G = C10.1212+ 1+4 order 320 = 26·5
30th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.1212+ 1+4, C5⋊D4⋊6D4, C4⋊C4⋊15D10, C5⋊8(D4⋊5D4), C4⋊D20⋊30C2, C22⋊C4⋊15D10, D10.45(C2×D4), (C22×C4)⋊23D10, D10⋊10(C4○D4), D10⋊D4⋊29C2, C22⋊D20⋊19C2, D20⋊8C4⋊31C2, C22.12(D4×D5), D10⋊Q8⋊27C2, (C2×D4).163D10, (C2×D20)⋊26C22, (C2×C20).71C23, Dic5.50(C2×D4), C22.D4⋊4D5, C10.83(C22×D4), Dic5⋊D4⋊20C2, Dic5⋊4D4⋊18C2, (C2×C10).198C24, (C22×C20)⋊17C22, (C4×Dic5)⋊31C22, D10.13D4⋊27C2, C2.41(D4⋊8D10), C23.D5⋊30C22, D10⋊C4⋊22C22, C23.26(C22×D5), Dic5.5D4⋊31C2, (C2×Dic10)⋊55C22, (D4×C10).136C22, C10.D4⋊22C22, (C22×C10).33C23, (C23×D5).57C22, C22.219(C23×D5), (C2×Dic5).102C23, (C22×Dic5)⋊24C22, (C22×D5).216C23, (C2×D4×D5)⋊15C2, C2.56(C2×D4×D5), C2.60(D5×C4○D4), (C2×C4×D5)⋊21C22, (C2×C4○D20)⋊11C2, (C5×C4⋊C4)⋊25C22, (D5×C22⋊C4)⋊12C2, (C2×C10).59(C2×D4), C10.172(C2×C4○D4), (C2×C5⋊D4)⋊41C22, (C2×D10⋊C4)⋊23C2, (C2×C4).61(C22×D5), (C5×C22⋊C4)⋊21C22, (C5×C22.D4)⋊6C2, SmallGroup(320,1326)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.1212+ 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a5b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
Subgroups: 1478 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, D4⋊5D4, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, D4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, D5×C22⋊C4, Dic5⋊4D4, C22⋊D20, D10⋊D4, Dic5.5D4, D20⋊8C4, D10.13D4, C4⋊D20, D10⋊Q8, C2×D10⋊C4, Dic5⋊D4, C5×C22.D4, C2×C4○D20, C2×D4×D5, C10.1212+ 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊5D4, D4×D5, C23×D5, C2×D4×D5, D5×C4○D4, D4⋊8D10, C10.1212+ 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 50 28 38)(2 41 29 39)(3 42 30 40)(4 43 21 31)(5 44 22 32)(6 45 23 33)(7 46 24 34)(8 47 25 35)(9 48 26 36)(10 49 27 37)(11 58 73 70)(12 59 74 61)(13 60 75 62)(14 51 76 63)(15 52 77 64)(16 53 78 65)(17 54 79 66)(18 55 80 67)(19 56 71 68)(20 57 72 69)
(1 50 6 45)(2 49 7 44)(3 48 8 43)(4 47 9 42)(5 46 10 41)(11 65 16 70)(12 64 17 69)(13 63 18 68)(14 62 19 67)(15 61 20 66)(21 35 26 40)(22 34 27 39)(23 33 28 38)(24 32 29 37)(25 31 30 36)(51 80 56 75)(52 79 57 74)(53 78 58 73)(54 77 59 72)(55 76 60 71)
(1 53 28 65)(2 52 29 64)(3 51 30 63)(4 60 21 62)(5 59 22 61)(6 58 23 70)(7 57 24 69)(8 56 25 68)(9 55 26 67)(10 54 27 66)(11 45 73 33)(12 44 74 32)(13 43 75 31)(14 42 76 40)(15 41 77 39)(16 50 78 38)(17 49 79 37)(18 48 80 36)(19 47 71 35)(20 46 72 34)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 61)(8 62)(9 63)(10 64)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
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), (1,50,28,38)(2,41,29,39)(3,42,30,40)(4,43,21,31)(5,44,22,32)(6,45,23,33)(7,46,24,34)(8,47,25,35)(9,48,26,36)(10,49,27,37)(11,58,73,70)(12,59,74,61)(13,60,75,62)(14,51,76,63)(15,52,77,64)(16,53,78,65)(17,54,79,66)(18,55,80,67)(19,56,71,68)(20,57,72,69), (1,50,6,45)(2,49,7,44)(3,48,8,43)(4,47,9,42)(5,46,10,41)(11,65,16,70)(12,64,17,69)(13,63,18,68)(14,62,19,67)(15,61,20,66)(21,35,26,40)(22,34,27,39)(23,33,28,38)(24,32,29,37)(25,31,30,36)(51,80,56,75)(52,79,57,74)(53,78,58,73)(54,77,59,72)(55,76,60,71), (1,53,28,65)(2,52,29,64)(3,51,30,63)(4,60,21,62)(5,59,22,61)(6,58,23,70)(7,57,24,69)(8,56,25,68)(9,55,26,67)(10,54,27,66)(11,45,73,33)(12,44,74,32)(13,43,75,31)(14,42,76,40)(15,41,77,39)(16,50,78,38)(17,49,79,37)(18,48,80,36)(19,47,71,35)(20,46,72,34), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,61)(8,62)(9,63)(10,64)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)>;
