direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C22⋊C8, D10.13M4(2), D10⋊7(C2×C8), (C2×C8)⋊24D10, C22⋊3(C8×D5), C4.193(D4×D5), (C22×D5)⋊3C8, (C2×C40)⋊19C22, (C4×D5).122D4, C20.352(C2×D4), D10⋊1C8⋊15C2, (C23×D5).7C4, C23.46(C4×D5), C2.4(D5×M4(2)), C10.30(C22×C8), C20.55D4⋊22C2, (C2×C20).819C23, (C22×C4).302D10, C10.55(C2×M4(2)), D10.51(C22⋊C4), (C22×Dic5).15C4, Dic5.53(C22⋊C4), (C22×C20).336C22, C2.8(D5×C2×C8), (D5×C2×C8)⋊12C2, C5⋊4(C2×C22⋊C8), (C2×C10)⋊5(C2×C8), (C2×C4×D5).21C4, C2.3(D5×C22⋊C4), C22.43(C2×C4×D5), (C5×C22⋊C8)⋊13C2, (C2×C4).131(C4×D5), (D5×C22×C4).16C2, (C2×C20).325(C2×C4), (C2×C5⋊2C8)⋊43C22, C10.48(C2×C22⋊C4), (C2×C4×D5).419C22, (C22×D5).98(C2×C4), (C2×C4).761(C22×D5), (C22×C10).105(C2×C4), (C2×C10).175(C22×C4), (C2×Dic5).139(C2×C4), SmallGroup(320,351)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C22⋊C8
G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 734 in 202 conjugacy classes, 73 normal (33 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×20], C5, C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×10], D5 [×4], D5 [×2], C10 [×3], C10 [×2], C2×C8 [×2], C2×C8 [×6], C22×C4, C22×C4 [×9], C24, Dic5 [×2], Dic5, C20 [×2], C20, D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, C22⋊C8 [×3], C22×C8 [×2], C23×C4, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C2×C22⋊C8, C8×D5 [×4], C2×C5⋊2C8 [×2], C2×C40 [×2], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D10⋊1C8 [×2], C20.55D4, C5×C22⋊C8, D5×C2×C8 [×2], D5×C22×C4, D5×C22⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], D10 [×3], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C4×D5 [×2], C22×D5, C2×C22⋊C8, C8×D5 [×2], C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, D5×C2×C8, D5×M4(2), D5×C22⋊C8
►On 80 points
(1 79 41 18 12)(2 80 42 19 13)(3 73 43 20 14)(4 74 44 21 15)(5 75 45 22 16)(6 76 46 23 9)(7 77 47 24 10)(8 78 48 17 11)(25 62 69 50 34)(26 63 70 51 35)(27 64 71 52 36)(28 57 72 53 37)(29 58 65 54 38)(30 59 66 55 39)(31 60 67 56 40)(32 61 68 49 33)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)
(1 5)(2 63)(3 7)(4 57)(6 59)(8 61)(9 30)(10 14)(11 32)(12 16)(13 26)(15 28)(17 33)(18 22)(19 35)(20 24)(21 37)(23 39)(25 29)(27 31)(34 38)(36 40)(41 45)(42 51)(43 47)(44 53)(46 55)(48 49)(50 54)(52 56)(58 62)(60 64)(65 69)(66 76)(67 71)(68 78)(70 80)(72 74)(73 77)(75 79)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)(65 79)(66 80)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(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)
G:=sub<Sym(80)| (1,79,41,18,12)(2,80,42,19,13)(3,73,43,20,14)(4,74,44,21,15)(5,75,45,22,16)(6,76,46,23,9)(7,77,47,24,10)(8,78,48,17,11)(25,62,69,50,34)(26,63,70,51,35)(27,64,71,52,36)(28,57,72,53,37)(29,58,65,54,38)(30,59,66,55,39)(31,60,67,56,40)(32,61,68,49,33), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,68)(34,69)(35,70)(36,71)(37,72)(38,65)(39,66)(40,67), (1,5)(2,63)(3,7)(4,57)(6,59)(8,61)(9,30)(10,14)(11,32)(12,16)(13,26)(15,28)(17,33)(18,22)(19,35)(20,24)(21,37)(23,39)(25,29)(27,31)(34,38)(36,40)(41,45)(42,51)(43,47)(44,53)(46,55)(48,49)(50,54)(52,56)(58,62)(60,64)(65,69)(66,76)(67,71)(68,78)(70,80)(72,74)(73,77)(75,79), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(65,79)(66,80)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (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)>;
