metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊3D20, C24.29D10, C10.42+ 1+4, C20⋊7D4⋊3C2, (C22×C4)⋊9D10, C22⋊D20⋊3C2, C22⋊C4⋊43D10, (C2×D20)⋊3C22, (C22×C10)⋊10D4, C5⋊1(C23⋊3D4), C4⋊Dic5⋊5C22, C10.8(C22×D4), (C2×C10).37C24, (C22×C20)⋊8C22, C22.18(C2×D20), C2.10(C22×D20), (C23×D5)⋊4C22, C2.8(D4⋊6D10), D10⋊C4⋊1C22, (C2×C20).130C23, C22.D20⋊2C2, (C22×D5).9C23, C22.76(C23×D5), (C23×C10).63C22, (C2×Dic5).10C23, (C22×Dic5)⋊7C22, C23.148(C22×D5), (C22×C10).127C23, (C2×C22⋊C4)⋊16D5, (C22×C5⋊D4)⋊6C2, (C10×C22⋊C4)⋊15C2, (C2×C10).173(C2×D4), (C2×C5⋊D4)⋊36C22, (C5×C22⋊C4)⋊48C22, (C2×C4).136(C22×D5), SmallGroup(320,1165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊3D20
G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, dad-1=eae=ac=ca, ebe=bc=cb, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1550 in 346 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22×D4, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23⋊3D4, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C2×D20, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C22⋊D20, C22.D20, C20⋊7D4, C10×C22⋊C4, C22×C5⋊D4, C23⋊3D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, D20, C22×D5, C23⋊3D4, C2×D20, C23×D5, C22×D20, D4⋊6D10, C23⋊3D20
►On 80 points
Generators in S80
(1 48)(2 12)(3 50)(4 14)(5 52)(6 16)(7 54)(8 18)(9 56)(10 20)(11 58)(13 60)(15 42)(17 44)(19 46)(21 77)(22 32)(23 79)(24 34)(25 61)(26 36)(27 63)(28 38)(29 65)(30 40)(31 67)(33 69)(35 71)(37 73)(39 75)(41 51)(43 53)(45 55)(47 57)(49 59)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 21)(18 22)(19 23)(20 24)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)
(1 58)(2 59)(3 60)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 74)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 80)(36 79)(37 78)(38 77)(39 76)(40 75)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)
G:=sub<Sym(80)| (1,48)(2,12)(3,50)(4,14)(5,52)(6,16)(7,54)(8,18)(9,56)(10,20)(11,58)(13,60)(15,42)(17,44)(19,46)(21,77)(22,32)(23,79)(24,34)(25,61)(26,36)(27,63)(28,38)(29,65)(30,40)(31,67)(33,69)(35,71)(37,73)(39,75)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,21)(18,22)(19,23)(20,24)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)>;
G:=Group( (1,48)(2,12)(3,50)(4,14)(5,52)(6,16)(7,54)(8,18)(9,56)(10,20)(11,58)(13,60)(15,42)(17,44)(19,46)(21,77)(22,32)(23,79)(24,34)(25,61)(26,36)(27,63)(28,38)(29,65)(30,40)(31,67)(33,69)(35,71)(37,73)(39,75)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,21)(18,22)(19,23)(20,24)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58) );
G=PermutationGroup([[(1,48),(2,12),(3,50),(4,14),(5,52),(6,16),(7,54),(8,18),(9,56),(10,20),(11,58),(13,60),(15,42),(17,44),(19,46),(21,77),(22,32),(23,79),(24,34),(25,61),(26,36),(27,63),(28,38),(29,65),(30,40),(31,67),(33,69),(35,71),(37,73),(39,75),(41,51),(43,53),(45,55),(47,57),(49,59),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,21),(18,22),(19,23),(20,24),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73)], [(1,58),(2,59),(3,60),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,74),(22,73),(23,72),(24,71),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,80),(36,79),(37,78),(38,77),(39,76),(40,75),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D20 | 2+ 1+4 | D4⋊6D10 |
| kernel | C23⋊3D20 | C22⋊D20 | C22.D20 | C20⋊7D4 | C10×C22⋊C4 | C22×C5⋊D4 | C22×C10 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C10 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 2 | 4 | 2 | 8 | 4 | 2 | 16 | 2 | 8 |
Matrix representation of C23⋊3D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 23 | 35 | 0 | 0 |
| 0 | 0 | 6 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 35 |
| 0 | 0 | 0 | 0 | 6 | 18 |
| 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 | 6 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,0,0,0,0,23,6,0,0,0,0,35,18],[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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,6,40,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
C23⋊3D20 in GAP, Magma, Sage, TeX
C_2^3\rtimes_3D_{20} % in TeX
G:=Group("C2^3:3D20"); // GroupNames label
G:=SmallGroup(320,1165);
// by ID
G=gap.SmallGroup(320,1165);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations