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## G = C4×D4×D5order 320 = 26·5

### Direct product of C4, D4 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×D4×D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C4×D4×D5
 Lower central C5 — C10 — C4×D4×D5
 Upper central C1 — C2×C4 — C4×D4

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 [×3], C2 [×12], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×34], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×35], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×4], D5 [×4], C10 [×3], C10 [×4], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×19], C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×4], Dic5 [×3], C20 [×4], C20 [×3], D10 [×10], D10 [×20], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4, C4×D4 [×7], C23×C4 [×2], C22×D4, C4×D5 [×8], C4×D5 [×14], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×D5, C22×D5 [×10], C22×D5 [×8], C22×C10 [×2], C2×C4×D4, C4×Dic5 [×3], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×4], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×3], C2×C4×D5 [×6], C2×C4×D5 [×8], C2×D20, D4×D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C23×D5 [×2], D5×C42, C4×D20, D5×C22⋊C4 [×2], Dic54D4 [×2], D5×C4⋊C4, D208C4, C4×C5⋊D4 [×2], D4×Dic5, D4×C20, D5×C22×C4 [×2], C2×D4×D5, C4×D4×D5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C4×D5 [×4], C22×D5 [×7], C2×C4×D4, C2×C4×D5 [×6], D4×D5 [×2], C23×D5, D5×C22×C4, C2×D4×D5, D5×C4○D4, C4×D4×D5

Smallest permutation representation of C4×D4×D5
On 80 points
Generators in S80
(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

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