direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C22×C4, C20⋊3C23, C10.2C24, Dic5⋊3C23, C23.34D10, D10.16C23, C5⋊2(C23×C4), C10⋊2(C22×C4), C2.1(C23×D5), (C2×C20)⋊14C22, (C22×C20)⋊10C2, (C23×D5).5C2, (C2×C10).63C23, (C22×Dic5)⋊10C2, (C2×Dic5)⋊12C22, C22.29(C22×D5), (C22×C10).44C22, (C22×D5).44C22, (C2×C10)⋊9(C2×C4), SmallGroup(160,214)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C22×C4 |
Generators and relations for D5×C22×C4
G = < a,b,c,d,e | a2=b2=c4=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 600 in 236 conjugacy classes, 145 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C23×C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D5×C22×C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, C4×D5, C22×D5, C2×C4×D5, C23×D5, D5×C22×C4
►On 80 points
Generators in S80
(1 49)(2 50)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)
(1 29)(2 30)(3 26)(4 27)(5 28)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 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 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)
G:=sub<Sym(80)| (1,49)(2,50)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75), (1,29)(2,30)(3,26)(4,27)(5,28)(6,21)(7,22)(8,23)(9,24)(10,25)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (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)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)>;
G:=Group( (1,49)(2,50)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75), (1,29)(2,30)(3,26)(4,27)(5,28)(6,21)(7,22)(8,23)(9,24)(10,25)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (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)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77) );
G=PermutationGroup([[(1,49),(2,50),(3,46),(4,47),(5,48),(6,41),(7,42),(8,43),(9,44),(10,45),(11,56),(12,57),(13,58),(14,59),(15,60),(16,51),(17,52),(18,53),(19,54),(20,55),(21,66),(22,67),(23,68),(24,69),(25,70),(26,61),(27,62),(28,63),(29,64),(30,65),(31,76),(32,77),(33,78),(34,79),(35,80),(36,71),(37,72),(38,73),(39,74),(40,75)], [(1,29),(2,30),(3,26),(4,27),(5,28),(6,21),(7,22),(8,23),(9,24),(10,25),(11,36),(12,37),(13,38),(14,39),(15,40),(16,31),(17,32),(18,33),(19,34),(20,35),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(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),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,66),(62,70),(63,69),(64,68),(65,67),(71,76),(72,80),(73,79),(74,78),(75,77)]])
D5×C22×C4 is a maximal subgroup of
C22.58(D4×D5) (C2×C4)⋊9D20 D10⋊2C42 D10⋊2(C4⋊C4) D10⋊3(C4⋊C4) D10⋊7M4(2) C24.12D10 D10⋊4(C4⋊C4) D10⋊5(C4⋊C4) D10⋊8M4(2) D10.11M4(2) D10⋊9M4(2) D10⋊10M4(2) (C22×C4)⋊7F5 D10⋊6(C4⋊C4) C42⋊8D10 C42⋊12D10 C4⋊C4⋊21D10 C4⋊C4⋊26D10 C4⋊C4⋊28D10 (C2×C20)⋊15D4
D5×C22×C4 is a maximal quotient of
C24.24D10 C10.82+ 1+4 C42.87D10 C42⋊7D10 C42.188D10 C42.91D10 C42⋊11D10 C42.108D10 C42.125D10 C42.126D10 C40.47C23 C20.72C24
64 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | C4×D5 |
| kernel | D5×C22×C4 | C2×C4×D5 | C22×Dic5 | C22×C20 | C23×D5 | C22×D5 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 16 | 2 | 12 | 2 | 16 |
Matrix representation of D5×C22×C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,34,40,0,0,1,0],[40,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0] >;
D5×C22×C4 in GAP, Magma, Sage, TeX
D_5\times C_2^2\times C_4
% in TeX
G:=Group("D5xC2^2xC4"); // GroupNames label
G:=SmallGroup(160,214);
// by ID
G=gap.SmallGroup(160,214);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,69,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations