direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xD8:D5, D8:9D10, C40:5C23, C20.2C24, D20.1C23, Dic10:1C23, (C2xC8):8D10, (C2xD8):11D5, C4.40(D4xD5), C8:3(C22xD5), (C10xD8):11C2, (C2xD4):28D10, C5:2C8:1C23, D4:D5:9C22, (C4xD5).14D4, C20.77(C2xD4), (D4xD5):5C22, D4:2(C22xD5), (C5xD4):2C23, C4.2(C23xD5), C10:2(C8:C22), (C2xC40):16C22, D10.83(C2xD4), (C5xD8):14C22, D4.D5:7C22, (C4xD5).1C23, (D4xC10):19C22, D4:2D5:5C22, C40:C2:13C22, C8:D5:12C22, Dic5.94(C2xD4), C22.136(D4xD5), (C2xC20).519C23, (C2xDic5).247D4, (C22xD5).134D4, C10.103(C22xD4), (C2xDic10):36C22, (C2xD20).182C22, (C2xD4xD5):22C2, C5:2(C2xC8:C22), C2.76(C2xD4xD5), (C2xD4:D5):26C2, (C2xC8:D5):8C2, (C2xC40:C2):24C2, (C2xD4.D5):25C2, (C2xD4:2D5):23C2, (C2xC5:2C8):14C22, (C2xC10).392(C2xD4), (C2xC4xD5).164C22, (C2xC4).609(C22xD5), SmallGroup(320,1427)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xD8:D5
G = < a,b,c,d,e | a2=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1310 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C23, D5, C10, C10, C10, C2xC8, C2xC8, M4(2), D8, D8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, Dic5, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xM4(2), C2xD8, C2xD8, C2xSD16, C8:C22, C22xD4, C2xC4oD4, C5:2C8, C40, Dic10, Dic10, C4xD5, D20, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C5xD4, C5xD4, C22xD5, C22xD5, C22xC10, C2xC8:C22, C8:D5, C40:C2, C2xC5:2C8, D4:D5, D4.D5, C2xC40, C5xD8, C2xDic10, C2xC4xD5, C2xD20, D4xD5, D4xD5, D4:2D5, D4:2D5, C22xDic5, C2xC5:D4, D4xC10, C23xD5, C2xC8:D5, C2xC40:C2, D8:D5, C2xD4:D5, C2xD4.D5, C10xD8, C2xD4xD5, C2xD4:2D5, C2xD8:D5
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C8:C22, C22xD4, C22xD5, C2xC8:C22, D4xD5, C23xD5, D8:D5, C2xD4xD5, C2xD8:D5
►On 80 points
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 65)(32 66)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(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 7)(2 6)(3 5)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 31)(26 30)(27 29)(33 39)(34 38)(35 37)(42 48)(43 47)(44 46)(49 51)(52 56)(53 55)(58 64)(59 63)(60 62)(65 67)(68 72)(69 71)(74 80)(75 79)(76 78)
(1 67 33 15 58)(2 68 34 16 59)(3 69 35 9 60)(4 70 36 10 61)(5 71 37 11 62)(6 72 38 12 63)(7 65 39 13 64)(8 66 40 14 57)(17 43 52 79 26)(18 44 53 80 27)(19 45 54 73 28)(20 46 55 74 29)(21 47 56 75 30)(22 48 49 76 31)(23 41 50 77 32)(24 42 51 78 25)
(1 55)(2 52)(3 49)(4 54)(5 51)(6 56)(7 53)(8 50)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)(41 66)(42 71)(43 68)(44 65)(45 70)(46 67)(47 72)(48 69)(57 77)(58 74)(59 79)(60 76)(61 73)(62 78)(63 75)(64 80)
G:=sub<Sym(80)| (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (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), (1,7)(2,6)(3,5)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(42,48)(43,47)(44,46)(49,51)(52,56)(53,55)(58,64)(59,63)(60,62)(65,67)(68,72)(69,71)(74,80)(75,79)(76,78), (1,67,33,15,58)(2,68,34,16,59)(3,69,35,9,60)(4,70,36,10,61)(5,71,37,11,62)(6,72,38,12,63)(7,65,39,13,64)(8,66,40,14,57)(17,43,52,79,26)(18,44,53,80,27)(19,45,54,73,28)(20,46,55,74,29)(21,47,56,75,30)(22,48,49,76,31)(23,41,50,77,32)(24,42,51,78,25), (1,55)(2,52)(3,49)(4,54)(5,51)(6,56)(7,53)(8,50)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(57,77)(58,74)(59,79)(60,76)(61,73)(62,78)(63,75)(64,80)>;
