direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22⋊D20, C23⋊5D20, C24.54D10, C10⋊1C22≀C2, D10⋊10(C2×D4), (C2×C20)⋊3C23, (D5×C24)⋊1C2, C22⋊3(C2×D20), (C22×C4)⋊7D10, (C22×C10)⋊9D4, C22⋊C4⋊37D10, (C22×D5)⋊13D4, (C22×D20)⋊5C2, C10.6(C22×D4), C2.8(C22×D20), (C2×D20)⋊43C22, (C2×C10).31C24, (C22×C20)⋊7C22, (C2×Dic5)⋊1C23, C22.125(D4×D5), (C23×D5)⋊3C22, (C22×D5)⋊1C23, D10⋊C4⋊44C22, C22.70(C23×D5), (C23×C10).57C22, (C22×Dic5)⋊6C22, C23.146(C22×D5), (C22×C10).123C23, C2.8(C2×D4×D5), C5⋊1(C2×C22≀C2), (C2×C10)⋊4(C2×D4), (C2×C4)⋊3(C22×D5), (C2×C22⋊C4)⋊10D5, (C22×C5⋊D4)⋊3C2, (C10×C22⋊C4)⋊13C2, (C2×C5⋊D4)⋊34C22, (C2×D10⋊C4)⋊16C2, (C5×C22⋊C4)⋊46C22, SmallGroup(320,1159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22⋊D20
G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 3038 in 662 conjugacy classes, 143 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C22×D4, C25, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C22≀C2, D10⋊C4, C5×C22⋊C4, C2×D20, C2×D20, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×D5, C23×D5, C23×C10, C22⋊D20, C2×D10⋊C4, C10×C22⋊C4, C22×D20, C22×C5⋊D4, D5×C24, C2×C22⋊D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22≀C2, C22×D4, D20, C22×D5, C2×C22≀C2, C2×D20, D4×D5, C23×D5, C22⋊D20, C22×D20, C2×D4×D5, C2×C22⋊D20
►On 80 points
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 47)(2 72)(3 49)(4 74)(5 51)(6 76)(7 53)(8 78)(9 55)(10 80)(11 57)(12 62)(13 59)(14 64)(15 41)(16 66)(17 43)(18 68)(19 45)(20 70)(21 46)(22 71)(23 48)(24 73)(25 50)(26 75)(27 52)(28 77)(29 54)(30 79)(31 56)(32 61)(33 58)(34 63)(35 60)(36 65)(37 42)(38 67)(39 44)(40 69)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 21)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 61)(58 62)(59 63)(60 64)
(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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)
G:=sub<Sym(80)| (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(40,45), (1,47)(2,72)(3,49)(4,74)(5,51)(6,76)(7,53)(8,78)(9,55)(10,80)(11,57)(12,62)(13,59)(14,64)(15,41)(16,66)(17,43)(18,68)(19,45)(20,70)(21,46)(22,71)(23,48)(24,73)(25,50)(26,75)(27,52)(28,77)(29,54)(30,79)(31,56)(32,61)(33,58)(34,63)(35,60)(36,65)(37,42)(38,67)(39,44)(40,69), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,21)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,61)(58,62)(59,63)(60,64), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)>;
G:=Group( (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(40,45), (1,47)(2,72)(3,49)(4,74)(5,51)(6,76)(7,53)(8,78)(9,55)(10,80)(11,57)(12,62)(13,59)(14,64)(15,41)(16,66)(17,43)(18,68)(19,45)(20,70)(21,46)(22,71)(23,48)(24,73)(25,50)(26,75)(27,52)(28,77)(29,54)(30,79)(31,56)(32,61)(33,58)(34,63)(35,60)(36,65)(37,42)(38,67)(39,44)(40,69), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,21)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,61)(58,62)(59,63)(60,64), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48) );
G=PermutationGroup([[(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,47),(2,72),(3,49),(4,74),(5,51),(6,76),(7,53),(8,78),(9,55),(10,80),(11,57),(12,62),(13,59),(14,64),(15,41),(16,66),(17,43),(18,68),(19,45),(20,70),(21,46),(22,71),(23,48),(24,73),(25,50),(26,75),(27,52),(28,77),(29,54),(30,79),(31,56),(32,61),(33,58),(34,63),(35,60),(36,65),(37,42),(38,67),(39,44),(40,69)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,21),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,60),(29,59),(30,58),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 2T | 2U | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D4×D5 |
| kernel | C2×C22⋊D20 | C22⋊D20 | C2×D10⋊C4 | C10×C22⋊C4 | C22×D20 | C22×C5⋊D4 | D5×C24 | C22×D5 | C22×C10 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 8 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 2 | 8 | 4 | 2 | 16 | 8 |
Matrix representation of C2×C22⋊D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 25 | 39 | 0 | 0 | 0 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 4 | 0 | 0 |
| 0 | 0 | 5 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 9 |
| 0 | 0 | 0 | 0 | 1 | 20 |
| 25 | 39 | 0 | 0 | 0 | 0 |
| 25 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 37 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 32 |
| 0 | 0 | 0 | 0 | 17 | 21 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[25,2,0,0,0,0,39,13,0,0,0,0,0,0,26,5,0,0,0,0,4,15,0,0,0,0,0,0,21,1,0,0,0,0,9,20],[25,25,0,0,0,0,39,16,0,0,0,0,0,0,15,15,0,0,0,0,37,26,0,0,0,0,0,0,20,17,0,0,0,0,32,21] >;
C2×C22⋊D20 in GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes D_{20} % in TeX
G:=Group("C2xC2^2:D20"); // GroupNames label
G:=SmallGroup(320,1159);
// by ID
G=gap.SmallGroup(320,1159);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations