direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D4×D5, C42⋊32D10, C4⋊C4⋊55D10, (D4×C20)⋊8C2, C20⋊13(C2×D4), D20⋊22(C2×C4), (C4×D20)⋊23C2, C20⋊2(C22×C4), (D5×C42)⋊3C2, (C4×C20)⋊15C22, C22⋊C4⋊52D10, Dic5⋊11(C2×D4), (D4×Dic5)⋊44C2, D10⋊5(C22×C4), (C22×C4)⋊36D10, D20⋊8C4⋊46C2, (C2×D4).244D10, D10.106(C2×D4), (C2×C10).88C24, C10.42(C23×C4), C4⋊Dic5⋊72C22, Dic5⋊3(C22×C4), C10.46(C22×D4), D10.60(C4○D4), Dic5⋊4D4⋊51C2, (C2×C20).586C23, (C22×C20)⋊35C22, (C4×Dic5)⋊78C22, C23.D5⋊47C22, D10⋊C4⋊61C22, C22.31(C23×D5), (D4×C10).252C22, (C2×D20).265C22, C10.D4⋊63C22, C23.167(C22×D5), (C22×C10).158C23, (C2×Dic5).375C23, (C22×Dic5)⋊43C22, (C22×D5).291C23, (C23×D5).117C22, C5⋊5(C2×C4×D4), C4⋊1(C2×C4×D5), C2.5(C2×D4×D5), C22⋊1(C2×C4×D5), (D5×C4⋊C4)⋊47C2, C5⋊D4⋊6(C2×C4), (C2×D4×D5).19C2, C2.4(D5×C4○D4), (C5×D4)⋊21(C2×C4), (C4×D5)⋊11(C2×C4), (C4×C5⋊D4)⋊39C2, (C2×C4×D5)⋊69C22, (D5×C22×C4)⋊21C2, (C5×C4⋊C4)⋊55C22, C2.23(D5×C22×C4), (C2×C10)⋊2(C22×C4), (D5×C22⋊C4)⋊30C2, C10.138(C2×C4○D4), (C22×D5)⋊17(C2×C4), (C5×C22⋊C4)⋊62C22, (C2×C4).819(C22×D5), (C2×C5⋊D4).117C22, SmallGroup(320,1216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D4×D5
G = < a,b,c,d,e | a4=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1486 in 426 conjugacy classes, 169 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, D5, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C2×C4×D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×C4×D5, C2×D20, D4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, D5×C42, C4×D20, D5×C22⋊C4, Dic5⋊4D4, D5×C4⋊C4, D20⋊8C4, C4×C5⋊D4, D4×Dic5, D4×C20, D5×C22×C4, C2×D4×D5, C4×D4×D5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, C24, D10, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×D5, C22×D5, C2×C4×D4, C2×C4×D5, D4×D5, C23×D5, D5×C22×C4, C2×D4×D5, D5×C4○D4, C4×D4×D5
►On 80 points
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(1 54 9 59)(2 55 10 60)(3 51 6 56)(4 52 7 57)(5 53 8 58)(11 41 16 46)(12 42 17 47)(13 43 18 48)(14 44 19 49)(15 45 20 50)(21 71 26 76)(22 72 27 77)(23 73 28 78)(24 74 29 79)(25 75 30 80)(31 61 36 66)(32 62 37 67)(33 63 38 68)(34 64 39 69)(35 65 40 70)
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 51)(42 55)(43 54)(44 53)(45 52)(46 56)(47 60)(48 59)(49 58)(50 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)
G:=sub<Sym(80)| (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,54,9,59)(2,55,10,60)(3,51,6,56)(4,52,7,57)(5,53,8,58)(11,41,16,46)(12,42,17,47)(13,43,18,48)(14,44,19,49)(15,45,20,50)(21,71,26,76)(22,72,27,77)(23,73,28,78)(24,74,29,79)(25,75,30,80)(31,61,36,66)(32,62,37,67)(33,63,38,68)(34,64,39,69)(35,65,40,70), (41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)(41,51)(42,55)(43,54)(44,53)(45,52)(46,56)(47,60)(48,59)(49,58)(50,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77)>;
G:=Group( (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,54,9,59)(2,55,10,60)(3,51,6,56)(4,52,7,57)(5,53,8,58)(11,41,16,46)(12,42,17,47)(13,43,18,48)(14,44,19,49)(15,45,20,50)(21,71,26,76)(22,72,27,77)(23,73,28,78)(24,74,29,79)(25,75,30,80)(31,61,36,66)(32,62,37,67)(33,63,38,68)(34,64,39,69)(35,65,40,70), (41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)(41,51)(42,55)(43,54)(44,53)(45,52)(46,56)(47,60)(48,59)(49,58)(50,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77) );
G=PermutationGroup([[(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70)], [(1,54,9,59),(2,55,10,60),(3,51,6,56),(4,52,7,57),(5,53,8,58),(11,41,16,46),(12,42,17,47),(13,43,18,48),(14,44,19,49),(15,45,20,50),(21,71,26,76),(22,72,27,77),(23,73,28,78),(24,74,29,79),(25,75,30,80),(31,61,36,66),(32,62,37,67),(33,63,38,68),(34,64,39,69),(35,65,40,70)], [(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(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)], [(1,13),(2,12),(3,11),(4,15),(5,14),(6,16),(7,20),(8,19),(9,18),(10,17),(21,31),(22,35),(23,34),(24,33),(25,32),(26,36),(27,40),(28,39),(29,38),(30,37),(41,51),(42,55),(43,54),(44,53),(45,52),(46,56),(47,60),(48,59),(49,58),(50,57),(61,71),(62,75),(63,74),(64,73),(65,72),(66,76),(67,80),(68,79),(69,78),(70,77)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 4Q | ··· | 4X | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4×D5 | D4×D5 | D5×C4○D4 |
| kernel | C4×D4×D5 | D5×C42 | C4×D20 | D5×C22⋊C4 | Dic5⋊4D4 | D5×C4⋊C4 | D20⋊8C4 | C4×C5⋊D4 | D4×Dic5 | D4×C20 | D5×C22×C4 | C2×D4×D5 | D4×D5 | C4×D5 | C4×D4 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 4 | 4 |
Matrix representation of C4×D4×D5 ►in GL5(𝔽41)
| 32 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 40 | 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 | 40 | 1 | 0 | 0 |
| 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(41))| [32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,40,0],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,0,0,0,0,40,5,0,0,0,1,35,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,5,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
C4×D4×D5 in GAP, Magma, Sage, TeX
C_4\times D_4\times D_5
% in TeX
G:=Group("C4xD4xD5"); // GroupNames label
G:=SmallGroup(320,1216);
// by ID
G=gap.SmallGroup(320,1216);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations