metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.5D20, C23⋊C4⋊4D5, (C2×C20).7D4, (C2×C4).5D20, C22⋊C4⋊2D10, (C2×D4).12D10, (C2×Dic5).1D4, (C22×D5).1D4, C22.25(D4×D5), C22.9(C2×D20), C10.14C22≀C2, D4⋊6D10.1C2, (D4×C10).9C22, (C22×C10).18D4, C5⋊1(C23.7D4), C23.D5⋊2C22, C23.3(C22×D5), C23.1D10⋊2C2, C22.D20⋊1C2, (C22×C10).3C23, C2.17(C22⋊D20), C23.18D10⋊1C2, (C22×Dic5)⋊1C22, (C5×C23⋊C4)⋊5C2, (C2×C10).18(C2×D4), (C5×C22⋊C4)⋊2C22, (C2×C5⋊D4).3C22, SmallGroup(320,369)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.5D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=c, ab=ba, ac=ca, dad-1=eae-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 766 in 160 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C23⋊C4, C23⋊C4, C22.D4, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C23.7D4, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C4○D20, D4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23.1D10, C5×C23⋊C4, C22.D20, C23.18D10, D4⋊6D10, C23.5D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, C23.7D4, C2×D20, D4×D5, C22⋊D20, C23.5D20
►On 80 points
Generators in S80
(2 33)(3 73)(4 57)(6 37)(7 77)(8 41)(10 21)(11 61)(12 45)(14 25)(15 65)(16 49)(18 29)(19 69)(20 53)(22 44)(23 62)(26 48)(27 66)(30 52)(31 70)(34 56)(35 74)(38 60)(39 78)(43 80)(47 64)(51 68)(55 72)(59 76)
(1 54)(2 33)(3 56)(4 35)(5 58)(6 37)(7 60)(8 39)(9 42)(10 21)(11 44)(12 23)(13 46)(14 25)(15 48)(16 27)(17 50)(18 29)(19 52)(20 31)(22 61)(24 63)(26 65)(28 67)(30 69)(32 71)(34 73)(36 75)(38 77)(40 79)(41 78)(43 80)(45 62)(47 64)(49 66)(51 68)(53 70)(55 72)(57 74)(59 76)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)
(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 71 20)(2 19 72 69)(3 68 73 18)(4 17 74 67)(5 66 75 16)(6 15 76 65)(7 64 77 14)(8 13 78 63)(9 62 79 12)(10 11 80 61)(21 44 43 22)(23 42 45 40)(24 39 46 41)(25 60 47 38)(26 37 48 59)(27 58 49 36)(28 35 50 57)(29 56 51 34)(30 33 52 55)(31 54 53 32)
G:=sub<Sym(80)| (2,33)(3,73)(4,57)(6,37)(7,77)(8,41)(10,21)(11,61)(12,45)(14,25)(15,65)(16,49)(18,29)(19,69)(20,53)(22,44)(23,62)(26,48)(27,66)(30,52)(31,70)(34,56)(35,74)(38,60)(39,78)(43,80)(47,64)(51,68)(55,72)(59,76), (1,54)(2,33)(3,56)(4,35)(5,58)(6,37)(7,60)(8,39)(9,42)(10,21)(11,44)(12,23)(13,46)(14,25)(15,48)(16,27)(17,50)(18,29)(19,52)(20,31)(22,61)(24,63)(26,65)(28,67)(30,69)(32,71)(34,73)(36,75)(38,77)(40,79)(41,78)(43,80)(45,62)(47,64)(49,66)(51,68)(53,70)(55,72)(57,74)(59,76), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42), (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), (1,70,71,20)(2,19,72,69)(3,68,73,18)(4,17,74,67)(5,66,75,16)(6,15,76,65)(7,64,77,14)(8,13,78,63)(9,62,79,12)(10,11,80,61)(21,44,43,22)(23,42,45,40)(24,39,46,41)(25,60,47,38)(26,37,48,59)(27,58,49,36)(28,35,50,57)(29,56,51,34)(30,33,52,55)(31,54,53,32)>;
G:=Group( (2,33)(3,73)(4,57)(6,37)(7,77)(8,41)(10,21)(11,61)(12,45)(14,25)(15,65)(16,49)(18,29)(19,69)(20,53)(22,44)(23,62)(26,48)(27,66)(30,52)(31,70)(34,56)(35,74)(38,60)(39,78)(43,80)(47,64)(51,68)(55,72)(59,76), (1,54)(2,33)(3,56)(4,35)(5,58)(6,37)(7,60)(8,39)(9,42)(10,21)(11,44)(12,23)(13,46)(14,25)(15,48)(16,27)(17,50)(18,29)(19,52)(20,31)(22,61)(24,63)(26,65)(28,67)(30,69)(32,71)(34,73)(36,75)(38,77)(40,79)(41,78)(43,80)(45,62)(47,64)(49,66)(51,68)(53,70)(55,72)(57,74)(59,76), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42), (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), (1,70,71,20)(2,19,72,69)(3,68,73,18)(4,17,74,67)(5,66,75,16)(6,15,76,65)(7,64,77,14)(8,13,78,63)(9,62,79,12)(10,11,80,61)(21,44,43,22)(23,42,45,40)(24,39,46,41)(25,60,47,38)(26,37,48,59)(27,58,49,36)(28,35,50,57)(29,56,51,34)(30,33,52,55)(31,54,53,32) );
G=PermutationGroup([[(2,33),(3,73),(4,57),(6,37),(7,77),(8,41),(10,21),(11,61),(12,45),(14,25),(15,65),(16,49),(18,29),(19,69),(20,53),(22,44),(23,62),(26,48),(27,66),(30,52),(31,70),(34,56),(35,74),(38,60),(39,78),(43,80),(47,64),(51,68),(55,72),(59,76)], [(1,54),(2,33),(3,56),(4,35),(5,58),(6,37),(7,60),(8,39),(9,42),(10,21),(11,44),(12,23),(13,46),(14,25),(15,48),(16,27),(17,50),(18,29),(19,52),(20,31),(22,61),(24,63),(26,65),(28,67),(30,69),(32,71),(34,73),(36,75),(38,77),(40,79),(41,78),(43,80),(45,62),(47,64),(49,66),(51,68),(53,70),(55,72),(57,74),(59,76)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,41),(40,42)], [(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)], [(1,70,71,20),(2,19,72,69),(3,68,73,18),(4,17,74,67),(5,66,75,16),(6,15,76,65),(7,64,77,14),(8,13,78,63),(9,62,79,12),(10,11,80,61),(21,44,43,22),(23,42,45,40),(24,39,46,41),(25,60,47,38),(26,37,48,59),(27,58,49,36),(28,35,50,57),(29,56,51,34),(30,33,52,55),(31,54,53,32)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | 10B | 10C | ··· | 10H | 10I | 10J | 20A | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 20 | 20 | 4 | 8 | 8 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D20 | D20 | C23.7D4 | D4×D5 | C23.5D20 |
| kernel | C23.5D20 | C23.1D10 | C5×C23⋊C4 | C22.D20 | C23.18D10 | D4⋊6D10 | C2×Dic5 | C2×C20 | C22×D5 | C22×C10 | C23⋊C4 | C22⋊C4 | C2×D4 | C2×C4 | C23 | C5 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 2 |
Matrix representation of C23.5D20 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 10 | 40 | 0 | 0 | 0 | 0 | 0 |
| 11 | 3 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 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 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 | 1 | 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 | 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 |
| 9 | 23 | 28 | 28 | 0 | 0 | 0 | 0 |
| 18 | 12 | 22 | 28 | 0 | 0 | 0 | 0 |
| 16 | 8 | 29 | 18 | 0 | 0 | 0 | 0 |
| 22 | 16 | 23 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 | 9 | 9 |
| 0 | 0 | 0 | 0 | 32 | 0 | 9 | 9 |
| 0 | 0 | 0 | 0 | 0 | 32 | 0 | 9 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 9 |
| 9 | 23 | 28 | 28 | 0 | 0 | 0 | 0 |
| 1 | 32 | 28 | 22 | 0 | 0 | 0 | 0 |
| 13 | 17 | 18 | 29 | 0 | 0 | 0 | 0 |
| 17 | 24 | 32 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 | 9 | 9 |
| 0 | 0 | 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 | 0 | 9 |
| 0 | 0 | 0 | 0 | 0 | 32 | 9 | 0 |
G:=sub<GL(8,GF(41))| [1,0,3,11,0,0,0,0,0,1,10,3,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,39,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[9,18,16,22,0,0,0,0,23,12,8,16,0,0,0,0,28,22,29,23,0,0,0,0,28,28,18,32,0,0,0,0,0,0,0,0,32,32,0,32,0,0,0,0,32,0,32,0,0,0,0,0,9,9,0,0,0,0,0,0,9,9,9,9],[9,1,13,17,0,0,0,0,23,32,17,24,0,0,0,0,28,28,18,32,0,0,0,0,28,22,29,23,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,32,32,32,0,0,0,0,9,0,0,9,0,0,0,0,9,0,9,0] >;
C23.5D20 in GAP, Magma, Sage, TeX
C_2^3._5D_{20} % in TeX
G:=Group("C2^3.5D20"); // GroupNames label
G:=SmallGroup(320,369);
// by ID
G=gap.SmallGroup(320,369);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,226,570,1684,438,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations