metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.4D20, C4.D4.D5, (C2×D4).6D10, (C2×C20).14D4, C23.D5⋊4C4, C23.4(C4×D5), (C22×Dic5)⋊2C4, (C22×C10).13D4, C5⋊4(C23.D4), C23⋊Dic5.3C2, C10.33(C23⋊C4), (D4×C10).171C22, C23.18D10.4C2, C22.13(D10⋊C4), C2.13(C23.1D10), (C2×C4).2(C5⋊D4), (C22×C10).4(C2×C4), (C5×C4.D4).1C2, (C2×C10).70(C22⋊C4), SmallGroup(320,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.4D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=ca=ac, ab=ba, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd19 >
Subgroups: 318 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×4], C5, C8, C2×C4, C2×C4 [×4], D4, C23 [×2], C10, C10 [×3], C22⋊C4 [×3], C4⋊C4, M4(2), C22×C4, C2×D4, Dic5 [×3], C20, C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22.D4, C40, C2×Dic5 [×4], C2×C20, C5×D4, C22×C10 [×2], C23.D4, C10.D4, C23.D5, C23.D5 [×2], C5×M4(2), C22×Dic5, D4×C10, C23⋊Dic5, C5×C4.D4, C23.18D10, C23.4D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C23.D4, D10⋊C4, C23.1D10, C23.4D20
►On 80 points
Generators in S80
(1 70)(3 52)(4 24)(5 74)(7 56)(8 28)(9 78)(11 60)(12 32)(13 42)(15 64)(16 36)(17 46)(19 68)(20 40)(21 50)(23 72)(25 54)(27 76)(29 58)(31 80)(33 62)(35 44)(37 66)(39 48)(41 61)(45 65)(49 69)(53 73)(57 77)
(1 50)(2 71)(3 52)(4 73)(5 54)(6 75)(7 56)(8 77)(9 58)(10 79)(11 60)(12 41)(13 62)(14 43)(15 64)(16 45)(17 66)(18 47)(19 68)(20 49)(21 70)(22 51)(23 72)(24 53)(25 74)(26 55)(27 76)(28 57)(29 78)(30 59)(31 80)(32 61)(33 42)(34 63)(35 44)(36 65)(37 46)(38 67)(39 48)(40 69)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(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 64 50 35)(2 63 22 43)(3 33 72 42)(4 12)(5 60 54 31)(6 59 26 79)(7 29 76 78)(9 56 58 27)(10 55 30 75)(11 25 80 74)(13 52 62 23)(14 51 34 71)(15 21 44 70)(16 40)(17 48 66 19)(18 47 38 67)(20 36)(24 32)(37 68 46 39)(41 53)(45 49)(57 77)(61 73)(65 69)
G:=sub<Sym(80)| (1,70)(3,52)(4,24)(5,74)(7,56)(8,28)(9,78)(11,60)(12,32)(13,42)(15,64)(16,36)(17,46)(19,68)(20,40)(21,50)(23,72)(25,54)(27,76)(29,58)(31,80)(33,62)(35,44)(37,66)(39,48)(41,61)(45,65)(49,69)(53,73)(57,77), (1,50)(2,71)(3,52)(4,73)(5,54)(6,75)(7,56)(8,77)(9,58)(10,79)(11,60)(12,41)(13,62)(14,43)(15,64)(16,45)(17,66)(18,47)(19,68)(20,49)(21,70)(22,51)(23,72)(24,53)(25,74)(26,55)(27,76)(28,57)(29,78)(30,59)(31,80)(32,61)(33,42)(34,63)(35,44)(36,65)(37,46)(38,67)(39,48)(40,69), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,64,50,35)(2,63,22,43)(3,33,72,42)(4,12)(5,60,54,31)(6,59,26,79)(7,29,76,78)(9,56,58,27)(10,55,30,75)(11,25,80,74)(13,52,62,23)(14,51,34,71)(15,21,44,70)(16,40)(17,48,66,19)(18,47,38,67)(20,36)(24,32)(37,68,46,39)(41,53)(45,49)(57,77)(61,73)(65,69)>;
G:=Group( (1,70)(3,52)(4,24)(5,74)(7,56)(8,28)(9,78)(11,60)(12,32)(13,42)(15,64)(16,36)(17,46)(19,68)(20,40)(21,50)(23,72)(25,54)(27,76)(29,58)(31,80)(33,62)(35,44)(37,66)(39,48)(41,61)(45,65)(49,69)(53,73)(57,77), (1,50)(2,71)(3,52)(4,73)(5,54)(6,75)(7,56)(8,77)(9,58)(10,79)(11,60)(12,41)(13,62)(14,43)(15,64)(16,45)(17,66)(18,47)(19,68)(20,49)(21,70)(22,51)(23,72)(24,53)(25,74)(26,55)(27,76)(28,57)(29,78)(30,59)(31,80)(32,61)(33,42)(34,63)(35,44)(36,65)(37,46)(38,67)(39,48)(40,69), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,64,50,35)(2,63,22,43)(3,33,72,42)(4,12)(5,60,54,31)(6,59,26,79)(7,29,76,78)(9,56,58,27)(10,55,30,75)(11,25,80,74)(13,52,62,23)(14,51,34,71)(15,21,44,70)(16,40)(17,48,66,19)(18,47,38,67)(20,36)(24,32)(37,68,46,39)(41,53)(45,49)(57,77)(61,73)(65,69) );
