direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C22⋊D8, D4⋊2(C5×D4), (C2×C10)⋊7D8, (C5×D4)⋊20D4, (C2×D8)⋊1C10, C2.4(C10×D8), C22⋊2(C5×D8), (C10×D8)⋊15C2, C4⋊D4⋊1C10, C22⋊C8⋊3C10, C4.21(D4×C10), C10.76(C2×D8), D4⋊C4⋊4C10, (C2×C40)⋊28C22, (C2×C20).317D4, C20.382(C2×D4), (C22×D4)⋊2C10, C23.41(C5×D4), C10.94C22≀C2, (D4×C10)⋊27C22, C22.77(D4×C10), (C2×C20).912C23, (C22×C10).163D4, C10.131(C8⋊C22), (C22×C20).419C22, C4⋊C4⋊1(C2×C10), (C2×C8)⋊1(C2×C10), (D4×C2×C10)⋊14C2, (C2×D4)⋊1(C2×C10), (C2×C4).26(C5×D4), C2.6(C5×C8⋊C22), (C5×C4⋊D4)⋊28C2, (C5×C4⋊C4)⋊35C22, (C5×C22⋊C8)⋊20C2, C2.8(C5×C22≀C2), (C5×D4⋊C4)⋊27C2, (C2×C10).633(C2×D4), (C2×C4).87(C22×C10), (C22×C4).37(C2×C10), SmallGroup(320,948)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22⋊D8
G = < a,b,c,d,e | a5=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 450 in 198 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, C22⋊D8, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×D8, C22×C20, D4×C10, D4×C10, D4×C10, C23×C10, C5×C22⋊C8, C5×D4⋊C4, C5×C4⋊D4, C10×D8, D4×C2×C10, C5×C22⋊D8
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C22≀C2, C2×D8, C8⋊C22, C5×D4, C22×C10, C22⋊D8, C5×D8, D4×C10, C5×C22≀C2, C10×D8, C5×C8⋊C22, C5×C22⋊D8
►On 80 points
(1 14 63 22 55)(2 15 64 23 56)(3 16 57 24 49)(4 9 58 17 50)(5 10 59 18 51)(6 11 60 19 52)(7 12 61 20 53)(8 13 62 21 54)(25 48 74 33 66)(26 41 75 34 67)(27 42 76 35 68)(28 43 77 36 69)(29 44 78 37 70)(30 45 79 38 71)(31 46 80 39 72)(32 47 73 40 65)
(1 30)(3 32)(5 26)(7 28)(10 41)(12 43)(14 45)(16 47)(18 34)(20 36)(22 38)(24 40)(49 65)(51 67)(53 69)(55 71)(57 73)(59 75)(61 77)(63 79)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(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 48)(42 47)(43 46)(44 45)(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)
G:=sub<Sym(80)| (1,14,63,22,55)(2,15,64,23,56)(3,16,57,24,49)(4,9,58,17,50)(5,10,59,18,51)(6,11,60,19,52)(7,12,61,20,53)(8,13,62,21,54)(25,48,74,33,66)(26,41,75,34,67)(27,42,76,35,68)(28,43,77,36,69)(29,44,78,37,70)(30,45,79,38,71)(31,46,80,39,72)(32,47,73,40,65), (1,30)(3,32)(5,26)(7,28)(10,41)(12,43)(14,45)(16,47)(18,34)(20,36)(22,38)(24,40)(49,65)(51,67)(53,69)(55,71)(57,73)(59,75)(61,77)(63,79), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(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)>;
G:=Group( (1,14,63,22,55)(2,15,64,23,56)(3,16,57,24,49)(4,9,58,17,50)(5,10,59,18,51)(6,11,60,19,52)(7,12,61,20,53)(8,13,62,21,54)(25,48,74,33,66)(26,41,75,34,67)(27,42,76,35,68)(28,43,77,36,69)(29,44,78,37,70)(30,45,79,38,71)(31,46,80,39,72)(32,47,73,40,65), (1,30)(3,32)(5,26)(7,28)(10,41)(12,43)(14,45)(16,47)(18,34)(20,36)(22,38)(24,40)(49,65)(51,67)(53,69)(55,71)(57,73)(59,75)(61,77)(63,79), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(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) );
G=PermutationGroup([[(1,14,63,22,55),(2,15,64,23,56),(3,16,57,24,49),(4,9,58,17,50),(5,10,59,18,51),(6,11,60,19,52),(7,12,61,20,53),(8,13,62,21,54),(25,48,74,33,66),(26,41,75,34,67),(27,42,76,35,68),(28,43,77,36,69),(29,44,78,37,70),(30,45,79,38,71),(31,46,80,39,72),(32,47,73,40,65)], [(1,30),(3,32),(5,26),(7,28),(10,41),(12,43),(14,45),(16,47),(18,34),(20,36),(22,38),(24,40),(49,65),(51,67),(53,69),(55,71),(57,73),(59,75),(61,77),(63,79)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,80)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,26),(27,32),(28,31),(29,30),(33,34),(35,40),(36,39),(37,38),(41,48),(42,47),(43,46),(44,45),(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)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 10AK | 10AL | 10AM | 10AN | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | 20N | 20O | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | D8 | C5×D4 | C5×D4 | C5×D4 | C5×D8 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C22⋊D8 | C5×C22⋊C8 | C5×D4⋊C4 | C5×C4⋊D4 | C10×D8 | D4×C2×C10 | C22⋊D8 | C22⋊C8 | D4⋊C4 | C4⋊D4 | C2×D8 | C22×D4 | C2×C20 | C5×D4 | C22×C10 | C2×C10 | C2×C4 | D4 | C23 | C22 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 8 | 4 | 1 | 4 | 1 | 4 | 4 | 16 | 4 | 16 | 1 | 4 |
Matrix representation of C5×C22⋊D8 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 12 | 17 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 29 | 0 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,0,0,0,0,0,0,12,0,0,17,17],[0,1,0,0,1,0,0,0,0,0,0,29,0,0,17,0] >;
C5×C22⋊D8 in GAP, Magma, Sage, TeX
C_5\times C_2^2\rtimes D_8
% in TeX
G:=Group("C5xC2^2:D8"); // GroupNames label
G:=SmallGroup(320,948);
// by ID
G=gap.SmallGroup(320,948);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations