direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D8⋊C22, C40.51C23, C20.84C24, C4○D8⋊3C10, D8⋊4(C2×C10), C8⋊C22⋊6C10, Q16⋊4(C2×C10), C4.68(D4×C10), (C2×C40)⋊30C22, SD16⋊3(C2×C10), (C2×C20).527D4, C20.473(C2×D4), (C5×D8)⋊20C22, C8.C22⋊6C10, C8.2(C22×C10), C4.7(C23×C10), C23.20(C5×D4), (D4×C10)⋊67C22, M4(2)⋊5(C2×C10), (C2×M4(2))⋊5C10, (Q8×C10)⋊56C22, (C5×Q16)⋊18C22, D4.4(C22×C10), (C5×D4).37C23, (C22×C10).38D4, C22.25(D4×C10), Q8.4(C22×C10), (C5×Q8).38C23, (C10×M4(2))⋊15C2, (C2×C20).686C23, (C5×SD16)⋊19C22, C10.205(C22×D4), (C5×M4(2))⋊31C22, (C22×C20).467C22, (C2×C8)⋊3(C2×C10), C2.29(D4×C2×C10), C4○D4⋊5(C2×C10), (C5×C4○D8)⋊10C2, (C2×C4○D4)⋊12C10, (C10×C4○D4)⋊28C2, (C2×D4)⋊16(C2×C10), (C5×C8⋊C22)⋊13C2, (C2×Q8)⋊16(C2×C10), (C2×C4).138(C5×D4), (C2×C10).421(C2×D4), (C5×C4○D4)⋊25C22, (C5×C8.C22)⋊13C2, (C2×C4).47(C22×C10), (C22×C4).78(C2×C10), SmallGroup(320,1577)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 262 conjugacy classes, 158 normal (22 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C10, C10 [×7], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×9], C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×10], C5×D4 [×4], C5×D4 [×10], C5×Q8 [×4], C5×Q8 [×2], C22×C10, C22×C10 [×2], D8⋊C22, C2×C40 [×2], C5×M4(2) [×4], C5×D8 [×4], C5×SD16 [×8], C5×Q16 [×4], C22×C20, C22×C20 [×2], D4×C10 [×2], D4×C10 [×2], Q8×C10 [×2], C5×C4○D4 [×8], C5×C4○D4 [×4], C10×M4(2), C5×C4○D8 [×4], C5×C8⋊C22 [×4], C5×C8.C22 [×4], C10×C4○D4 [×2], C5×D8⋊C22
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C22×D4, C5×D4 [×4], C22×C10 [×15], D8⋊C22, D4×C10 [×6], C23×C10, D4×C2×C10, C5×D8⋊C22
Generators and relations
G = < a,b,c,d,e | a5=b8=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd=b5, be=eb, dcd=ece=b4c, de=ed >
►On 80 points
(1 22 67 55 26)(2 23 68 56 27)(3 24 69 49 28)(4 17 70 50 29)(5 18 71 51 30)(6 19 72 52 31)(7 20 65 53 32)(8 21 66 54 25)(9 79 59 38 42)(10 80 60 39 43)(11 73 61 40 44)(12 74 62 33 45)(13 75 63 34 46)(14 76 64 35 47)(15 77 57 36 48)(16 78 58 37 41)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)(33 34)(35 40)(36 39)(37 38)(41 42)(43 48)(44 47)(45 46)(49 52)(50 51)(53 56)(54 55)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)
(1 40)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 73)(56 74)
G:=sub<Sym(80)| (1,22,67,55,26)(2,23,68,56,27)(3,24,69,49,28)(4,17,70,50,29)(5,18,71,51,30)(6,19,72,52,31)(7,20,65,53,32)(8,21,66,54,25)(9,79,59,38,42)(10,80,60,39,43)(11,73,61,40,44)(12,74,62,33,45)(13,75,63,34,46)(14,76,64,35,47)(15,77,57,36,48)(16,78,58,37,41), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80), (1,40)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,73)(56,74)>;
G:=Group( (1,22,67,55,26)(2,23,68,56,27)(3,24,69,49,28)(4,17,70,50,29)(5,18,71,51,30)(6,19,72,52,31)(7,20,65,53,32)(8,21,66,54,25)(9,79,59,38,42)(10,80,60,39,43)(11,73,61,40,44)(12,74,62,33,45)(13,75,63,34,46)(14,76,64,35,47)(15,77,57,36,48)(16,78,58,37,41), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80), (1,40)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,73)(56,74) );
G=PermutationGroup([(1,22,67,55,26),(2,23,68,56,27),(3,24,69,49,28),(4,17,70,50,29),(5,18,71,51,30),(6,19,72,52,31),(7,20,65,53,32),(8,21,66,54,25),(9,79,59,38,42),(10,80,60,39,43),(11,73,61,40,44),(12,74,62,33,45),(13,75,63,34,46),(14,76,64,35,47),(15,77,57,36,48),(16,78,58,37,41)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,18),(19,24),(20,23),(21,22),(25,26),(27,32),(28,31),(29,30),(33,34),(35,40),(36,39),(37,38),(41,42),(43,48),(44,47),(45,46),(49,52),(50,51),(53,56),(54,55),(57,60),(58,59),(61,64),(62,63),(65,68),(66,67),(69,72),(70,71),(73,76),(74,75),(77,80),(78,79)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80)], [(1,40),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,47),(18,48),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,73),(56,74)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 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 | 1 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 1 | 16 |
| 0 | 0 | 22 | 0 | 0 | 17 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 39 | 0 | 0 | 19 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 1 | 16 |
| 0 | 0 | 0 | 22 | 0 | 16 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 39 | 0 | 19 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 22 |
| 0 | 0 | 0 | 1 | 0 | 22 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 7 | 0 |
| 0 | 0 | 9 | 0 | 7 | 34 |
| 0 | 0 | 0 | 0 | 9 | 32 |
| 0 | 0 | 0 | 0 | 18 | 32 |
G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,39,0,0,0,0,1,1,0,0,0,0,0,0,22,22,40,39,0,0,0,0,1,0,0,0,1,0,0,0,0,0,16,17,0,19],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,22,22,40,39,0,0,1,0,0,0,0,0,16,16,0,19],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,22,22,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,7,7,9,18,0,0,0,34,32,32] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | ··· | 10AF | 20A | ··· | 20H | 20I | ··· | 20T | 20U | ··· | 20AJ | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C5×D4 | C5×D4 | D8⋊C22 | C5×D8⋊C22 |
| kernel | C5×D8⋊C22 | C10×M4(2) | C5×C4○D8 | C5×C8⋊C22 | C5×C8.C22 | C10×C4○D4 | D8⋊C22 | C2×M4(2) | C4○D8 | C8⋊C22 | C8.C22 | C2×C4○D4 | C2×C20 | C22×C10 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 16 | 16 | 8 | 3 | 1 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_5\times D_8\rtimes C_2^2
% in TeX
G:=Group("C5xD8:C2^2"); // GroupNames label
G:=SmallGroup(320,1577);
// by ID
G=gap.SmallGroup(320,1577);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,584,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d=b^5,b*e=e*b,d*c*d=e*c*e=b^4*c,d*e=e*d>;
// generators/relations