(8,24,76,46)(9,47,77,25)(10,26,78,48)(11,49,79,27)(12,28,80,50)(13,51,61,29)(14,30,62,52)(15,53,63,31)(16,32,64,54)(17,55,65,33)(18,34,66,56)(19,57,67,35)(20,36,68,58), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)>;
G:=Group( (1,59,69,37)(2,38,70,60)(3,41,71,39)(4,40,72,42)(5,43,73,21)(6,22,74,44)(7,45,75,23)(8,24,76,46)(9,47,77,25)(10,26,78,48)(11,49,79,27)(12,28,80,50)(13,51,61,29)(14,30,62,52)(15,53,63,31)(16,32,64,54)(17,55,65,33)(18,34,66,56)(19,57,67,35)(20,36,68,58), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77) );
G=PermutationGroup([(1,59,69,37),(2,38,70,60),(3,41,71,39),(4,40,72,42),(5,43,73,21),(6,22,74,44),(7,45,75,23),(8,24,76,46),(9,47,77,25),(10,26,78,48),(11,49,79,27),(12,28,80,50),(13,51,61,29),(14,30,62,52),(15,53,63,31),(16,32,64,54),(17,55,65,33),(18,34,66,56),(19,57,67,35),(20,36,68,58)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,60),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D20 | D4×D5 | Q8⋊2D5 |
| kernel | C4⋊2D20 | D10⋊C4 | C5×C4⋊C4 | C2×C4×D5 | C2×D20 | C20 | D10 | C4⋊C4 | C10 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of C4⋊2D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 14 | 39 | 0 | 0 |
| 16 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[14,16,0,0,39,30,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,1,40,0,0,0,0,1,1,0,0,0,40] >;
C4⋊2D20 in GAP, Magma, Sage, TeX
C_4\rtimes_2D_{20} % in TeX
G:=Group("C4:2D20"); // GroupNames label
G:=SmallGroup(160,116);
// by ID
G=gap.SmallGroup(160,116);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,50,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations