metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.1D20, (C4×Dic5)⋊1C4, (C2×D4).3D10, C23⋊C4.2D5, C5⋊3(C42⋊3C4), (C2×Dic10)⋊4C4, (D4×C10).3C22, (C22×C10).10D4, C23.3(C5⋊D4), C23⋊Dic5.1C2, C10.30(C23⋊C4), C20.17D4.1C2, C22.10(D10⋊C4), C2.10(C23.1D10), (C2×C4).1(C4×D5), (C2×C20).1(C2×C4), (C5×C23⋊C4).2C2, (C2×C10).67(C22⋊C4), SmallGroup(320,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=ca=ac, dad-1=ab=ba, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad-1 >
Subgroups: 334 in 70 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C23⋊C4, C4.4D4, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C42⋊3C4, C4×Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, D4×C10, C23⋊Dic5, C5×C23⋊C4, C20.17D4, C23.D20
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C42⋊3C4, D10⋊C4, C23.1D10, C23.D20
►On 80 points
Generators in S80
(1 23)(3 64)(4 42)(5 27)(7 68)(8 46)(9 31)(11 72)(12 50)(13 35)(15 76)(16 54)(17 39)(19 80)(20 58)(21 57)(22 61)(25 41)(26 65)(29 45)(30 69)(33 49)(34 73)(37 53)(38 77)(43 66)(47 70)(51 74)(55 78)(59 62)
(1 23)(2 63)(3 25)(4 65)(5 27)(6 67)(7 29)(8 69)(9 31)(10 71)(11 33)(12 73)(13 35)(14 75)(15 37)(16 77)(17 39)(18 79)(19 21)(20 61)(22 58)(24 60)(26 42)(28 44)(30 46)(32 48)(34 50)(36 52)(38 54)(40 56)(41 64)(43 66)(45 68)(47 70)(49 72)(51 74)(53 76)(55 78)(57 80)(59 62)
(1 59)(2 60)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 80)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)
(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 76 62 53)(2 75 60 36)(3 51 25 35)(4 12)(5 72 66 49)(6 71 44 32)(7 47 29 31)(9 68 70 45)(10 67 48 28)(11 43 33 27)(13 64 74 41)(14 63 52 24)(15 59 37 23)(16 20)(17 80 78 57)(18 79 56 40)(19 55 21 39)(22 77)(26 73)(30 69)(34 65)(38 61)(42 50)(54 58)
G:=sub<Sym(80)| (1,23)(3,64)(4,42)(5,27)(7,68)(8,46)(9,31)(11,72)(12,50)(13,35)(15,76)(16,54)(17,39)(19,80)(20,58)(21,57)(22,61)(25,41)(26,65)(29,45)(30,69)(33,49)(34,73)(37,53)(38,77)(43,66)(47,70)(51,74)(55,78)(59,62), (1,23)(2,63)(3,25)(4,65)(5,27)(6,67)(7,29)(8,69)(9,31)(10,71)(11,33)(12,73)(13,35)(14,75)(15,37)(16,77)(17,39)(18,79)(19,21)(20,61)(22,58)(24,60)(26,42)(28,44)(30,46)(32,48)(34,50)(36,52)(38,54)(40,56)(41,64)(43,66)(45,68)(47,70)(49,72)(51,74)(53,76)(55,78)(57,80)(59,62), (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,80)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79), (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), (1,76,62,53)(2,75,60,36)(3,51,25,35)(4,12)(5,72,66,49)(6,71,44,32)(7,47,29,31)(9,68,70,45)(10,67,48,28)(11,43,33,27)(13,64,74,41)(14,63,52,24)(15,59,37,23)(16,20)(17,80,78,57)(18,79,56,40)(19,55,21,39)(22,77)(26,73)(30,69)(34,65)(38,61)(42,50)(54,58)>;
G:=Group( (1,23)(3,64)(4,42)(5,27)(7,68)(8,46)(9,31)(11,72)(12,50)(13,35)(15,76)(16,54)(17,39)(19,80)(20,58)(21,57)(22,61)(25,41)(26,65)(29,45)(30,69)(33,49)(34,73)(37,53)(38,77)(43,66)(47,70)(51,74)(55,78)(59,62), (1,23)(2,63)(3,25)(4,65)(5,27)(6,67)(7,29)(8,69)(9,31)(10,71)(11,33)(12,73)(13,35)(14,75)(15,37)(16,77)(17,39)(18,79)(19,21)(20,61)(22,58)(24,60)(26,42)(28,44)(30,46)(32,48)(34,50)(36,52)(38,54)(40,56)(41,64)(43,66)(45,68)(47,70)(49,72)(51,74)(53,76)(55,78)(57,80)(59,62), (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,80)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79), (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), (1,76,62,53)(2,75,60,36)(3,51,25,35)(4,12)(5,72,66,49)(6,71,44,32)(7,47,29,31)(9,68,70,45)(10,67,48,28)(11,43,33,27)(13,64,74,41)(14,63,52,24)(15,59,37,23)(16,20)(17,80,78,57)(18,79,56,40)(19,55,21,39)(22,77)(26,73)(30,69)(34,65)(38,61)(42,50)(54,58) );
