metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.8D20, C24.1D10, C23.1Dic10, (C22×C20)⋊1C4, C23.8(C4×D5), C23.D5⋊11C4, (C22×C4)⋊1Dic5, (C22×C10).6Q8, (C2×C10).39C42, (C22×C10).41D4, C5⋊3(C23.9D4), C22.8(C4×Dic5), C10.38(C23⋊C4), C23.46(C5⋊D4), C22.8(C4⋊Dic5), C2.2(C23⋊Dic5), C23.21(C2×Dic5), (C23×C10).22C22, C22.1(C10.D4), C22.24(C23.D5), C22.16(D10⋊C4), C10.22(C2.C42), C2.4(C10.10C42), (C2×C22⋊C4).2D5, (C2×C10).29(C4⋊C4), (C10×C22⋊C4).1C2, (C2×C23.D5).1C2, (C22×C10).96(C2×C4), (C2×C10).73(C22⋊C4), SmallGroup(320,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.8D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >
Subgroups: 518 in 142 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×12], C23 [×3], C23 [×4], C23 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×8], C22×C4 [×2], C22×C4 [×2], C24, Dic5 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×Dic5 [×8], C2×C20 [×4], C22×C10 [×3], C22×C10 [×4], C22×C10 [×2], C23.9D4, C23.D5 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×2], C22×C20 [×2], C23×C10, C2×C23.D5 [×2], C10×C22⋊C4, C23.8D20
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, C23⋊C4 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C23.9D4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C10.10C42, C23⋊Dic5 [×2], C23.8D20
►On 80 points
(2 68)(4 70)(6 72)(8 74)(10 76)(12 78)(14 80)(16 62)(18 64)(20 66)(22 45)(24 47)(26 49)(28 51)(30 53)(32 55)(34 57)(36 59)(38 41)(40 43)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 79)(22 80)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 41)(39 42)(40 43)
(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 41 29 10)(2 9 53 37)(3 59 31 8)(4 7 55 35)(5 57 33 6)(11 51 39 20)(12 19 43 27)(13 49 21 18)(14 17 45 25)(15 47 23 16)(22 48 80 63)(24 46 62 61)(26 44 64 79)(28 42 66 77)(30 60 68 75)(32 58 70 73)(34 56 72 71)(36 54 74 69)(38 52 76 67)(40 50 78 65)
G:=sub<Sym(80)| (2,68)(4,70)(6,72)(8,74)(10,76)(12,78)(14,80)(16,62)(18,64)(20,66)(22,45)(24,47)(26,49)(28,51)(30,53)(32,55)(34,57)(36,59)(38,41)(40,43), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,79)(22,80)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43), (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), (1,41,29,10)(2,9,53,37)(3,59,31,8)(4,7,55,35)(5,57,33,6)(11,51,39,20)(12,19,43,27)(13,49,21,18)(14,17,45,25)(15,47,23,16)(22,48,80,63)(24,46,62,61)(26,44,64,79)(28,42,66,77)(30,60,68,75)(32,58,70,73)(34,56,72,71)(36,54,74,69)(38,52,76,67)(40,50,78,65)>;
G:=Group( (2,68)(4,70)(6,72)(8,74)(10,76)(12,78)(14,80)(16,62)(18,64)(20,66)(22,45)(24,47)(26,49)(28,51)(30,53)(32,55)(34,57)(36,59)(38,41)(40,43), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,79)(22,80)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43), (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), (1,41,29,10)(2,9,53,37)(3,59,31,8)(4,7,55,35)(5,57,33,6)(11,51,39,20)(12,19,43,27)(13,49,21,18)(14,17,45,25)(15,47,23,16)(22,48,80,63)(24,46,62,61)(26,44,64,79)(28,42,66,77)(30,60,68,75)(32,58,70,73)(34,56,72,71)(36,54,74,69)(38,52,76,67)(40,50,78,65) );
G=PermutationGroup([(2,68),(4,70),(6,72),(8,74),(10,76),(12,78),(14,80),(16,62),(18,64),(20,66),(22,45),(24,47),(26,49),(28,51),(30,53),(32,55),(34,57),(36,59),(38,41),(40,43)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,79),(22,80),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,41),(39,42),(40,43)], [(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)], [(1,41,29,10),(2,9,53,37),(3,59,31,8),(4,7,55,35),(5,57,33,6),(11,51,39,20),(12,19,43,27),(13,49,21,18),(14,17,45,25),(15,47,23,16),(22,48,80,63),(24,46,62,61),(26,44,64,79),(28,42,66,77),(30,60,68,75),(32,58,70,73),(34,56,72,71),(36,54,74,69),(38,52,76,67),(40,50,78,65)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | Dic5 | D10 | Dic10 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | C23⋊Dic5 |
| kernel | C23.8D20 | C2×C23.D5 | C10×C22⋊C4 | C23.D5 | C22×C20 | C22×C10 | C22×C10 | C2×C22⋊C4 | C22×C4 | C24 | C23 | C23 | C23 | C23 | C10 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 2 | 4 | 2 | 4 | 8 | 4 | 8 | 2 | 8 |
Matrix representation of C23.8D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 24 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,34,40,0,0,0,0,1,0,0,0],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,0,40,17,0,0,0,0,24,1,0,0,0,1,0,0,0,0,1,0,0,0] >;
C23.8D20 in GAP, Magma, Sage, TeX
C_2^3._8D_{20} % in TeX
G:=Group("C2^3.8D20"); // GroupNames label
G:=SmallGroup(320,84);
// by ID
G=gap.SmallGroup(320,84);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=a*b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;
// generators/relations