metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.70C24, C40.47C23, M4(2)⋊26D10, (C2×C8)⋊22D10, C4○D20.9C4, (C2×C40)⋊24C22, D20.43(C2×C4), (C2×D20).29C4, (C8×D5)⋊11C22, C5⋊4(Q8○M4(2)), C4.69(C23×D5), C8.44(C22×D5), C23.22(C4×D5), C8⋊D5⋊19C22, (C2×M4(2))⋊16D5, (D5×M4(2))⋊10C2, C10.54(C23×C4), C5⋊2C8.32C23, (C4×D5).72C23, D20.3C4⋊15C2, D20.2C4⋊12C2, (C10×M4(2))⋊10C2, (C2×C20).510C23, C20.152(C22×C4), Dic10.45(C2×C4), (C2×Dic10).30C4, C4○D20.59C22, D10.23(C22×C4), (C22×C4).264D10, C4.Dic5⋊40C22, (C5×M4(2))⋊26C22, Dic5.22(C22×C4), (C22×C20).265C22, C4.95(C2×C4×D5), (C2×C4).58(C4×D5), C5⋊D4.8(C2×C4), C22.28(C2×C4×D5), C2.34(D5×C22×C4), (C4×D5).10(C2×C4), (C2×C5⋊D4).24C4, (C2×C20).305(C2×C4), (C2×C5⋊2C8)⋊12C22, (C2×C4○D20).22C2, (C2×C4.Dic5)⋊25C2, (C2×C4×D5).162C22, (C2×Dic5).39(C2×C4), (C22×D5).32(C2×C4), (C2×C4).605(C22×D5), (C2×C10).127(C22×C4), (C22×C10).147(C2×C4), SmallGroup(320,1417)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 718 in 258 conjugacy classes, 147 normal (41 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×4], C22 [×3], C22 [×7], C5, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×3], C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×4], D10 [×4], D10 [×2], C2×C10 [×3], C2×C10, C2×M4(2), C2×M4(2) [×5], C8○D4 [×8], C2×C4○D4, C5⋊2C8 [×4], C40 [×4], Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×6], C22×D5 [×2], C22×C10, Q8○M4(2), C8×D5 [×8], C8⋊D5 [×8], C2×C5⋊2C8 [×2], C4.Dic5 [×4], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, D20.3C4 [×4], D5×M4(2) [×4], D20.2C4 [×4], C2×C4.Dic5, C10×M4(2), C2×C4○D20, C20.70C24
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, C4×D5 [×4], C22×D5 [×7], Q8○M4(2), C2×C4×D5 [×6], C23×D5, D5×C22×C4, C20.70C24
Generators and relations
G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a5, bab=a9, ac=ca, ad=da, ae=ea, bc=cb, dbd=a10b, be=eb, cd=dc, ece-1=a10c, de=ed >
►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)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(22 30)(23 39)(24 28)(25 37)(27 35)(29 33)(32 40)(34 38)(41 51)(42 60)(43 49)(44 58)(45 47)(46 56)(48 54)(50 52)(53 59)(55 57)(61 71)(62 80)(63 69)(64 78)(65 67)(66 76)(68 74)(70 72)(73 79)(75 77)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)
(1 31 6 36 11 21 16 26)(2 32 7 37 12 22 17 27)(3 33 8 38 13 23 18 28)(4 34 9 39 14 24 19 29)(5 35 10 40 15 25 20 30)(41 61 46 66 51 71 56 76)(42 62 47 67 52 72 57 77)(43 63 48 68 53 73 58 78)(44 64 49 69 54 74 59 79)(45 65 50 70 55 75 60 80)
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), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(22,30)(23,39)(24,28)(25,37)(27,35)(29,33)(32,40)(34,38)(41,51)(42,60)(43,49)(44,58)(45,47)(46,56)(48,54)(50,52)(53,59)(55,57)(61,71)(62,80)(63,69)(64,78)(65,67)(66,76)(68,74)(70,72)(73,79)(75,77), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70), (1,31,6,36,11,21,16,26)(2,32,7,37,12,22,17,27)(3,33,8,38,13,23,18,28)(4,34,9,39,14,24,19,29)(5,35,10,40,15,25,20,30)(41,61,46,66,51,71,56,76)(42,62,47,67,52,72,57,77)(43,63,48,68,53,73,58,78)(44,64,49,69,54,74,59,79)(45,65,50,70,55,75,60,80)>;
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), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(22,30)(23,39)(24,28)(25,37)(27,35)(29,33)(32,40)(34,38)(41,51)(42,60)(43,49)(44,58)(45,47)(46,56)(48,54)(50,52)(53,59)(55,57)(61,71)(62,80)(63,69)(64,78)(65,67)(66,76)(68,74)(70,72)(73,79)(75,77), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70), (1,31,6,36,11,21,16,26)(2,32,7,37,12,22,17,27)(3,33,8,38,13,23,18,28)(4,34,9,39,14,24,19,29)(5,35,10,40,15,25,20,30)(41,61,46,66,51,71,56,76)(42,62,47,67,52,72,57,77)(43,63,48,68,53,73,58,78)(44,64,49,69,54,74,59,79)(45,65,50,70,55,75,60,80) );
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)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(22,30),(23,39),(24,28),(25,37),(27,35),(29,33),(32,40),(34,38),(41,51),(42,60),(43,49),(44,58),(45,47),(46,56),(48,54),(50,52),(53,59),(55,57),(61,71),(62,80),(63,69),(64,78),(65,67),(66,76),(68,74),(70,72),(73,79),(75,77)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70)], [(1,31,6,36,11,21,16,26),(2,32,7,37,12,22,17,27),(3,33,8,38,13,23,18,28),(4,34,9,39,14,24,19,29),(5,35,10,40,15,25,20,30),(41,61,46,66,51,71,56,76),(42,62,47,67,52,72,57,77),(43,63,48,68,53,73,58,78),(44,64,49,69,54,74,59,79),(45,65,50,70,55,75,60,80)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 19 | 32 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 19 | 32 |
| 0 | 0 | 9 | 0 |
| 1 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 17 | 40 |
| 0 | 0 | 1 | 24 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
G:=sub<GL(4,GF(41))| [19,9,0,0,32,0,0,0,0,0,19,9,0,0,32,0],[1,34,0,0,0,40,0,0,0,0,1,34,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[17,1,0,0,40,24,0,0,0,0,17,1,0,0,40,24],[0,0,9,0,0,0,0,9,1,0,0,0,0,1,0,0] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | D10 | C4×D5 | C4×D5 | Q8○M4(2) | C20.70C24 |
| kernel | C20.70C24 | D20.3C4 | D5×M4(2) | D20.2C4 | C2×C4.Dic5 | C10×M4(2) | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C5⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 8 | 2 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{20}._{70}C_2^4 % in TeX
G:=Group("C20.70C2^4"); // GroupNames label
G:=SmallGroup(320,1417);
// by ID
G=gap.SmallGroup(320,1417);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,570,80,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=1,e^2=a^5,b*a*b=a^9,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=a^10*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^10*c,d*e=e*d>;
// generators/relations