metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.30D20, C22.2Dic20, C4⋊Dic5⋊3C4, C10.20C4≀C2, C22⋊C8.1D5, (C2×C10).1Q16, (C2×Dic10)⋊2C4, (C2×C20).437D4, (C2×C10).1SD16, (C22×C4).54D10, (C22×C10).39D4, C2.6(D20⋊7C4), C10.24(C23⋊C4), C20.48D4.1C2, C22.4(C40⋊C2), C5⋊4(C23.31D4), C10.13(Q8⋊C4), C2.3(C20.44D4), (C22×C20).40C22, C2.6(C23.1D10), C22.58(D10⋊C4), C10.10C42.21C2, (C2×C4).12(C4×D5), (C5×C22⋊C8).1C2, (C2×C20).197(C2×C4), (C2×C4).208(C5⋊D4), (C2×C10).103(C22⋊C4), SmallGroup(320,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.30D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=cb=bc, e2=b, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=acd19 >
Subgroups: 350 in 80 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2.C42, C22⋊C8, C22⋊Q8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.31D4, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C22×Dic5, C22×C20, C10.10C42, C5×C22⋊C8, C20.48D4, C23.30D20
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, SD16, Q16, D10, C23⋊C4, Q8⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, C23.31D4, C40⋊C2, Dic20, D10⋊C4, C23.1D10, C20.44D4, D20⋊7C4, C23.30D20
►On 80 points
Generators in S80
(2 54)(4 56)(6 58)(8 60)(10 62)(12 64)(14 66)(16 68)(18 70)(20 72)(22 74)(24 76)(26 78)(28 80)(30 42)(32 44)(34 46)(36 48)(38 50)(40 52)
(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 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)
(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 20 73 52)(2 39 74 71)(3 70 75 38)(4 49 76 17)(5 16 77 48)(6 35 78 67)(7 66 79 34)(8 45 80 13)(9 12 41 44)(10 31 42 63)(11 62 43 30)(14 27 46 59)(15 58 47 26)(18 23 50 55)(19 54 51 22)(21 40 53 72)(24 69 56 37)(25 36 57 68)(28 65 60 33)(29 32 61 64)
G:=sub<Sym(80)| (2,54)(4,56)(6,58)(8,60)(10,62)(12,64)(14,66)(16,68)(18,70)(20,72)(22,74)(24,76)(26,78)(28,80)(30,42)(32,44)(34,46)(36,48)(38,50)(40,52), (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), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52), (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), (1,20,73,52)(2,39,74,71)(3,70,75,38)(4,49,76,17)(5,16,77,48)(6,35,78,67)(7,66,79,34)(8,45,80,13)(9,12,41,44)(10,31,42,63)(11,62,43,30)(14,27,46,59)(15,58,47,26)(18,23,50,55)(19,54,51,22)(21,40,53,72)(24,69,56,37)(25,36,57,68)(28,65,60,33)(29,32,61,64)>;
G:=Group( (2,54)(4,56)(6,58)(8,60)(10,62)(12,64)(14,66)(16,68)(18,70)(20,72)(22,74)(24,76)(26,78)(28,80)(30,42)(32,44)(34,46)(36,48)(38,50)(40,52), (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), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52), (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), (1,20,73,52)(2,39,74,71)(3,70,75,38)(4,49,76,17)(5,16,77,48)(6,35,78,67)(7,66,79,34)(8,45,80,13)(9,12,41,44)(10,31,42,63)(11,62,43,30)(14,27,46,59)(15,58,47,26)(18,23,50,55)(19,54,51,22)(21,40,53,72)(24,69,56,37)(25,36,57,68)(28,65,60,33)(29,32,61,64) );
G=PermutationGroup([[(2,54),(4,56),(6,58),(8,60),(10,62),(12,64),(14,66),(16,68),(18,70),(20,72),(22,74),(24,76),(26,78),(28,80),(30,42),(32,44),(34,46),(36,48),(38,50),(40,52)], [(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)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52)], [(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)], [(1,20,73,52),(2,39,74,71),(3,70,75,38),(4,49,76,17),(5,16,77,48),(6,35,78,67),(7,66,79,34),(8,45,80,13),(9,12,41,44),(10,31,42,63),(11,62,43,30),(14,27,46,59),(15,58,47,26),(18,23,50,55),(19,54,51,22),(21,40,53,72),(24,69,56,37),(25,36,57,68),(28,65,60,33),(29,32,61,64)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 4 | 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 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | SD16 | Q16 | D10 | C4≀C2 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | Dic20 | C23⋊C4 | C23.1D10 | D20⋊7C4 |
| kernel | C23.30D20 | C10.10C42 | C5×C22⋊C8 | C20.48D4 | C4⋊Dic5 | C2×Dic10 | C2×C20 | C22×C10 | C22⋊C8 | C2×C10 | C2×C10 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C22 | C22 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of C23.30D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 14 | 4 | 0 | 0 |
| 37 | 31 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 5 | 40 |
| 26 | 2 | 0 | 0 |
| 10 | 15 | 0 | 0 |
| 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[14,37,0,0,4,31,0,0,0,0,1,5,0,0,39,40],[26,10,0,0,2,15,0,0,0,0,40,0,0,0,2,1] >;
C23.30D20 in GAP, Magma, Sage, TeX
C_2^3._{30}D_{20} % in TeX
G:=Group("C2^3.30D20"); // GroupNames label
G:=SmallGroup(320,25);
// by ID
G=gap.SmallGroup(320,25);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,85,92,422,387,268,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c*b=b*c,e^2=b,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=a*c*d^19>;
// generators/relations