G:=Group( (1,79,41,18,12)(2,80,42,19,13)(3,73,43,20,14)(4,74,44,21,15)(5,75,45,22,16)(6,76,46,23,9)(7,77,47,24,10)(8,78,48,17,11)(25,62,69,50,34)(26,63,70,51,35)(27,64,71,52,36)(28,57,72,53,37)(29,58,65,54,38)(30,59,66,55,39)(31,60,67,56,40)(32,61,68,49,33), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,68)(34,69)(35,70)(36,71)(37,72)(38,65)(39,66)(40,67), (1,5)(2,63)(3,7)(4,57)(6,59)(8,61)(9,30)(10,14)(11,32)(12,16)(13,26)(15,28)(17,33)(18,22)(19,35)(20,24)(21,37)(23,39)(25,29)(27,31)(34,38)(36,40)(41,45)(42,51)(43,47)(44,53)(46,55)(48,49)(50,54)(52,56)(58,62)(60,64)(65,69)(66,76)(67,71)(68,78)(70,80)(72,74)(73,77)(75,79), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(65,79)(66,80)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (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) );
G=PermutationGroup([(1,79,41,18,12),(2,80,42,19,13),(3,73,43,20,14),(4,74,44,21,15),(5,75,45,22,16),(6,76,46,23,9),(7,77,47,24,10),(8,78,48,17,11),(25,62,69,50,34),(26,63,70,51,35),(27,64,71,52,36),(28,57,72,53,37),(29,58,65,54,38),(30,59,66,55,39),(31,60,67,56,40),(32,61,68,49,33)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,78),(18,79),(19,80),(20,73),(21,74),(22,75),(23,76),(24,77),(25,62),(26,63),(27,64),(28,57),(29,58),(30,59),(31,60),(32,61),(33,68),(34,69),(35,70),(36,71),(37,72),(38,65),(39,66),(40,67)], [(1,5),(2,63),(3,7),(4,57),(6,59),(8,61),(9,30),(10,14),(11,32),(12,16),(13,26),(15,28),(17,33),(18,22),(19,35),(20,24),(21,37),(23,39),(25,29),(27,31),(34,38),(36,40),(41,45),(42,51),(43,47),(44,53),(46,55),(48,49),(50,54),(52,56),(58,62),(60,64),(65,69),(66,76),(67,71),(68,78),(70,80),(72,74),(73,77),(75,79)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,57),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53),(65,79),(66,80),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)], [(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)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 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 | C4 | C4 | C4 | C8 | D4 | D5 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | C8×D5 | D4×D5 | D5×M4(2) |
| kernel | D5×C22⋊C8 | D10⋊1C8 | C20.55D4 | C5×C22⋊C8 | D5×C2×C8 | D5×C22×C4 | C2×C4×D5 | C22×Dic5 | C23×D5 | C22×D5 | C4×D5 | C22⋊C8 | D10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 16 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 16 | 4 | 4 |
Matrix representation of D5×C22⋊C8 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 1 | 0 | 0 |
| 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 38 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,33,0,0,0,1,7,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,33,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[38,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,0] >;
D5×C22⋊C8 in GAP, Magma, Sage, TeX
D_5\times C_2^2\rtimes C_8
% in TeX
G:=Group("D5xC2^2:C8"); // GroupNames label
G:=SmallGroup(320,351);
// by ID
G=gap.SmallGroup(320,351);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations