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, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C4.D4, C22.D4, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, 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, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C23⋊C4, C2×Dic5, C5⋊D4, C23.D4, C23.D5, C23⋊Dic5, (C22×C20)⋊C4
►On 80 points
Generators in S80
(1 78)(2 79)(3 80)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 40)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(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)
(2 67 21 52)(3 19)(4 65 23 50)(5 17)(6 63 25 48)(7 15)(8 61 27 46)(9 13)(10 79 29 44)(12 77 31 42)(14 75 33 60)(16 73 35 58)(18 71 37 56)(20 69 39 54)(22 28)(24 26)(30 40)(32 38)(34 36)(41 80 51 70)(43 78 53 68)(45 76 55 66)(47 74 57 64)(49 72 59 62)
G:=sub<Sym(80)| (1,78)(2,79)(3,80)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(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), (2,67,21,52)(3,19)(4,65,23,50)(5,17)(6,63,25,48)(7,15)(8,61,27,46)(9,13)(10,79,29,44)(12,77,31,42)(14,75,33,60)(16,73,35,58)(18,71,37,56)(20,69,39,54)(22,28)(24,26)(30,40)(32,38)(34,36)(41,80,51,70)(43,78,53,68)(45,76,55,66)(47,74,57,64)(49,72,59,62)>;
G:=Group( (1,78)(2,79)(3,80)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(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), (2,67,21,52)(3,19)(4,65,23,50)(5,17)(6,63,25,48)(7,15)(8,61,27,46)(9,13)(10,79,29,44)(12,77,31,42)(14,75,33,60)(16,73,35,58)(18,71,37,56)(20,69,39,54)(22,28)(24,26)(30,40)(32,38)(34,36)(41,80,51,70)(43,78,53,68)(45,76,55,66)(47,74,57,64)(49,72,59,62) );
G=PermutationGroup([[(1,78),(2,79),(3,80),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,73),(17,74),(18,75),(19,76),(20,77),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(40,53)], [(1,40),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(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)], [(2,67,21,52),(3,19),(4,65,23,50),(5,17),(6,63,25,48),(7,15),(8,61,27,46),(9,13),(10,79,29,44),(12,77,31,42),(14,75,33,60),(16,73,35,58),(18,71,37,56),(20,69,39,54),(22,28),(24,26),(30,40),(32,38),(34,36),(41,80,51,70),(43,78,53,68),(45,76,55,66),(47,74,57,64),(49,72,59,62)]])
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