metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.48D20, (C2×C8)⋊21D10, (C2×D20)⋊26C4, (C2×C40)⋊36C22, D20.38(C2×C4), (C2×C4).153D20, C20.417(C2×D4), (C2×C20).173D4, D20⋊5C4⋊39C2, C2.4(C8⋊D10), (C2×M4(2))⋊11D5, C4⋊Dic5⋊48C22, C22.57(C2×D20), C10.20(C8⋊C22), C20.74(C22⋊C4), (C10×M4(2))⋊19C2, C20.174(C22×C4), (C2×C20).773C23, (C22×D20).16C2, (C22×C4).139D10, (C22×C10).101D4, C5⋊4(C23.37D4), C4.13(D10⋊C4), (C2×D20).206C22, C23.21D10⋊16C2, (C22×C20).188C22, C22.28(D10⋊C4), C4.73(C2×C4×D5), (C2×C4).53(C4×D5), C4.110(C2×C5⋊D4), (C2×C20).281(C2×C4), (C2×C10).163(C2×D4), (C2×C4).76(C5⋊D4), C10.99(C2×C22⋊C4), C2.30(C2×D10⋊C4), (C2×C4).722(C22×D5), (C2×C10).85(C22⋊C4), SmallGroup(320,758)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.48D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >
Subgroups: 1006 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C40, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, C23.37D4, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C2×D20, C2×D20, C22×C20, C23×D5, D20⋊5C4, C23.21D10, C10×M4(2), C22×D20, C23.48D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8⋊C22, C4×D5, D20, C5⋊D4, C22×D5, C23.37D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C8⋊D10, C2×D10⋊C4, C23.48D20
►On 80 points
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 52 53 40)(2 39 54 51)(3 50 55 38)(4 37 56 49)(5 48 57 36)(6 35 58 47)(7 46 59 34)(8 33 60 45)(9 44 61 32)(10 31 62 43)(11 42 63 30)(12 29 64 41)(13 80 65 28)(14 27 66 79)(15 78 67 26)(16 25 68 77)(17 76 69 24)(18 23 70 75)(19 74 71 22)(20 21 72 73)
G:=sub<Sym(80)| (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,52,53,40)(2,39,54,51)(3,50,55,38)(4,37,56,49)(5,48,57,36)(6,35,58,47)(7,46,59,34)(8,33,60,45)(9,44,61,32)(10,31,62,43)(11,42,63,30)(12,29,64,41)(13,80,65,28)(14,27,66,79)(15,78,67,26)(16,25,68,77)(17,76,69,24)(18,23,70,75)(19,74,71,22)(20,21,72,73)>;
G:=Group( (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,52,53,40)(2,39,54,51)(3,50,55,38)(4,37,56,49)(5,48,57,36)(6,35,58,47)(7,46,59,34)(8,33,60,45)(9,44,61,32)(10,31,62,43)(11,42,63,30)(12,29,64,41)(13,80,65,28)(14,27,66,79)(15,78,67,26)(16,25,68,77)(17,76,69,24)(18,23,70,75)(19,74,71,22)(20,21,72,73) );
G=PermutationGroup([[(2,22),(4,24),(6,26),(8,28),(10,30),(12,32),(14,34),(16,36),(18,38),(20,40),(42,62),(44,64),(46,66),(48,68),(50,70),(52,72),(54,74),(56,76),(58,78),(60,80)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,52,53,40),(2,39,54,51),(3,50,55,38),(4,37,56,49),(5,48,57,36),(6,35,58,47),(7,46,59,34),(8,33,60,45),(9,44,61,32),(10,31,62,43),(11,42,63,30),(12,29,64,41),(13,80,65,28),(14,27,66,79),(15,78,67,26),(16,25,68,77),(17,76,69,24),(18,23,70,75),(19,74,71,22),(20,21,72,73)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D20 | C8⋊C22 | C8⋊D10 |
| kernel | C23.48D20 | D20⋊5C4 | C23.21D10 | C10×M4(2) | C22×D20 | C2×D20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 4 | 2 | 8 |
Matrix representation of C23.48D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 23 | 40 | 0 |
| 0 | 0 | 38 | 38 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 23 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 6 | 33 |
| 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 18 | 7 | 6 | 40 |
| 0 | 0 | 17 | 19 | 1 | 34 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 7 | 8 |
| 0 | 0 | 40 | 40 | 2 | 1 |
| 0 | 0 | 22 | 21 | 1 | 1 |
| 0 | 0 | 30 | 31 | 7 | 7 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,23,38,0,0,0,1,23,38,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,18,17,0,0,1,0,7,19,0,0,6,7,6,1,0,0,33,40,40,34],[1,9,0,0,0,0,18,40,0,0,0,0,0,0,34,40,22,30,0,0,34,40,21,31,0,0,7,2,1,7,0,0,8,1,1,7] >;
C23.48D20 in GAP, Magma, Sage, TeX
C_2^3._{48}D_{20} % in TeX
G:=Group("C2^3.48D20"); // GroupNames label
G:=SmallGroup(320,758);
// by ID
G=gap.SmallGroup(320,758);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,387,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^19>;
// generators/relations