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
Subgroups: 1550 in 346 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×6], C22 [×30], C5, C2×C4 [×4], C2×C4 [×10], D4 [×20], C23, C23 [×6], C23 [×14], D5 [×4], C10, C10 [×2], C10 [×6], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4 [×20], C24, C24 [×2], Dic5 [×4], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], D20 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×4], C2×C20 [×2], C22×D5 [×4], C22×D5 [×8], C22×C10, C22×C10 [×6], C22×C10 [×2], C23⋊3D4, C4⋊Dic5 [×4], D10⋊C4 [×8], C5×C22⋊C4 [×4], C2×D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20 [×2], C23×D5 [×2], C23×C10, C22⋊D20 [×4], C22.D20 [×4], C20⋊7D4 [×4], C10×C22⋊C4, C22×C5⋊D4 [×2], C23⋊3D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], D20 [×4], C22×D5 [×7], C23⋊3D4, C2×D20 [×6], C23×D5, C22×D20, D4⋊6D10 [×2], C23⋊3D20
Generators and relations
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 >
►On 80 points
(1 38)(2 12)(3 40)(4 14)(5 22)(6 16)(7 24)(8 18)(9 26)(10 20)(11 28)(13 30)(15 32)(17 34)(19 36)(21 31)(23 33)(25 35)(27 37)(29 39)(41 69)(42 52)(43 71)(44 54)(45 73)(46 56)(47 75)(48 58)(49 77)(50 60)(51 79)(53 61)(55 63)(57 65)(59 67)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(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 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 62)(42 61)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)
G:=sub<Sym(80)| (1,38)(2,12)(3,40)(4,14)(5,22)(6,16)(7,24)(8,18)(9,26)(10,20)(11,28)(13,30)(15,32)(17,34)(19,36)(21,31)(23,33)(25,35)(27,37)(29,39)(41,69)(42,52)(43,71)(44,54)(45,73)(46,56)(47,75)(48,58)(49,77)(50,60)(51,79)(53,61)(55,63)(57,65)(59,67)(62,72)(64,74)(66,76)(68,78)(70,80), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(40,45), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,62)(42,61)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)>;
G:=Group( (1,38)(2,12)(3,40)(4,14)(5,22)(6,16)(7,24)(8,18)(9,26)(10,20)(11,28)(13,30)(15,32)(17,34)(19,36)(21,31)(23,33)(25,35)(27,37)(29,39)(41,69)(42,52)(43,71)(44,54)(45,73)(46,56)(47,75)(48,58)(49,77)(50,60)(51,79)(53,61)(55,63)(57,65)(59,67)(62,72)(64,74)(66,76)(68,78)(70,80), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(40,45), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,62)(42,61)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63) );
G=PermutationGroup([(1,38),(2,12),(3,40),(4,14),(5,22),(6,16),(7,24),(8,18),(9,26),(10,20),(11,28),(13,30),(15,32),(17,34),(19,36),(21,31),(23,33),(25,35),(27,37),(29,39),(41,69),(42,52),(43,71),(44,54),(45,73),(46,56),(47,75),(48,58),(49,77),(50,60),(51,79),(53,61),(55,63),(57,65),(59,67),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,21),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(41,79),(42,80),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)], [(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),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,62),(42,61),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63)])
Matrix representation ►G ⊆ 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] >;
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 |
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