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, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C2×C8, C2×C8, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C22⋊C8, C22×C8, C23×C4, C5⋊2C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, C2×C22⋊C8, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D10⋊1C8, C20.55D4, C5×C22⋊C8, D5×C2×C8, D5×C22×C4, D5×C22⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D5, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, D10, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C4×D5, C22×D5, C2×C22⋊C8, C8×D5, C2×C4×D5, D4×D5, D5×C22⋊C4, D5×C2×C8, D5×M4(2), D5×C22⋊C8
►On 80 points
(1 73 29 43 35)(2 74 30 44 36)(3 75 31 45 37)(4 76 32 46 38)(5 77 25 47 39)(6 78 26 48 40)(7 79 27 41 33)(8 80 28 42 34)(9 51 59 21 68)(10 52 60 22 69)(11 53 61 23 70)(12 54 62 24 71)(13 55 63 17 72)(14 56 64 18 65)(15 49 57 19 66)(16 50 58 20 67)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(41 79)(42 80)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 65)
(1 5)(2 23)(3 7)(4 17)(6 19)(8 21)(9 28)(10 14)(11 30)(12 16)(13 32)(15 26)(18 22)(20 24)(25 29)(27 31)(33 37)(34 59)(35 39)(36 61)(38 63)(40 57)(41 45)(42 51)(43 47)(44 53)(46 55)(48 49)(50 54)(52 56)(58 62)(60 64)(65 69)(66 78)(67 71)(68 80)(70 74)(72 76)(73 77)(75 79)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 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)
G:=sub<Sym(80)| (1,73,29,43,35)(2,74,30,44,36)(3,75,31,45,37)(4,76,32,46,38)(5,77,25,47,39)(6,78,26,48,40)(7,79,27,41,33)(8,80,28,42,34)(9,51,59,21,68)(10,52,60,22,69)(11,53,61,23,70)(12,54,62,24,71)(13,55,63,17,72)(14,56,64,18,65)(15,49,57,19,66)(16,50,58,20,67), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,65), (1,5)(2,23)(3,7)(4,17)(6,19)(8,21)(9,28)(10,14)(11,30)(12,16)(13,32)(15,26)(18,22)(20,24)(25,29)(27,31)(33,37)(34,59)(35,39)(36,61)(38,63)(40,57)(41,45)(42,51)(43,47)(44,53)(46,55)(48,49)(50,54)(52,56)(58,62)(60,64)(65,69)(66,78)(67,71)(68,80)(70,74)(72,76)(73,77)(75,79), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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)>;
G:=Group( (1,73,29,43,35)(2,74,30,44,36)(3,75,31,45,37)(4,76,32,46,38)(5,77,25,47,39)(6,78,26,48,40)(7,79,27,41,33)(8,80,28,42,34)(9,51,59,21,68)(10,52,60,22,69)(11,53,61,23,70)(12,54,62,24,71)(13,55,63,17,72)(14,56,64,18,65)(15,49,57,19,66)(16,50,58,20,67), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,65), (1,5)(2,23)(3,7)(4,17)(6,19)(8,21)(9,28)(10,14)(11,30)(12,16)(13,32)(15,26)(18,22)(20,24)(25,29)(27,31)(33,37)(34,59)(35,39)(36,61)(38,63)(40,57)(41,45)(42,51)(43,47)(44,53)(46,55)(48,49)(50,54)(52,56)(58,62)(60,64)(65,69)(66,78)(67,71)(68,80)(70,74)(72,76)(73,77)(75,79), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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) );
G=PermutationGroup([[(1,73,29,43,35),(2,74,30,44,36),(3,75,31,45,37),(4,76,32,46,38),(5,77,25,47,39),(6,78,26,48,40),(7,79,27,41,33),(8,80,28,42,34),(9,51,59,21,68),(10,52,60,22,69),(11,53,61,23,70),(12,54,62,24,71),(13,55,63,17,72),(14,56,64,18,65),(15,49,57,19,66),(16,50,58,20,67)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(41,79),(42,80),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,65)], [(1,5),(2,23),(3,7),(4,17),(6,19),(8,21),(9,28),(10,14),(11,30),(12,16),(13,32),(15,26),(18,22),(20,24),(25,29),(27,31),(33,37),(34,59),(35,39),(36,61),(38,63),(40,57),(41,45),(42,51),(43,47),(44,53),(46,55),(48,49),(50,54),(52,56),(58,62),(60,64),(65,69),(66,78),(67,71),(68,80),(70,74),(72,76),(73,77),(75,79)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(33,62),(34,63),(35,64),(36,57),(37,58),(38,59),(39,60),(40,61),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80)], [(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)]])
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