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), (1,50,28,38)(2,41,29,39)(3,42,30,40)(4,43,21,31)(5,44,22,32)(6,45,23,33)(7,46,24,34)(8,47,25,35)(9,48,26,36)(10,49,27,37)(11,58,73,70)(12,59,74,61)(13,60,75,62)(14,51,76,63)(15,52,77,64)(16,53,78,65)(17,54,79,66)(18,55,80,67)(19,56,71,68)(20,57,72,69), (1,50,6,45)(2,49,7,44)(3,48,8,43)(4,47,9,42)(5,46,10,41)(11,65,16,70)(12,64,17,69)(13,63,18,68)(14,62,19,67)(15,61,20,66)(21,35,26,40)(22,34,27,39)(23,33,28,38)(24,32,29,37)(25,31,30,36)(51,80,56,75)(52,79,57,74)(53,78,58,73)(54,77,59,72)(55,76,60,71), (1,53,28,65)(2,52,29,64)(3,51,30,63)(4,60,21,62)(5,59,22,61)(6,58,23,70)(7,57,24,69)(8,56,25,68)(9,55,26,67)(10,54,27,66)(11,45,73,33)(12,44,74,32)(13,43,75,31)(14,42,76,40)(15,41,77,39)(16,50,78,38)(17,49,79,37)(18,48,80,36)(19,47,71,35)(20,46,72,34), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,61)(8,62)(9,63)(10,64)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80) );
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)], [(1,50,28,38),(2,41,29,39),(3,42,30,40),(4,43,21,31),(5,44,22,32),(6,45,23,33),(7,46,24,34),(8,47,25,35),(9,48,26,36),(10,49,27,37),(11,58,73,70),(12,59,74,61),(13,60,75,62),(14,51,76,63),(15,52,77,64),(16,53,78,65),(17,54,79,66),(18,55,80,67),(19,56,71,68),(20,57,72,69)], [(1,50,6,45),(2,49,7,44),(3,48,8,43),(4,47,9,42),(5,46,10,41),(11,65,16,70),(12,64,17,69),(13,63,18,68),(14,62,19,67),(15,61,20,66),(21,35,26,40),(22,34,27,39),(23,33,28,38),(24,32,29,37),(25,31,30,36),(51,80,56,75),(52,79,57,74),(53,78,58,73),(54,77,59,72),(55,76,60,71)], [(1,53,28,65),(2,52,29,64),(3,51,30,63),(4,60,21,62),(5,59,22,61),(6,58,23,70),(7,57,24,69),(8,56,25,68),(9,55,26,67),(10,54,27,66),(11,45,73,33),(12,44,74,32),(13,43,75,31),(14,42,76,40),(15,41,77,39),(16,50,78,38),(17,49,79,37),(18,48,80,36),(19,47,71,35),(20,46,72,34)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,61),(8,62),(9,63),(10,64),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,41),(18,42),(19,43),(20,44),(21,56),(22,57),(23,58),(24,59),(25,60),(26,51),(27,52),(28,53),(29,54),(30,55),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | C10.1212+ 1+4 | D5×C22⋊C4 | Dic5⋊4D4 | C22⋊D20 | D10⋊D4 | Dic5.5D4 | D20⋊8C4 | D10.13D4 | C4⋊D20 | D10⋊Q8 | C2×D10⋊C4 | Dic5⋊D4 | C5×C22.D4 | C2×C4○D20 | C2×D4×D5 | C5⋊D4 | C22.D4 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 4 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C10.1212+ 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 23 | 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 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 40 | 18 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 5 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 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 | 5 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,23,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,9,0,0,0,0,18,1,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,40,16,0,0,0,0,5,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,5,1] >;
C10.1212+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{121}2_+^{1+4} % in TeX
G:=Group("C10.121ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1326);
// by ID
G=gap.SmallGroup(320,1326);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,346,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^5*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations