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 [×6], C4 [×7], C22, C22 [×2], C22 [×8], C5, C2×C4, C2×C4 [×11], D4 [×9], Q8, C23 [×2], C23 [×3], D5 [×2], C10, C10 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C22×C4, C2×D4, C2×D4 [×5], C4○D4 [×3], Dic5 [×4], C20 [×3], D10 [×5], C2×C10, C2×C10 [×2], C2×C10 [×3], C23⋊C4, C23⋊C4 [×2], C22.D4 [×3], 2+ 1+4, Dic10, C4×D5 [×2], D20, C2×Dic5 [×2], C2×Dic5 [×5], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×D5, C22×C10 [×2], C23.7D4, C10.D4, C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5, C23.D5, C5×C22⋊C4 [×2], C4○D20, D4×D5 [×2], D4⋊2D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], C2×C5⋊D4, D4×C10, C23.1D10 [×2], C5×C23⋊C4, C22.D20 [×2], C23.18D10, D4⋊6D10, C23.5D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, C23.7D4, C2×D20, D4×D5 [×2], C22⋊D20, C23.5D20
►On 80 points
Generators in S80
(2 65)(3 55)(4 28)(6 69)(7 59)(8 32)(10 73)(11 43)(12 36)(14 77)(15 47)(16 40)(18 61)(19 51)(20 24)(22 50)(23 62)(26 54)(27 66)(30 58)(31 70)(34 42)(35 74)(38 46)(39 78)(44 75)(48 79)(52 63)(56 67)(60 71)
(1 25)(2 65)(3 27)(4 67)(5 29)(6 69)(7 31)(8 71)(9 33)(10 73)(11 35)(12 75)(13 37)(14 77)(15 39)(16 79)(17 21)(18 61)(19 23)(20 63)(22 50)(24 52)(26 54)(28 56)(30 58)(32 60)(34 42)(36 44)(38 46)(40 48)(41 72)(43 74)(45 76)(47 78)(49 80)(51 62)(53 64)(55 66)(57 68)(59 70)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(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 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 52 53 20)(2 19 54 51)(3 50 55 18)(4 17 56 49)(5 48 57 16)(6 15 58 47)(7 46 59 14)(8 13 60 45)(9 44 41 12)(10 11 42 43)(21 28 80 67)(22 66 61 27)(23 26 62 65)(24 64 63 25)(29 40 68 79)(30 78 69 39)(31 38 70 77)(32 76 71 37)(33 36 72 75)(34 74 73 35)
G:=sub<Sym(80)| (2,65)(3,55)(4,28)(6,69)(7,59)(8,32)(10,73)(11,43)(12,36)(14,77)(15,47)(16,40)(18,61)(19,51)(20,24)(22,50)(23,62)(26,54)(27,66)(30,58)(31,70)(34,42)(35,74)(38,46)(39,78)(44,75)(48,79)(52,63)(56,67)(60,71), (1,25)(2,65)(3,27)(4,67)(5,29)(6,69)(7,31)(8,71)(9,33)(10,73)(11,35)(12,75)(13,37)(14,77)(15,39)(16,79)(17,21)(18,61)(19,23)(20,63)(22,50)(24,52)(26,54)(28,56)(30,58)(32,60)(34,42)(36,44)(38,46)(40,48)(41,72)(43,74)(45,76)(47,78)(49,80)(51,62)(53,64)(55,66)(57,68)(59,70), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,80)(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), (1,52,53,20)(2,19,54,51)(3,50,55,18)(4,17,56,49)(5,48,57,16)(6,15,58,47)(7,46,59,14)(8,13,60,45)(9,44,41,12)(10,11,42,43)(21,28,80,67)(22,66,61,27)(23,26,62,65)(24,64,63,25)(29,40,68,79)(30,78,69,39)(31,38,70,77)(32,76,71,37)(33,36,72,75)(34,74,73,35)>;
G:=Group( (2,65)(3,55)(4,28)(6,69)(7,59)(8,32)(10,73)(11,43)(12,36)(14,77)(15,47)(16,40)(18,61)(19,51)(20,24)(22,50)(23,62)(26,54)(27,66)(30,58)(31,70)(34,42)(35,74)(38,46)(39,78)(44,75)(48,79)(52,63)(56,67)(60,71), (1,25)(2,65)(3,27)(4,67)(5,29)(6,69)(7,31)(8,71)(9,33)(10,73)(11,35)(12,75)(13,37)(14,77)(15,39)(16,79)(17,21)(18,61)(19,23)(20,63)(22,50)(24,52)(26,54)(28,56)(30,58)(32,60)(34,42)(36,44)(38,46)(40,48)(41,72)(43,74)(45,76)(47,78)(49,80)(51,62)(53,64)(55,66)(57,68)(59,70), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,80)(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), (1,52,53,20)(2,19,54,51)(3,50,55,18)(4,17,56,49)(5,48,57,16)(6,15,58,47)(7,46,59,14)(8,13,60,45)(9,44,41,12)(10,11,42,43)(21,28,80,67)(22,66,61,27)(23,26,62,65)(24,64,63,25)(29,40,68,79)(30,78,69,39)(31,38,70,77)(32,76,71,37)(33,36,72,75)(34,74,73,35) );
G=PermutationGroup([(2,65),(3,55),(4,28),(6,69),(7,59),(8,32),(10,73),(11,43),(12,36),(14,77),(15,47),(16,40),(18,61),(19,51),(20,24),(22,50),(23,62),(26,54),(27,66),(30,58),(31,70),(34,42),(35,74),(38,46),(39,78),(44,75),(48,79),(52,63),(56,67),(60,71)], [(1,25),(2,65),(3,27),(4,67),(5,29),(6,69),(7,31),(8,71),(9,33),(10,73),(11,35),(12,75),(13,37),(14,77),(15,39),(16,79),(17,21),(18,61),(19,23),(20,63),(22,50),(24,52),(26,54),(28,56),(30,58),(32,60),(34,42),(36,44),(38,46),(40,48),(41,72),(43,74),(45,76),(47,78),(49,80),(51,62),(53,64),(55,66),(57,68),(59,70)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,80),(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)], [(1,52,53,20),(2,19,54,51),(3,50,55,18),(4,17,56,49),(5,48,57,16),(6,15,58,47),(7,46,59,14),(8,13,60,45),(9,44,41,12),(10,11,42,43),(21,28,80,67),(22,66,61,27),(23,26,62,65),(24,64,63,25),(29,40,68,79),(30,78,69,39),(31,38,70,77),(32,76,71,37),(33,36,72,75),(34,74,73,35)])
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