G=PermutationGroup([(1,70),(3,52),(4,24),(5,74),(7,56),(8,28),(9,78),(11,60),(12,32),(13,42),(15,64),(16,36),(17,46),(19,68),(20,40),(21,50),(23,72),(25,54),(27,76),(29,58),(31,80),(33,62),(35,44),(37,66),(39,48),(41,61),(45,65),(49,69),(53,73),(57,77)], [(1,50),(2,71),(3,52),(4,73),(5,54),(6,75),(7,56),(8,77),(9,58),(10,79),(11,60),(12,41),(13,62),(14,43),(15,64),(16,45),(17,66),(18,47),(19,68),(20,49),(21,70),(22,51),(23,72),(24,53),(25,74),(26,55),(27,76),(28,57),(29,78),(30,59),(31,80),(32,61),(33,42),(34,63),(35,44),(36,65),(37,46),(38,67),(39,48),(40,69)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,64,50,35),(2,63,22,43),(3,33,72,42),(4,12),(5,60,54,31),(6,59,26,79),(7,29,76,78),(9,56,58,27),(10,55,30,75),(11,25,80,74),(13,52,62,23),(14,51,34,71),(15,21,44,70),(16,40),(17,48,66,19),(18,47,38,67),(20,36),(24,32),(37,68,46,39),(41,53),(45,49),(57,77),(61,73),(65,69)])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 20 | 20 | 40 | 40 | 40 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D10 | C5⋊D4 | C4×D5 | D20 | C23⋊C4 | C23.D4 | C23.1D10 | C23.4D20 |
| kernel | C23.4D20 | C23⋊Dic5 | C5×C4.D4 | C23.18D10 | C23.D5 | C22×Dic5 | C2×C20 | C22×C10 | C4.D4 | C2×D4 | C2×C4 | C23 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 4 | 2 |
Matrix representation of C23.4D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 32 | 0 | 0 |
| 0 | 0 | 1 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 |
| 0 | 0 | 0 | 0 | 40 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 9 | 0 | 0 |
| 0 | 0 | 40 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 |
| 0 | 0 | 0 | 0 | 40 | 16 |
| 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 |
| 19 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 21 | 26 | 20 |
| 0 | 0 | 10 | 35 | 40 | 15 |
| 0 | 0 | 26 | 20 | 35 | 20 |
| 0 | 0 | 40 | 15 | 31 | 6 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 32 |
| 0 | 0 | 0 | 0 | 24 | 25 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,1,0,0,0,0,32,25,0,0,0,0,0,0,25,40,0,0,0,0,9,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,40,0,0,0,0,9,16,0,0,0,0,0,0,25,40,0,0,0,0,9,16],[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],[19,32,0,0,0,0,9,0,0,0,0,0,0,0,6,10,26,40,0,0,21,35,20,15,0,0,26,40,35,31,0,0,20,15,20,6],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,16,24,0,0,0,0,32,25,0,0] >;
C23.4D20 in GAP, Magma, Sage, TeX
C_2^3._4D_{20} % in TeX
G:=Group("C2^3.4D20"); // GroupNames label
G:=SmallGroup(320,34);
// by ID
G=gap.SmallGroup(320,34);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,184,346,297,851,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*a=a*c,a*b=b*a,d*a*d^-1=a*b*c,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*c*d^19>;
// generators/relations