G=PermutationGroup([[(1,23),(3,64),(4,42),(5,27),(7,68),(8,46),(9,31),(11,72),(12,50),(13,35),(15,76),(16,54),(17,39),(19,80),(20,58),(21,57),(22,61),(25,41),(26,65),(29,45),(30,69),(33,49),(34,73),(37,53),(38,77),(43,66),(47,70),(51,74),(55,78),(59,62)], [(1,23),(2,63),(3,25),(4,65),(5,27),(6,67),(7,29),(8,69),(9,31),(10,71),(11,33),(12,73),(13,35),(14,75),(15,37),(16,77),(17,39),(18,79),(19,21),(20,61),(22,58),(24,60),(26,42),(28,44),(30,46),(32,48),(34,50),(36,52),(38,54),(40,56),(41,64),(43,66),(45,68),(47,70),(49,72),(51,74),(53,76),(55,78),(57,80),(59,62)], [(1,59),(2,60),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,80),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79)], [(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)], [(1,76,62,53),(2,75,60,36),(3,51,25,35),(4,12),(5,72,66,49),(6,71,44,32),(7,47,29,31),(9,68,70,45),(10,67,48,28),(11,43,33,27),(13,64,74,41),(14,63,52,24),(15,59,37,23),(16,20),(17,80,78,57),(18,79,56,40),(19,55,21,39),(22,77),(26,73),(30,69),(34,65),(38,61),(42,50),(54,58)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | 10B | 10C | ··· | 10H | 10I | 10J | 20A | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 20 | 20 | 40 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | C42⋊3C4 | C23.1D10 | C23.D20 |
| kernel | C23.D20 | C23⋊Dic5 | C5×C23⋊C4 | C20.17D4 | C4×Dic5 | C2×Dic10 | C22×C10 | C23⋊C4 | C2×D4 | C2×C4 | C23 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 4 | 2 |
Matrix representation of C23.D20 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 24 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 33 | 13 | 36 |
| 0 | 0 | 0 | 0 | 27 | 12 | 9 | 28 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 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 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 37 | 26 | 4 | 9 | 0 | 0 | 0 | 0 |
| 0 | 33 | 28 | 13 | 0 | 0 | 0 | 0 |
| 29 | 30 | 17 | 28 | 0 | 0 | 0 | 0 |
| 13 | 8 | 9 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 3 | 9 |
| 0 | 0 | 0 | 0 | 40 | 32 | 27 | 40 |
| 0 | 0 | 0 | 0 | 37 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 15 | 39 | 27 | 9 |
| 24 | 38 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 18 | 40 | 0 | 0 | 0 | 0 | 0 |
| 3 | 31 | 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 | 3 | 2 |
| 0 | 0 | 0 | 0 | 9 | 40 | 3 | 9 |
| 0 | 0 | 0 | 0 | 0 | 37 | 13 | 36 |
| 0 | 0 | 0 | 0 | 0 | 38 | 34 | 28 |
G:=sub<GL(8,GF(41))| [40,0,12,12,0,0,0,0,0,40,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,36,27,0,0,0,0,18,40,33,12,0,0,0,0,0,0,13,9,0,0,0,0,0,0,36,28],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,40,0,0,0,0,0,0,0,0,28,32,0,0,0,0,0,0,5,13],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[37,0,29,13,0,0,0,0,26,33,30,8,0,0,0,0,4,28,17,9,0,0,0,0,9,13,28,36,0,0,0,0,0,0,0,0,9,40,37,15,0,0,0,0,0,32,0,39,0,0,0,0,3,27,32,27,0,0,0,0,9,40,0,9],[24,1,14,3,0,0,0,0,38,17,18,31,0,0,0,0,0,0,40,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,18,40,37,38,0,0,0,0,3,3,13,34,0,0,0,0,2,9,36,28] >;
C23.D20 in GAP, Magma, Sage, TeX
C_2^3.D_{20} % in TeX
G:=Group("C2^3.D20"); // GroupNames label
G:=SmallGroup(320,31);
// by ID
G=gap.SmallGroup(320,31);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,1123,794,297,136,851,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=c*a=a*c,d*a*d^-1=a*b=b*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^-1>;
// generators/relations