direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22.2D20, C24.11D10, C23.15D20, (C23×D5)⋊4C4, C10⋊3(C23⋊C4), C22⋊C4⋊36D10, C23.17(C4×D5), (C22×Dic5)⋊5C4, C22.14(C2×D20), (C22×C10).66D4, C23.74(C5⋊D4), C23.D5⋊42C22, C23.72(C22×D5), (C23×C10).37C22, (C22×C10).111C23, C22.44(D10⋊C4), C5⋊5(C2×C23⋊C4), (C2×C5⋊D4)⋊8C4, (C2×C22⋊C4)⋊1D5, (C10×C22⋊C4)⋊1C2, C22.18(C2×C4×D5), (C2×Dic5)⋊2(C2×C4), (C22×D5)⋊2(C2×C4), (C2×C23.D5)⋊1C2, (C2×C10).433(C2×D4), C2.8(C2×D10⋊C4), C10.76(C2×C22⋊C4), (C22×C5⋊D4).1C2, C22.26(C2×C5⋊D4), (C5×C22⋊C4)⋊44C22, (C2×C5⋊D4).91C22, (C2×C10).113(C22×C4), (C22×C10).120(C2×C4), (C2×C10).120(C22⋊C4), SmallGroup(320,581)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22.2D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 926 in 210 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×6], C22 [×7], C22 [×18], C5, C2×C4 [×12], D4 [×8], C23 [×7], C23 [×8], D5 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×2], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×8], C24, C24, Dic5 [×4], C20 [×2], D10 [×8], C2×C10 [×7], C2×C10 [×10], C23⋊C4 [×4], C2×C22⋊C4, C2×C22⋊C4, C22×D4, C2×Dic5 [×2], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×4], C22×D5 [×2], C22×D5 [×4], C22×C10 [×7], C22×C10 [×2], C2×C23⋊C4, C23.D5 [×2], C23.D5, C5×C22⋊C4 [×2], C5×C22⋊C4, C22×Dic5, C22×Dic5, C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20, C23×D5, C23×C10, C22.2D20 [×4], C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C2×C22.2D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C23⋊C4 [×2], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C23⋊C4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C22.2D20 [×2], C2×D10⋊C4, C2×C22.2D20
►On 80 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)
(1 11)(2 63)(3 13)(4 65)(5 15)(6 67)(7 17)(8 69)(9 19)(10 71)(12 73)(14 75)(16 77)(18 79)(20 61)(21 42)(22 32)(23 44)(24 34)(25 46)(26 36)(27 48)(28 38)(29 50)(30 40)(31 52)(33 54)(35 56)(37 58)(39 60)(41 51)(43 53)(45 55)(47 57)(49 59)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(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)
(1 61 11 20)(2 70 63 80)(3 8 13 69)(4 17 65 7)(5 77 15 16)(6 66 67 76)(9 73 19 12)(10 62 71 72)(14 78 75 68)(18 74 79 64)(21 49 42 59)(22 27 32 48)(23 36 44 26)(24 56 34 35)(25 45 46 55)(28 52 38 31)(29 41 50 51)(30 39 40 60)(33 57 54 47)(37 53 58 43)
G:=sub<Sym(80)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72), (1,11)(2,63)(3,13)(4,65)(5,15)(6,67)(7,17)(8,69)(9,19)(10,71)(12,73)(14,75)(16,77)(18,79)(20,61)(21,42)(22,32)(23,44)(24,34)(25,46)(26,36)(27,48)(28,38)(29,50)(30,40)(31,52)(33,54)(35,56)(37,58)(39,60)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(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), (1,61,11,20)(2,70,63,80)(3,8,13,69)(4,17,65,7)(5,77,15,16)(6,66,67,76)(9,73,19,12)(10,62,71,72)(14,78,75,68)(18,74,79,64)(21,49,42,59)(22,27,32,48)(23,36,44,26)(24,56,34,35)(25,45,46,55)(28,52,38,31)(29,41,50,51)(30,39,40,60)(33,57,54,47)(37,53,58,43)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72), (1,11)(2,63)(3,13)(4,65)(5,15)(6,67)(7,17)(8,69)(9,19)(10,71)(12,73)(14,75)(16,77)(18,79)(20,61)(21,42)(22,32)(23,44)(24,34)(25,46)(26,36)(27,48)(28,38)(29,50)(30,40)(31,52)(33,54)(35,56)(37,58)(39,60)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(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), (1,61,11,20)(2,70,63,80)(3,8,13,69)(4,17,65,7)(5,77,15,16)(6,66,67,76)(9,73,19,12)(10,62,71,72)(14,78,75,68)(18,74,79,64)(21,49,42,59)(22,27,32,48)(23,36,44,26)(24,56,34,35)(25,45,46,55)(28,52,38,31)(29,41,50,51)(30,39,40,60)(33,57,54,47)(37,53,58,43) );
G=PermutationGroup([(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72)], [(1,11),(2,63),(3,13),(4,65),(5,15),(6,67),(7,17),(8,69),(9,19),(10,71),(12,73),(14,75),(16,77),(18,79),(20,61),(21,42),(22,32),(23,44),(24,34),(25,46),(26,36),(27,48),(28,38),(29,50),(30,40),(31,52),(33,54),(35,56),(37,58),(39,60),(41,51),(43,53),(45,55),(47,57),(49,59),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(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)], [(1,61,11,20),(2,70,63,80),(3,8,13,69),(4,17,65,7),(5,77,15,16),(6,66,67,76),(9,73,19,12),(10,62,71,72),(14,78,75,68),(18,74,79,64),(21,49,42,59),(22,27,32,48),(23,36,44,26),(24,56,34,35),(25,45,46,55),(28,52,38,31),(29,41,50,51),(30,39,40,60),(33,57,54,47),(37,53,58,43)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 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 | 1 | 2 | ··· | 2 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | C22.2D20 |
| kernel | C2×C22.2D20 | C22.2D20 | C2×C23.D5 | C10×C22⋊C4 | C22×C5⋊D4 | C22×Dic5 | C2×C5⋊D4 | C23×D5 | C22×C10 | C2×C22⋊C4 | C22⋊C4 | C24 | C23 | C23 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 8 | 8 | 8 | 2 | 8 |
Matrix representation of C2×C22.2D20 ►in GL6(𝔽41)
| 40 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 39 | 9 | 0 | 40 |
| 0 | 0 | 2 | 32 | 40 | 0 |
| 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 |
| 22 | 3 | 0 | 0 | 0 | 0 |
| 38 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 0 | 0 | 0 | 40 | 32 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 3 | 19 | 0 | 0 | 0 | 0 |
| 19 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 32 | 0 |
| 0 | 0 | 2 | 0 | 40 | 1 |
| 0 | 0 | 0 | 40 | 32 | 0 |
| 0 | 0 | 23 | 0 | 9 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,2,0,0,9,1,9,32,0,0,0,0,0,40,0,0,0,0,40,0],[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],[22,38,0,0,0,0,3,22,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,32,40,32,9,0,0,0,40,0,0],[3,19,0,0,0,0,19,38,0,0,0,0,0,0,9,2,0,23,0,0,0,0,40,0,0,0,32,40,32,9,0,0,0,1,0,0] >;
C2×C22.2D20 in GAP, Magma, Sage, TeX
C_2\times C_2^2._2D_{20} % in TeX
G:=Group("C2xC2^2.2D20"); // GroupNames label
G:=SmallGroup(320,581);
// by ID
G=gap.SmallGroup(320,581);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,438,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations