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 [×3], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×4], D4, Q8, C23 [×2], C10, C10 [×3], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C23⋊C4, C23⋊C4, C4.4D4, Dic10, C2×Dic5 [×3], C2×C20, C2×C20, C5×D4, C22×C10 [×2], C42⋊3C4, C4×Dic5, C23.D5 [×3], C5×C22⋊C4, C2×Dic10, D4×C10, C23⋊Dic5, C5×C23⋊C4, C20.17D4, C23.D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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 39)(3 56)(4 77)(5 23)(7 60)(8 61)(9 27)(11 44)(12 65)(13 31)(15 48)(16 69)(17 35)(19 52)(20 73)(21 76)(22 57)(25 80)(26 41)(29 64)(30 45)(33 68)(34 49)(37 72)(38 53)(42 62)(46 66)(50 70)(54 74)(58 78)
(1 39)(2 55)(3 21)(4 57)(5 23)(6 59)(7 25)(8 41)(9 27)(10 43)(11 29)(12 45)(13 31)(14 47)(15 33)(16 49)(17 35)(18 51)(19 37)(20 53)(22 77)(24 79)(26 61)(28 63)(30 65)(32 67)(34 69)(36 71)(38 73)(40 75)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 56)(22 57)(23 58)(24 59)(25 60)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)
(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 48 54 68)(2 47 75 32)(3 66 21 31)(4 12)(5 44 58 64)(6 43 79 28)(7 62 25 27)(9 60 42 80)(10 59 63 24)(11 78 29 23)(13 56 46 76)(14 55 67 40)(15 74 33 39)(16 20)(17 52 50 72)(18 51 71 36)(19 70 37 35)(22 45)(26 41)(30 57)(34 53)(38 49)(65 77)(69 73)
G:=sub<Sym(80)| (1,39)(3,56)(4,77)(5,23)(7,60)(8,61)(9,27)(11,44)(12,65)(13,31)(15,48)(16,69)(17,35)(19,52)(20,73)(21,76)(22,57)(25,80)(26,41)(29,64)(30,45)(33,68)(34,49)(37,72)(38,53)(42,62)(46,66)(50,70)(54,74)(58,78), (1,39)(2,55)(3,21)(4,57)(5,23)(6,59)(7,25)(8,41)(9,27)(10,43)(11,29)(12,45)(13,31)(14,47)(15,33)(16,49)(17,35)(18,51)(19,37)(20,53)(22,77)(24,79)(26,61)(28,63)(30,65)(32,67)(34,69)(36,71)(38,73)(40,75)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (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), (1,48,54,68)(2,47,75,32)(3,66,21,31)(4,12)(5,44,58,64)(6,43,79,28)(7,62,25,27)(9,60,42,80)(10,59,63,24)(11,78,29,23)(13,56,46,76)(14,55,67,40)(15,74,33,39)(16,20)(17,52,50,72)(18,51,71,36)(19,70,37,35)(22,45)(26,41)(30,57)(34,53)(38,49)(65,77)(69,73)>;
G:=Group( (1,39)(3,56)(4,77)(5,23)(7,60)(8,61)(9,27)(11,44)(12,65)(13,31)(15,48)(16,69)(17,35)(19,52)(20,73)(21,76)(22,57)(25,80)(26,41)(29,64)(30,45)(33,68)(34,49)(37,72)(38,53)(42,62)(46,66)(50,70)(54,74)(58,78), (1,39)(2,55)(3,21)(4,57)(5,23)(6,59)(7,25)(8,41)(9,27)(10,43)(11,29)(12,45)(13,31)(14,47)(15,33)(16,49)(17,35)(18,51)(19,37)(20,53)(22,77)(24,79)(26,61)(28,63)(30,65)(32,67)(34,69)(36,71)(38,73)(40,75)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (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), (1,48,54,68)(2,47,75,32)(3,66,21,31)(4,12)(5,44,58,64)(6,43,79,28)(7,62,25,27)(9,60,42,80)(10,59,63,24)(11,78,29,23)(13,56,46,76)(14,55,67,40)(15,74,33,39)(16,20)(17,52,50,72)(18,51,71,36)(19,70,37,35)(22,45)(26,41)(30,57)(34,53)(38,49)(65,77)(69,73) );
G=PermutationGroup([(1,39),(3,56),(4,77),(5,23),(7,60),(8,61),(9,27),(11,44),(12,65),(13,31),(15,48),(16,69),(17,35),(19,52),(20,73),(21,76),(22,57),(25,80),(26,41),(29,64),(30,45),(33,68),(34,49),(37,72),(38,53),(42,62),(46,66),(50,70),(54,74),(58,78)], [(1,39),(2,55),(3,21),(4,57),(5,23),(6,59),(7,25),(8,41),(9,27),(10,43),(11,29),(12,45),(13,31),(14,47),(15,33),(16,49),(17,35),(18,51),(19,37),(20,53),(22,77),(24,79),(26,61),(28,63),(30,65),(32,67),(34,69),(36,71),(38,73),(40,75),(42,62),(44,64),(46,66),(48,68),(50,70),(52,72),(54,74),(56,76),(58,78),(60,80)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,56),(22,57),(23,58),(24,59),(25,60),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55)], [(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)], [(1,48,54,68),(2,47,75,32),(3,66,21,31),(4,12),(5,44,58,64),(6,43,79,28),(7,62,25,27),(9,60,42,80),(10,59,63,24),(11,78,29,23),(13,56,46,76),(14,55,67,40),(15,74,33,39),(16,20),(17,52,50,72),(18,51,71,36),(19,70,37,35),(22,45),(26,41),(30,57),(34,53),(38,49),(65,77),(69,73)])
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