metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).4D4, (C22×C20)⋊2C4, (C2×D4).8D10, C22⋊C4⋊2Dic5, (C22×C4)⋊2Dic5, C20.D4.2C2, (D4×C10).6C22, (C22×C10).15D4, C5⋊5(C23.D4), C23.6(C5⋊D4), C23⋊Dic5.2C2, C23.2(C2×Dic5), C10.43(C23⋊C4), C2.7(C23⋊Dic5), C22.D4.1D5, C22.13(C23.D5), (C5×C22⋊C4)⋊8C4, (C2×C4).6(C5⋊D4), (C22×C10).39(C2×C4), (C5×C22.D4).1C2, (C2×C10).161(C22⋊C4), SmallGroup(320,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C22×C20)⋊C4
G = < a,b,c,d | a2=b2=c20=d4=1, ab=ba, ac=ca, dad-1=abc10, bc=cb, dbd-1=bc10, dcd-1=abc-1 >
Subgroups: 254 in 68 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×4], C5, C8, C2×C4, C2×C4 [×4], D4, C23 [×2], C10, C10 [×3], C22⋊C4, C22⋊C4 [×2], C4⋊C4, M4(2), C22×C4, C2×D4, Dic5, C20 [×3], C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22.D4, C5⋊2C8, C2×Dic5, C2×C20, C2×C20 [×3], C5×D4, C22×C10 [×2], C23.D4, C4.Dic5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C20.D4, C23⋊Dic5, C5×C22.D4, (C22×C20)⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C23.D4, C23.D5, C23⋊Dic5, (C22×C20)⋊C4
►On 80 points
Generators in S80
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 61)(20 62)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)
(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)
(2 57 64 36)(3 19)(4 55 66 34)(5 17)(6 53 68 32)(7 15)(8 51 70 30)(9 13)(10 49 72 28)(12 47 74 26)(14 45 76 24)(16 43 78 22)(18 41 80 40)(20 59 62 38)(21 54 31 44)(23 52 33 42)(25 50 35 60)(27 48 37 58)(29 46 39 56)(61 75)(63 73)(65 71)(67 69)(77 79)
G:=sub<Sym(80)| (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,61)(20,62)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51), (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), (2,57,64,36)(3,19)(4,55,66,34)(5,17)(6,53,68,32)(7,15)(8,51,70,30)(9,13)(10,49,72,28)(12,47,74,26)(14,45,76,24)(16,43,78,22)(18,41,80,40)(20,59,62,38)(21,54,31,44)(23,52,33,42)(25,50,35,60)(27,48,37,58)(29,46,39,56)(61,75)(63,73)(65,71)(67,69)(77,79)>;
G:=Group( (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,61)(20,62)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51), (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), (2,57,64,36)(3,19)(4,55,66,34)(5,17)(6,53,68,32)(7,15)(8,51,70,30)(9,13)(10,49,72,28)(12,47,74,26)(14,45,76,24)(16,43,78,22)(18,41,80,40)(20,59,62,38)(21,54,31,44)(23,52,33,42)(25,50,35,60)(27,48,37,58)(29,46,39,56)(61,75)(63,73)(65,71)(67,69)(77,79) );
G=PermutationGroup([(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,61),(20,62),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51)], [(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)], [(2,57,64,36),(3,19),(4,55,66,34),(5,17),(6,53,68,32),(7,15),(8,51,70,30),(9,13),(10,49,72,28),(12,47,74,26),(14,45,76,24),(16,43,78,22),(18,41,80,40),(20,59,62,38),(21,54,31,44),(23,52,33,42),(25,50,35,60),(27,48,37,58),(29,46,39,56),(61,75),(63,73),(65,71),(67,69),(77,79)])
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 40 | 40 | 2 | 2 | 40 | 40 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | Dic5 | Dic5 | D10 | C5⋊D4 | C5⋊D4 | C23⋊C4 | C23.D4 | C23⋊Dic5 | (C22×C20)⋊C4 |
| kernel | (C22×C20)⋊C4 | C20.D4 | C23⋊Dic5 | C5×C22.D4 | C5×C22⋊C4 | C22×C20 | C2×C20 | C22×C10 | C22.D4 | C22⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 4 | 8 |
Matrix representation of (C22×C20)⋊C4 ►in GL4(𝔽41) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 17 | 40 |
| 0 | 0 | 1 | 24 |
| 23 | 1 | 37 | 31 |
| 40 | 16 | 10 | 25 |
| 37 | 31 | 23 | 1 |
| 10 | 25 | 40 | 16 |
| 1 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 38 | 17 |
G:=sub<GL(4,GF(41))| [0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[17,1,0,0,40,24,0,0,0,0,17,1,0,0,40,24],[23,40,37,10,1,16,31,25,37,10,23,40,31,25,1,16],[1,34,0,0,0,40,0,0,0,0,24,38,0,0,1,17] >;
(C22×C20)⋊C4 in GAP, Magma, Sage, TeX
(C_2^2\times C_{20})\rtimes C_4 % in TeX
G:=Group("(C2^2xC20):C4"); // GroupNames label
G:=SmallGroup(320,97);
// by ID
G=gap.SmallGroup(320,97);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,675,297,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^20=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c^10,b*c=c*b,d*b*d^-1=b*c^10,d*c*d^-1=a*b*c^-1>;
// generators/relations