metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.16C25, C20.51C24, D10.9C24, D20.37C23, 2+ (1+4)⋊5D5, Dic5.11C24, Dic10.38C23, C4○D4⋊11D10, (C2×D4)⋊32D10, (D4×D5)⋊14C22, (C2×C10).7C24, D4⋊6D10⋊10C2, (Q8×D5)⋊16C22, C4.48(C23×D5), C2.17(D5×C24), C5⋊D4.3C23, C4○D20⋊13C22, (D4×C10)⋊26C22, C5⋊2(C2.C25), D4.31(C22×D5), (C5×D4).31C23, (C4×D5).20C23, (C5×Q8).32C23, Q8.32(C22×D5), D4⋊2D5⋊16C22, C22.4(C23×D5), (C2×C20).122C23, Q8⋊2D5⋊19C22, (C5×2+ (1+4))⋊5C2, D4.10D10⋊11C2, C23.71(C22×D5), (C2×Dic10)⋊44C22, (C22×C10).79C23, (C2×Dic5).169C23, (C22×Dic5)⋊39C22, (C22×D5).143C23, (D5×C4○D4)⋊8C2, (C2×C4×D5)⋊37C22, (C2×D4⋊2D5)⋊31C2, (C5×C4○D4)⋊11C22, (C2×C5⋊D4)⋊33C22, (C2×C4).106(C22×D5), SmallGroup(320,1623)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2350 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C4 [×6], C4 [×10], C22 [×9], C22 [×21], C5, C2×C4 [×9], C2×C4 [×51], D4 [×18], D4 [×42], Q8 [×2], Q8 [×18], C23 [×6], C23 [×9], D5 [×6], C10, C10 [×9], C22×C4 [×15], C2×D4 [×9], C2×D4 [×36], C2×Q8 [×15], C4○D4 [×6], C4○D4 [×74], Dic5, Dic5 [×9], C20 [×6], D10 [×6], D10 [×9], C2×C10 [×9], C2×C10 [×6], C2×C4○D4 [×15], 2+ (1+4), 2+ (1+4) [×9], 2- (1+4) [×6], Dic10 [×18], C4×D5 [×24], D20 [×6], C2×Dic5 [×27], C5⋊D4 [×36], C2×C20 [×9], C5×D4 [×18], C5×Q8 [×2], C22×D5 [×9], C22×C10 [×6], C2.C25, C2×Dic10 [×9], C2×C4×D5 [×9], C4○D20 [×18], D4×D5 [×18], D4⋊2D5 [×54], Q8×D5 [×6], Q8⋊2D5 [×2], C22×Dic5 [×6], C2×C5⋊D4 [×18], D4×C10 [×9], C5×C4○D4 [×6], C2×D4⋊2D5 [×9], D4⋊6D10 [×9], D5×C4○D4 [×6], D4.10D10 [×6], C5×2+ (1+4), D20.37C23
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], C25, C22×D5 [×35], C2.C25, C23×D5 [×15], D5×C24, D20.37C23
Generators and relations
G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a10, bab=a-1, ac=ca, ad=da, eae-1=a9, cbc=dbd=a10b, ebe-1=a18b, dcd=ece-1=a10c, ede-1=a10d >
►On 80 points
(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 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 41)(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 77)(2 78)(3 79)(4 80)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 41)
(1 77 11 67)(2 66 12 76)(3 75 13 65)(4 64 14 74)(5 73 15 63)(6 62 16 72)(7 71 17 61)(8 80 18 70)(9 69 19 79)(10 78 20 68)(21 52 31 42)(22 41 32 51)(23 50 33 60)(24 59 34 49)(25 48 35 58)(26 57 36 47)(27 46 37 56)(28 55 38 45)(29 44 39 54)(30 53 40 43)
G:=sub<Sym(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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(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), (1,77)(2,78)(3,79)(4,80)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,41), (1,77,11,67)(2,66,12,76)(3,75,13,65)(4,64,14,74)(5,73,15,63)(6,62,16,72)(7,71,17,61)(8,80,18,70)(9,69,19,79)(10,78,20,68)(21,52,31,42)(22,41,32,51)(23,50,33,60)(24,59,34,49)(25,48,35,58)(26,57,36,47)(27,46,37,56)(28,55,38,45)(29,44,39,54)(30,53,40,43)>;
G:=Group( (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)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(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), (1,77)(2,78)(3,79)(4,80)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,41), (1,77,11,67)(2,66,12,76)(3,75,13,65)(4,64,14,74)(5,73,15,63)(6,62,16,72)(7,71,17,61)(8,80,18,70)(9,69,19,79)(10,78,20,68)(21,52,31,42)(22,41,32,51)(23,50,33,60)(24,59,34,49)(25,48,35,58)(26,57,36,47)(27,46,37,56)(28,55,38,45)(29,44,39,54)(30,53,40,43) );
G=PermutationGroup([(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),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,41),(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)], [(1,77),(2,78),(3,79),(4,80),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,41)], [(1,77,11,67),(2,66,12,76),(3,75,13,65),(4,64,14,74),(5,73,15,63),(6,62,16,72),(7,71,17,61),(8,80,18,70),(9,69,19,79),(10,78,20,68),(21,52,31,42),(22,41,32,51),(23,50,33,60),(24,59,34,49),(25,48,35,58),(26,57,36,47),(27,46,37,56),(28,55,38,45),(29,44,39,54),(30,53,40,43)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 1 | 0 | 0 | 0 | 0 |
| 5 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 1 | 40 | 40 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 39 | 40 | 1 | 1 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 23 | 32 | 9 | 9 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 1 |
| 0 | 0 | 2 | 1 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,2,0,0,0,0,0,1,40,0,0,40,0,40,0,0,0,40,1,40,0],[40,5,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,0,0,0,40,0,40,40,0,0,0,0,1,0,0,0,1,40,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,23,0,0,0,32,0,32,32,0,0,9,0,9,0,0,0,0,9,9,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,2,0,0,0,0,0,1,0,0,0,0,0,40,0,40,0,0,0,40,40,0],[6,6,0,0,0,0,1,35,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,40,0,40,0,0,1,40,1,0] >;
68 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2J | 2K | ··· | 2P | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4Q | 5A | 5B | 10A | 10B | 10C | ··· | 10T | 20A | ··· | 20L |
| order | 1 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | C2.C25 | D20.37C23 |
| kernel | D20.37C23 | C2×D4⋊2D5 | D4⋊6D10 | D5×C4○D4 | D4.10D10 | C5×2+ (1+4) | 2+ (1+4) | C2×D4 | C4○D4 | C5 | C1 |
| # reps | 1 | 9 | 9 | 6 | 6 | 1 | 2 | 18 | 12 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{20}._{37}C_2^3 % in TeX
G:=Group("D20.37C2^3"); // GroupNames label
G:=SmallGroup(320,1623);
// by ID
G=gap.SmallGroup(320,1623);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,1684,438,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=1,e^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^9,c*b*c=d*b*d=a^10*b,e*b*e^-1=a^18*b,d*c*d=e*c*e^-1=a^10*c,e*d*e^-1=a^10*d>;
// generators/relations