G:=Group( (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (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), (1,7)(2,6)(3,5)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(42,48)(43,47)(44,46)(49,51)(52,56)(53,55)(58,64)(59,63)(60,62)(65,67)(68,72)(69,71)(74,80)(75,79)(76,78), (1,67,33,15,58)(2,68,34,16,59)(3,69,35,9,60)(4,70,36,10,61)(5,71,37,11,62)(6,72,38,12,63)(7,65,39,13,64)(8,66,40,14,57)(17,43,52,79,26)(18,44,53,80,27)(19,45,54,73,28)(20,46,55,74,29)(21,47,56,75,30)(22,48,49,76,31)(23,41,50,77,32)(24,42,51,78,25), (1,55)(2,52)(3,49)(4,54)(5,51)(6,56)(7,53)(8,50)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(57,77)(58,74)(59,79)(60,76)(61,73)(62,78)(63,75)(64,80) );
G=PermutationGroup([[(1,78),(2,79),(3,80),(4,73),(5,74),(6,75),(7,76),(8,77),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,65),(32,66),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63)], [(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)], [(1,7),(2,6),(3,5),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24),(25,31),(26,30),(27,29),(33,39),(34,38),(35,37),(42,48),(43,47),(44,46),(49,51),(52,56),(53,55),(58,64),(59,63),(60,62),(65,67),(68,72),(69,71),(74,80),(75,79),(76,78)], [(1,67,33,15,58),(2,68,34,16,59),(3,69,35,9,60),(4,70,36,10,61),(5,71,37,11,62),(6,72,38,12,63),(7,65,39,13,64),(8,66,40,14,57),(17,43,52,79,26),(18,44,53,80,27),(19,45,54,73,28),(20,46,55,74,29),(21,47,56,75,30),(22,48,49,76,31),(23,41,50,77,32),(24,42,51,78,25)], [(1,55),(2,52),(3,49),(4,54),(5,51),(6,56),(7,53),(8,50),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26),(17,34),(18,39),(19,36),(20,33),(21,38),(22,35),(23,40),(24,37),(41,66),(42,71),(43,68),(44,65),(45,70),(46,67),(47,72),(48,69),(57,77),(58,74),(59,79),(60,76),(61,73),(62,78),(63,75),(64,80)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C8:C22 | D4xD5 | D4xD5 | D8:D5 |
| kernel | C2xD8:D5 | C2xC8:D5 | C2xC40:C2 | D8:D5 | C2xD4:D5 | C2xD4.D5 | C10xD8 | C2xD4xD5 | C2xD4:2D5 | C4xD5 | C2xDic5 | C22xD5 | C2xD8 | C2xC8 | D8 | C2xD4 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C2xD8:D5 ►in GL8(F41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 2 | 0 | 0 | 0 | 0 |
| 40 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 31 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 39 |
| 0 | 0 | 0 | 0 | 4 | 21 | 19 | 20 |
| 0 | 0 | 0 | 0 | 0 | 21 | 21 | 22 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 21 | 21 |
| 0 | 0 | 0 | 0 | 0 | 40 | 40 | 38 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 6 |
| 35 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 35 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,2,0,1,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,21,21,0,0,0,0,10,0,19,21,0,0,0,0,31,39,20,22],[40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,21,40,1,0,0,0,0,0,21,38,0,1],[6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,8,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6],[35,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,35,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35] >;
C2xD8:D5 in GAP, Magma, Sage, TeX
C_2\times D_8\rtimes D_5
% in TeX
G:=Group("C2xD8:D5"); // GroupNames label
G:=SmallGroup(320,1427);
// by ID
G=gap.SmallGroup(320,1427);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations