metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.76C24, C4○D4.45D10, (D4×C10).26C4, Q8.Dic5⋊9C2, C5⋊6(Q8○M4(2)), (Q8×C10).23C4, C4○D4.2Dic5, (C2×Q8).8Dic5, D4.9(C2×Dic5), C4.75(C23×D5), C10.70(C23×C4), C5⋊2C8.34C23, Q8.10(C2×Dic5), (C2×D4).10Dic5, C20.157(C22×C4), (C2×C20).554C23, (C22×C4).280D10, C4.Dic5⋊35C22, C2.11(C23×Dic5), C4.20(C22×Dic5), C23.11(C2×Dic5), C22.2(C22×Dic5), (C22×C20).289C22, (C5×C4○D4).8C4, (C2×C4○D4).9D5, (C5×D4).40(C2×C4), (C5×Q8).43(C2×C4), (C2×C20).309(C2×C4), (C2×C5⋊2C8)⋊21C22, (C10×C4○D4).10C2, (C2×C4.Dic5)⋊29C2, (C2×C4).31(C2×Dic5), (C5×C4○D4).49C22, (C2×C4).635(C22×D5), (C22×C10).149(C2×C4), (C2×C10).130(C22×C4), SmallGroup(320,1491)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 494 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×7], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C5⋊2C8 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], Q8○M4(2), C2×C5⋊2C8 [×12], C4.Dic5 [×16], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4.Dic5 [×6], Q8.Dic5 [×8], C10×C4○D4, C20.76C24
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, C2×Dic5 [×28], C22×D5 [×7], Q8○M4(2), C22×Dic5 [×14], C23×D5, C23×Dic5, C20.76C24
Generators and relations
G = < a,b,c,d,e | a20=c2=d2=e2=1, b2=a5, bab-1=a9, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a10b, dcd=a10c, ce=ec, 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)
(1 50 6 55 11 60 16 45)(2 59 7 44 12 49 17 54)(3 48 8 53 13 58 18 43)(4 57 9 42 14 47 19 52)(5 46 10 51 15 56 20 41)(21 71 26 76 31 61 36 66)(22 80 27 65 32 70 37 75)(23 69 28 74 33 79 38 64)(24 78 29 63 34 68 39 73)(25 67 30 72 35 77 40 62)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 21)(17 22)(18 23)(19 24)(20 25)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 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), (1,50,6,55,11,60,16,45)(2,59,7,44,12,49,17,54)(3,48,8,53,13,58,18,43)(4,57,9,42,14,47,19,52)(5,46,10,51,15,56,20,41)(21,71,26,76,31,61,36,66)(22,80,27,65,32,70,37,75)(23,69,28,74,33,79,38,64)(24,78,29,63,34,68,39,73)(25,67,30,72,35,77,40,62), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)>;
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), (1,50,6,55,11,60,16,45)(2,59,7,44,12,49,17,54)(3,48,8,53,13,58,18,43)(4,57,9,42,14,47,19,52)(5,46,10,51,15,56,20,41)(21,71,26,76,31,61,36,66)(22,80,27,65,32,70,37,75)(23,69,28,74,33,79,38,64)(24,78,29,63,34,68,39,73)(25,67,30,72,35,77,40,62), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80) );
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)], [(1,50,6,55,11,60,16,45),(2,59,7,44,12,49,17,54),(3,48,8,53,13,58,18,43),(4,57,9,42,14,47,19,52),(5,46,10,51,15,56,20,41),(21,71,26,76,31,61,36,66),(22,80,27,65,32,70,37,75),(23,69,28,74,33,79,38,64),(24,78,29,63,34,68,39,73),(25,67,30,72,35,77,40,62)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,21),(17,22),(18,23),(19,24),(20,25),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71)], [(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 23 | 0 | 0 | 0 | 0 | 0 |
| 38 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 5 | 24 | 0 | 0 | 0 | 0 |
| 16 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 1 | 23 |
| 0 | 0 | 0 | 17 | 0 | 26 |
| 0 | 0 | 9 | 40 | 0 | 1 |
| 0 | 0 | 0 | 5 | 0 | 24 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 26 | 32 |
| 0 | 0 | 32 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 36 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 29 | 27 |
| 0 | 0 | 1 | 0 | 11 | 32 |
| 0 | 0 | 0 | 0 | 9 | 20 |
| 0 | 0 | 0 | 0 | 37 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 35 |
| 0 | 0 | 0 | 1 | 0 | 26 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [23,38,0,0,0,0,0,25,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[5,16,0,0,0,0,24,36,0,0,0,0,0,0,0,0,9,0,0,0,15,17,40,5,0,0,1,0,0,0,0,0,23,26,1,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,26,24,1,36,0,0,32,40,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,29,11,9,37,0,0,27,32,20,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,35,26,0,40] >;
74 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 4A | 4B | 4C | ··· | 4I | 5A | 5B | 8A | ··· | 8P | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | D10 | Dic5 | Dic5 | Dic5 | D10 | Q8○M4(2) | C20.76C24 |
| kernel | C20.76C24 | C2×C4.Dic5 | Q8.Dic5 | C10×C4○D4 | D4×C10 | Q8×C10 | C5×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C5 | C1 |
| # reps | 1 | 6 | 8 | 1 | 6 | 2 | 8 | 2 | 6 | 6 | 2 | 8 | 8 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{20}._{76}C_2^4 % in TeX
G:=Group("C20.76C2^4"); // GroupNames label
G:=SmallGroup(320,1491);
// by ID
G=gap.SmallGroup(320,1491);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=c^2=d^2=e^2=1,b^2=a^5,b*a*b^-1=a^9,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^10*b,d*c*d=a^10*c,c*e=e*c,d*e=e*d>;
// generators/relations