metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.35C24, D20.31C23, Dic10.30C23, C4oD4:16D10, (C2xD4):42D10, (C2xQ8):31D10, D4:D5:19C22, (C2xC20).218D4, C20.427(C2xD4), Q8:D5:18C22, D4:D10:13C2, C4.35(C23xD5), D4.8D10:7C2, C4oD20:21C22, (D4xC10):46C22, (C2xD20):59C22, C5:5(D8:C22), C5:2C8.14C23, D4.D5:17C22, (Q8xC10):38C22, D4.23(C22xD5), (C5xD4).23C23, C5:Q16:16C22, D4.9D10:13C2, D4.D10:13C2, (C5xQ8).23C23, Q8.23(C22xD5), C20.C23:13C2, (C2xC20).557C23, (C22xC10).124D4, C10.160(C22xD4), (C22xC4).282D10, C23.34(C5:D4), C4.Dic5:37C22, (C2xDic10):69C22, (C22xC20).292C22, (C2xC4oD4):4D5, (C10xC4oD4):4C2, (C2xC4oD20):31C2, C4.121(C2xC5:D4), (C2xC5:2C8):23C22, (C2xC10).591(C2xD4), (C5xC4oD4):18C22, (C2xC4.Dic5):31C2, C2.33(C22xC5:D4), C22.21(C2xC5:D4), (C2xC4).203(C5:D4), (C2xC4).246(C22xD5), SmallGroup(320,1494)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.C24
G = < a,b,c,d,e | a20=b2=c2=e2=1, d2=a10, bab=a-1, ac=ca, ad=da, eae=a11, bc=cb, bd=db, ebe=a5b, cd=dc, ece=a10c, de=ed >
Subgroups: 830 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4oD4, Dic5, C20, C20, D10, C2xC10, C2xC10, C2xM4(2), C4oD8, C8:C22, C8.C22, C2xC4oD4, C2xC4oD4, C5:2C8, Dic10, Dic10, C4xD5, D20, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C5xQ8, C22xD5, C22xC10, C22xC10, D8:C22, C2xC5:2C8, C4.Dic5, D4:D5, D4.D5, Q8:D5, C5:Q16, C2xDic10, C2xC4xD5, C2xD20, C4oD20, C4oD20, C2xC5:D4, C22xC20, C22xC20, D4xC10, D4xC10, Q8xC10, C5xC4oD4, C5xC4oD4, C2xC4.Dic5, D4.D10, C20.C23, D4:D10, D4.8D10, D4.9D10, C2xC4oD20, C10xC4oD4, C20.C24
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C22xD4, C5:D4, C22xD5, D8:C22, C2xC5:D4, C23xD5, C22xC5:D4, C20.C24
►On 80 points
(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 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 56 11 46)(2 57 12 47)(3 58 13 48)(4 59 14 49)(5 60 15 50)(6 41 16 51)(7 42 17 52)(8 43 18 53)(9 44 19 54)(10 45 20 55)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(1 26)(2 37)(3 28)(4 39)(5 30)(6 21)(7 32)(8 23)(9 34)(10 25)(11 36)(12 27)(13 38)(14 29)(15 40)(16 31)(17 22)(18 33)(19 24)(20 35)(41 61)(42 72)(43 63)(44 74)(45 65)(46 76)(47 67)(48 78)(49 69)(50 80)(51 71)(52 62)(53 73)(54 64)(55 75)(56 66)(57 77)(58 68)(59 79)(60 70)
G:=sub<Sym(80)| (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,56,11,46)(2,57,12,47)(3,58,13,48)(4,59,14,49)(5,60,15,50)(6,41,16,51)(7,42,17,52)(8,43,18,53)(9,44,19,54)(10,45,20,55)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80), (1,26)(2,37)(3,28)(4,39)(5,30)(6,21)(7,32)(8,23)(9,34)(10,25)(11,36)(12,27)(13,38)(14,29)(15,40)(16,31)(17,22)(18,33)(19,24)(20,35)(41,61)(42,72)(43,63)(44,74)(45,65)(46,76)(47,67)(48,78)(49,69)(50,80)(51,71)(52,62)(53,73)(54,64)(55,75)(56,66)(57,77)(58,68)(59,79)(60,70)>;
G:=Group( (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,56,11,46)(2,57,12,47)(3,58,13,48)(4,59,14,49)(5,60,15,50)(6,41,16,51)(7,42,17,52)(8,43,18,53)(9,44,19,54)(10,45,20,55)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80), (1,26)(2,37)(3,28)(4,39)(5,30)(6,21)(7,32)(8,23)(9,34)(10,25)(11,36)(12,27)(13,38)(14,29)(15,40)(16,31)(17,22)(18,33)(19,24)(20,35)(41,61)(42,72)(43,63)(44,74)(45,65)(46,76)(47,67)(48,78)(49,69)(50,80)(51,71)(52,62)(53,73)(54,64)(55,75)(56,66)(57,77)(58,68)(59,79)(60,70) );
G=PermutationGroup([[(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)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,56,11,46),(2,57,12,47),(3,58,13,48),(4,59,14,49),(5,60,15,50),(6,41,16,51),(7,42,17,52),(8,43,18,53),(9,44,19,54),(10,45,20,55),(21,61,31,71),(22,62,32,72),(23,63,33,73),(24,64,34,74),(25,65,35,75),(26,66,36,76),(27,67,37,77),(28,68,38,78),(29,69,39,79),(30,70,40,80)], [(1,26),(2,37),(3,28),(4,39),(5,30),(6,21),(7,32),(8,23),(9,34),(10,25),(11,36),(12,27),(13,38),(14,29),(15,40),(16,31),(17,22),(18,33),(19,24),(20,35),(41,61),(42,72),(43,63),(44,74),(45,65),(46,76),(47,67),(48,78),(49,69),(50,80),(51,71),(52,62),(53,73),(54,64),(55,75),(56,66),(57,77),(58,68),(59,79),(60,70)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | C5:D4 | C5:D4 | D8:C22 | C20.C24 |
| kernel | C20.C24 | C2xC4.Dic5 | D4.D10 | C20.C23 | D4:D10 | D4.8D10 | D4.9D10 | C2xC4oD20 | C10xC4oD4 | C2xC20 | C22xC10 | C2xC4oD4 | C22xC4 | C2xD4 | C2xQ8 | C4oD4 | C2xC4 | C23 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 1 | 3 | 1 | 2 | 2 | 2 | 2 | 8 | 12 | 4 | 2 | 8 |
Matrix representation of C20.C24 ►in GL6(F41)
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C20.C24 in GAP, Magma, Sage, TeX
C_{20}.C_2^4 % in TeX
G:=Group("C20.C2^4"); // GroupNames label
G:=SmallGroup(320,1494);
// by ID
G=gap.SmallGroup(320,1494);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=e^2=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^11,b*c=c*b,b*d=d*b,e*b*e=a^5*b,c*d=d*c,e*c*e=a^10*c,d*e=e*d>;
// generators/relations