metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.20D20, M4(2)⋊24D10, C4○D20⋊14C4, (C2×D20)⋊28C4, (C2×C4).55D20, D20.41(C2×C4), C20.446(C2×D4), (C2×C20).178D4, D20⋊7C4⋊13C2, (C2×Dic10)⋊27C4, (C2×M4(2))⋊15D5, (C4×Dic5)⋊4C22, C22.17(C2×D20), C20.76(C22⋊C4), (C10×M4(2))⋊23C2, C20.129(C22×C4), (C2×C20).420C23, Dic10.43(C2×C4), C5⋊6(C42⋊C22), C4○D20.43C22, (C22×C4).146D10, (C22×C10).107D4, C4.30(D10⋊C4), (C5×M4(2))⋊36C22, C23.21D10⋊17C2, (C22×C20).193C22, C22.29(D10⋊C4), C4.55(C2×C4×D5), (C2×C4).56(C4×D5), (C2×C10).33(C2×D4), C4.137(C2×C5⋊D4), (C2×C20).286(C2×C4), (C2×C4○D20).15C2, C2.35(C2×D10⋊C4), (C2×C4).258(C5⋊D4), C10.104(C2×C22⋊C4), (C2×C4).513(C22×D5), (C2×C10).87(C22⋊C4), SmallGroup(320,766)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.20D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=b, ab=ba, dad-1=eae-1=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd19 >
Subgroups: 622 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C42⋊C22, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, D20⋊7C4, C23.21D10, C10×M4(2), C2×C4○D20, C23.20D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C42⋊C22, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, C23.20D20
►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 51)(2 72)(3 53)(4 74)(5 55)(6 76)(7 57)(8 78)(9 59)(10 80)(11 61)(12 42)(13 63)(14 44)(15 65)(16 46)(17 67)(18 48)(19 69)(20 50)(21 71)(22 52)(23 73)(24 54)(25 75)(26 56)(27 77)(28 58)(29 79)(30 60)(31 41)(32 62)(33 43)(34 64)(35 45)(36 66)(37 47)(38 68)(39 49)(40 70)
(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 70 51 40)(2 19 72 69)(3 48 53 18)(4 37 74 47)(5 66 55 36)(6 15 76 65)(7 44 57 14)(8 33 78 43)(9 62 59 32)(10 11 80 61)(12 29 42 79)(13 58 63 28)(16 25 46 75)(17 54 67 24)(20 21 50 71)(22 39 52 49)(23 68 73 38)(26 35 56 45)(27 64 77 34)(30 31 60 41)
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), (1,51)(2,72)(3,53)(4,74)(5,55)(6,76)(7,57)(8,78)(9,59)(10,80)(11,61)(12,42)(13,63)(14,44)(15,65)(16,46)(17,67)(18,48)(19,69)(20,50)(21,71)(22,52)(23,73)(24,54)(25,75)(26,56)(27,77)(28,58)(29,79)(30,60)(31,41)(32,62)(33,43)(34,64)(35,45)(36,66)(37,47)(38,68)(39,49)(40,70), (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), (1,70,51,40)(2,19,72,69)(3,48,53,18)(4,37,74,47)(5,66,55,36)(6,15,76,65)(7,44,57,14)(8,33,78,43)(9,62,59,32)(10,11,80,61)(12,29,42,79)(13,58,63,28)(16,25,46,75)(17,54,67,24)(20,21,50,71)(22,39,52,49)(23,68,73,38)(26,35,56,45)(27,64,77,34)(30,31,60,41)>;
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), (1,51)(2,72)(3,53)(4,74)(5,55)(6,76)(7,57)(8,78)(9,59)(10,80)(11,61)(12,42)(13,63)(14,44)(15,65)(16,46)(17,67)(18,48)(19,69)(20,50)(21,71)(22,52)(23,73)(24,54)(25,75)(26,56)(27,77)(28,58)(29,79)(30,60)(31,41)(32,62)(33,43)(34,64)(35,45)(36,66)(37,47)(38,68)(39,49)(40,70), (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), (1,70,51,40)(2,19,72,69)(3,48,53,18)(4,37,74,47)(5,66,55,36)(6,15,76,65)(7,44,57,14)(8,33,78,43)(9,62,59,32)(10,11,80,61)(12,29,42,79)(13,58,63,28)(16,25,46,75)(17,54,67,24)(20,21,50,71)(22,39,52,49)(23,68,73,38)(26,35,56,45)(27,64,77,34)(30,31,60,41) );
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)], [(1,51),(2,72),(3,53),(4,74),(5,55),(6,76),(7,57),(8,78),(9,59),(10,80),(11,61),(12,42),(13,63),(14,44),(15,65),(16,46),(17,67),(18,48),(19,69),(20,50),(21,71),(22,52),(23,73),(24,54),(25,75),(26,56),(27,77),(28,58),(29,79),(30,60),(31,41),(32,62),(33,43),(34,64),(35,45),(36,66),(37,47),(38,68),(39,49),(40,70)], [(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)], [(1,70,51,40),(2,19,72,69),(3,48,53,18),(4,37,74,47),(5,66,55,36),(6,15,76,65),(7,44,57,14),(8,33,78,43),(9,62,59,32),(10,11,80,61),(12,29,42,79),(13,58,63,28),(16,25,46,75),(17,54,67,24),(20,21,50,71),(22,39,52,49),(23,68,73,38),(26,35,56,45),(27,64,77,34),(30,31,60,41)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 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 | 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 | 2 | 2 | 2 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D20 | C42⋊C22 | C23.20D20 |
| kernel | C23.20D20 | D20⋊7C4 | C23.21D10 | C10×M4(2) | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C20 | C22×C10 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 4 | 2 | 8 |
Matrix representation of C23.20D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 14 |
| 0 | 1 | 27 | 27 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 24 | 1 | 27 | 35 |
| 40 | 17 | 8 | 35 |
| 0 | 0 | 23 | 5 |
| 0 | 0 | 1 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 12 | 12 |
| 40 | 34 | 39 | 10 |
| 35 | 5 | 6 | 5 |
| 0 | 35 | 1 | 1 |
| 11 | 9 | 19 | 36 |
| 14 | 30 | 17 | 2 |
| 27 | 29 | 2 | 27 |
| 16 | 28 | 9 | 39 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,27,40,0,14,27,0,40],[24,40,0,0,1,17,0,0,27,8,23,1,35,35,5,18],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,40,35,0,1,34,5,35,12,39,6,1,12,10,5,1],[11,14,27,16,9,30,29,28,19,17,2,9,36,2,27,39] >;
C23.20D20 in GAP, Magma, Sage, TeX
C_2^3._{20}D_{20} % in TeX
G:=Group("C2^3.20D20"); // GroupNames label
G:=SmallGroup(320,766);
// by ID
G=gap.SmallGroup(320,766);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,136,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^19>;
// generators/relations