G = C10.482+ 1+4 order 320 = 26·5
48th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.482+ 1+4, C4⋊C4⋊9D10, (C2×D4)⋊11D10, C4⋊D4⋊23D5, C20⋊7D4⋊45C2, C20⋊D4⋊20C2, C20⋊2D4⋊28C2, C22⋊C4⋊13D10, (C22×C4)⋊21D10, C22⋊D20⋊14C2, D10⋊D4⋊23C2, C23⋊D10⋊14C2, (D4×C10)⋊16C22, (C2×C20).46C23, C4⋊Dic5⋊13C22, Dic5⋊D4⋊18C2, (C2×C10).164C24, (C22×C20)⋊31C22, (C4×Dic5)⋊26C22, D10.13D4⋊14C2, C2.32(D4⋊8D10), C23.D5⋊27C22, C2.50(D4⋊6D10), D10⋊C4⋊32C22, C5⋊2(C22.54C24), (C2×D20).152C22, C22.D20⋊14C2, C23.D10⋊21C2, C10.D4⋊35C22, (C2×Dic5).81C23, (C22×D5).71C23, (C23×D5).51C22, C23.114(C22×D5), C22.185(C23×D5), C23.23D10⋊12C2, (C22×C10).192C23, (C22×Dic5)⋊23C22, (C2×C4×D5)⋊16C22, C4⋊C4⋊D5⋊15C2, (C5×C4⋊D4)⋊26C2, (C5×C4⋊C4)⋊16C22, (C2×C5⋊D4)⋊17C22, (C2×C4).42(C22×D5), (C5×C22⋊C4)⋊18C22, SmallGroup(320,1292)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.482+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=a5b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, dcd-1=ece=a5c, ede=a5b2d >
Subgroups: 1126 in 252 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22.54C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C23.D10, C22⋊D20, D10⋊D4, C22.D20, D10.13D4, C4⋊C4⋊D5, C23.23D10, C20⋊7D4, C23⋊D10, C20⋊2D4, Dic5⋊D4, C20⋊D4, C5×C4⋊D4, C10.482+ 1+4
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C22.54C24, C23×D5, D4⋊6D10, D4⋊8D10, C10.482+ 1+4
►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)
(1 48 13 58)(2 49 14 59)(3 50 15 60)(4 41 16 51)(5 42 17 52)(6 43 18 53)(7 44 19 54)(8 45 20 55)(9 46 11 56)(10 47 12 57)(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)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 38 18 23)(2 37 19 22)(3 36 20 21)(4 35 11 30)(5 34 12 29)(6 33 13 28)(7 32 14 27)(8 31 15 26)(9 40 16 25)(10 39 17 24)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)(46 75 51 70)(47 74 52 69)(48 73 53 68)(49 72 54 67)(50 71 55 66)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
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), (1,48,13,58)(2,49,14,59)(3,50,15,60)(4,41,16,51)(5,42,17,52)(6,43,18,53)(7,44,19,54)(8,45,20,55)(9,46,11,56)(10,47,12,57)(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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,38,18,23)(2,37,19,22)(3,36,20,21)(4,35,11,30)(5,34,12,29)(6,33,13,28)(7,32,14,27)(8,31,15,26)(9,40,16,25)(10,39,17,24)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)(46,75,51,70)(47,74,52,69)(48,73,53,68)(49,72,54,67)(50,71,55,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)>;
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), (1,48,13,58)(2,49,14,59)(3,50,15,60)(4,41,16,51)(5,42,17,52)(6,43,18,53)(7,44,19,54)(8,45,20,55)(9,46,11,56)(10,47,12,57)(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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,38,18,23)(2,37,19,22)(3,36,20,21)(4,35,11,30)(5,34,12,29)(6,33,13,28)(7,32,14,27)(8,31,15,26)(9,40,16,25)(10,39,17,24)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)(46,75,51,70)(47,74,52,69)(48,73,53,68)(49,72,54,67)(50,71,55,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75) );
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)], [(1,48,13,58),(2,49,14,59),(3,50,15,60),(4,41,16,51),(5,42,17,52),(6,43,18,53),(7,44,19,54),(8,45,20,55),(9,46,11,56),(10,47,12,57),(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)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,38,18,23),(2,37,19,22),(3,36,20,21),(4,35,11,30),(5,34,12,29),(6,33,13,28),(7,32,14,27),(8,31,15,26),(9,40,16,25),(10,39,17,24),(41,80,56,65),(42,79,57,64),(43,78,58,63),(44,77,59,62),(45,76,60,61),(46,75,51,70),(47,74,52,69),(48,73,53,68),(49,72,54,67),(50,71,55,66)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊6D10 | D4⋊8D10 |
| kernel | C10.482+ 1+4 | C23.D10 | C22⋊D20 | D10⋊D4 | C22.D20 | D10.13D4 | C4⋊C4⋊D5 | C23.23D10 | C20⋊7D4 | C23⋊D10 | C20⋊2D4 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 6 | 3 | 8 | 4 |
Matrix representation of C10.482+ 1+4 ►in GL8(𝔽41)
| 1 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 34 |
| 24 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 38 | 21 |
| 0 | 0 | 0 | 0 | 0 | 1 | 40 | 21 |
| 0 | 0 | 0 | 0 | 1 | 40 | 40 | 0 |
| 0 | 0 | 0 | 0 | 2 | 35 | 0 | 40 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 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 | 1 | 40 | 40 | 0 |
| 0 | 0 | 0 | 0 | 2 | 35 | 0 | 40 |
| 40 | 7 | 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 2 | 27 | 1 | 34 | 0 | 0 | 0 | 0 |
| 0 | 39 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 38 | 23 |
| 0 | 0 | 0 | 0 | 0 | 0 | 37 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 39 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 39 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 24 |
G:=sub<GL(8,GF(41))| [1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34],[24,1,0,0,0,0,0,0,40,17,0,0,0,0,0,0,0,0,24,1,0,0,0,0,0,0,40,17,0,0,0,0,0,0,0,0,1,0,1,2,0,0,0,0,0,1,40,35,0,0,0,0,38,40,40,0,0,0,0,0,21,21,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,1,2,0,0,0,0,0,1,40,35,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,2,0,0,0,0,0,7,1,27,39,0,0,0,0,40,0,1,0,0,0,0,0,7,1,34,40,0,0,0,0,0,0,0,0,23,21,0,0,0,0,0,0,6,18,0,0,0,0,0,0,0,0,38,37,0,0,0,0,0,0,23,3],[1,0,39,0,0,0,0,0,0,1,0,39,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,23,35,0,0,0,0,0,0,6,18,0,0,0,0,0,0,0,0,17,34,0,0,0,0,0,0,6,24] >;
C10.482+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{48}2_+^{1+4} % in TeX
G:=Group("C10.48ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1292);
// by ID
G=gap.SmallGroup(320,1292);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=a^5*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*b^2*d>;
// generators/relations