direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xD10:C4, C23.31D10, C22.16D20, (C2xC4):8D10, D10:7(C2xC4), C2.3(C2xD20), (C22xC4):1D5, (C22xC20):1C2, C10.40(C2xD4), (C2xC10).36D4, (C22xD5):4C4, C10:2(C22:C4), (C2xC20):10C22, (C23xD5).2C2, C22.17(C4xD5), (C2xC10).45C23, C10.31(C22xC4), (C2xDic5):6C22, (C22xDic5):3C2, C22.20(C5:D4), C22.23(C22xD5), (C22xC10).37C22, (C22xD5).26C22, C5:3(C2xC22:C4), C2.19(C2xC4xD5), C2.2(C2xC5:D4), (C2xC10).38(C2xC4), SmallGroup(160,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xD10:C4
G = < a,b,c,d | a2=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >
Subgroups: 456 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2xC4, C2xC4, C23, C23, D5, C10, C10, C22:C4, C22xC4, C22xC4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC22:C4, C2xDic5, C2xDic5, C2xC20, C2xC20, C22xD5, C22xD5, C22xC10, D10:C4, C22xDic5, C22xC20, C23xD5, C2xD10:C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22:C4, C22xC4, C2xD4, D10, C2xC22:C4, C4xD5, D20, C5:D4, C22xD5, D10:C4, C2xC4xD5, C2xD20, C2xC5:D4, C2xD10:C4
►On 80 points
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 76)(22 77)(23 78)(24 79)(25 80)(26 71)(27 72)(28 73)(29 74)(30 75)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 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)
(1 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 60)(8 59)(9 58)(10 57)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 80)(29 79)(30 78)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 70)(39 69)(40 68)
(1 37 17 27)(2 38 18 28)(3 39 19 29)(4 40 20 30)(5 31 11 21)(6 32 12 22)(7 33 13 23)(8 34 14 24)(9 35 15 25)(10 36 16 26)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
G:=sub<Sym(80)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(21,76)(22,77)(23,78)(24,79)(25,80)(26,71)(27,72)(28,73)(29,74)(30,75)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,60)(8,59)(9,58)(10,57)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,80)(29,79)(30,78)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,61)(38,70)(39,69)(40,68), (1,37,17,27)(2,38,18,28)(3,39,19,29)(4,40,20,30)(5,31,11,21)(6,32,12,22)(7,33,13,23)(8,34,14,24)(9,35,15,25)(10,36,16,26)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(21,76)(22,77)(23,78)(24,79)(25,80)(26,71)(27,72)(28,73)(29,74)(30,75)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,60)(8,59)(9,58)(10,57)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,80)(29,79)(30,78)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,61)(38,70)(39,69)(40,68), (1,37,17,27)(2,38,18,28)(3,39,19,29)(4,40,20,30)(5,31,11,21)(6,32,12,22)(7,33,13,23)(8,34,14,24)(9,35,15,25)(10,36,16,26)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,51),(11,46),(12,47),(13,48),(14,49),(15,50),(16,41),(17,42),(18,43),(19,44),(20,45),(21,76),(22,77),(23,78),(24,79),(25,80),(26,71),(27,72),(28,73),(29,74),(30,75),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65)], [(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,60),(8,59),(9,58),(10,57),(11,42),(12,41),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,77),(22,76),(23,75),(24,74),(25,73),(26,72),(27,71),(28,80),(29,79),(30,78),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62),(37,61),(38,70),(39,69),(40,68)], [(1,37,17,27),(2,38,18,28),(3,39,19,29),(4,40,20,30),(5,31,11,21),(6,32,12,22),(7,33,13,23),(8,34,14,24),(9,35,15,25),(10,36,16,26),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70)]])
C2xD10:C4 is a maximal subgroup of
C5:3(C23:C8) C5:(C23:C8) C22:F5:C4 C22.58(D4xD5) (C2xC4):9D20 D10:2C42 D10:2(C4:C4) D10:3(C4:C4) C10.54(C4xD4) C10.55(C4xD4) (C2xC20):5D4 (C2xDic5):3D4 (C2xC4).20D20 (C2xC4).21D20 C10.(C4:D4) (C22xD5).Q8 (C2xC20).33D4 (C2xC4):6D20 (C2xC42):D5 C24.48D10 C24.12D10 C24.13D10 C23.45D20 C24.14D10 C23:2D20 C24.16D10 D10:4(C4:C4) (C2xD20):22C4 D10:5(C4:C4) C10.90(C4xD4) (C2xC4):3D20 (C2xC20).289D4 (C2xC20).290D4 (C2xC20).56D4 C24.65D10 C24.21D10 (C22xD5):Q8 C2xC4xD20 C2xD5xC22:C4 C42:10D10 C42:11D10 D4:5D20 C42:16D10 C42:17D10 C10.372+ 1+4 C10.402+ 1+4 C10.462+ 1+4 C10.512+ 1+4 C10.532+ 1+4 C10.562+ 1+4 C10.1212+ 1+4 C10.1222+ 1+4 C2xC4xC5:D4 C10.1452+ 1+4
C2xD10:C4 is a maximal quotient of
(C2xC20):10Q8 (C2xC4):6D20 C23.42D20 C24.48D10 C23.45D20 C4oD20:9C4 (C2xDic5):6Q8 D10:4(C4:C4) (C2xD20):22C4 C4:C4:36D10 C4oD20:10C4 C4.(C2xD20) C42:4D10 (C2xD20):25C4 (C22xC8):D5 C23.23D20 C23.46D20 D10:8M4(2) C4.89(C2xD20) C23.48D20 M4(2).31D10 C23.49D20 C23.20D20 C24.65D10
52 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | D10 | D10 | C4xD5 | D20 | C5:D4 |
| kernel | C2xD10:C4 | D10:C4 | C22xDic5 | C22xC20 | C23xD5 | C22xD5 | C2xC10 | C22xC4 | C2xC4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 2 | 4 | 2 | 8 | 8 | 8 |
Matrix representation of C2xD10:C4 ►in GL5(F41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 6 | 34 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,6,6,0,0,0,34,0,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,40],[32,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0] >;
C2xD10:C4 in GAP, Magma, Sage, TeX
C_2\times D_{10}\rtimes C_4 % in TeX
G:=Group("C2xD10:C4"); // GroupNames label
G:=SmallGroup(160,148);
// by ID
G=gap.SmallGroup(160,148);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations