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## G = D5×D4⋊C4order 320 = 26·5

### Direct product of D5 and D4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5×D4⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×D4×D5 — D5×D4⋊C4
 Lower central C5 — C10 — C20 — D5×D4⋊C4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D5×D4⋊C4
G = < a,b,c,d,e | a5=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=ece-1=c-1, ede-1=cd >

Subgroups: 1022 in 202 conjugacy classes, 63 normal (37 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×22], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×2], D4 [×8], C23 [×11], D5 [×4], D5 [×2], C10 [×3], C10 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8 [×3], C22×C4 [×2], C2×D4, C2×D4 [×8], C24, Dic5 [×2], Dic5, C20 [×2], C20, D10 [×6], D10 [×12], C2×C10, C2×C10 [×4], D4⋊C4, D4⋊C4 [×3], C2×C4⋊C4, C22×C8, C22×D4, C52C8, C40, C4×D5 [×4], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×D5 [×9], C22×C10, C2×D4⋊C4, C8×D5 [×2], C2×C52C8, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], C2×C5⋊D4, D4×C10, C23×D5, D206C4, D205C4, D4⋊Dic5, C5×D4⋊C4, D5×C4⋊C4, D5×C2×C8, C2×D4×D5, D5×D4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D10 [×3], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C4×D5 [×2], C22×D5, C2×D4⋊C4, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, D5×D8, D5×SD16, D5×D4⋊C4

Smallest permutation representation of D5×D4⋊C4
On 80 points
Generators in S80
(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 28)(2 27)(3 26)(4 30)(5 29)(6 21)(7 25)(8 24)(9 23)(10 22)(11 36)(12 40)(13 39)(14 38)(15 37)(16 31)(17 35)(18 34)(19 33)(20 32)(41 66)(42 70)(43 69)(44 68)(45 67)(46 61)(47 65)(48 64)(49 63)(50 62)(51 76)(52 80)(53 79)(54 78)(55 77)(56 71)(57 75)(58 74)(59 73)(60 72)
(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 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 46)(42 47)(43 48)(44 49)(45 50)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 64 24 44)(2 65 25 45)(3 61 21 41)(4 62 22 42)(5 63 23 43)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)

G:=sub<Sym(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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,21)(7,25)(8,24)(9,23)(10,22)(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32)(41,66)(42,70)(43,69)(44,68)(45,67)(46,61)(47,65)(48,64)(49,63)(50,62)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72), (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)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,46)(42,47)(43,48)(44,49)(45,50)(61,66)(62,67)(63,68)(64,69)(65,70), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)>;

G:=Group( (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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,21)(7,25)(8,24)(9,23)(10,22)(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32)(41,66)(42,70)(43,69)(44,68)(45,67)(46,61)(47,65)(48,64)(49,63)(50,62)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72), (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)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,46)(42,47)(43,48)(44,49)(45,50)(61,66)(62,67)(63,68)(64,69)(65,70), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60) );

G=PermutationGroup([(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)], [(1,28),(2,27),(3,26),(4,30),(5,29),(6,21),(7,25),(8,24),(9,23),(10,22),(11,36),(12,40),(13,39),(14,38),(15,37),(16,31),(17,35),(18,34),(19,33),(20,32),(41,66),(42,70),(43,69),(44,68),(45,67),(46,61),(47,65),(48,64),(49,63),(50,62),(51,76),(52,80),(53,79),(54,78),(55,77),(56,71),(57,75),(58,74),(59,73),(60,72)], [(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),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,11),(7,12),(8,13),(9,14),(10,15),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35),(41,46),(42,47),(43,48),(44,49),(45,50),(61,66),(62,67),(63,68),(64,69),(65,70)], [(1,64,24,44),(2,65,25,45),(3,61,21,41),(4,62,22,42),(5,63,23,43),(6,66,26,46),(7,67,27,47),(8,68,28,48),(9,69,29,49),(10,70,30,50),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60)])

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 5 5 5 5 20 20 2 2 4 4 10 10 20 20 2 2 2 2 2 2 10 10 10 10 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D5 D8 SD16 D10 D10 D10 C4×D5 D4×D5 D4×D5 D5×D8 D5×SD16 kernel D5×D4⋊C4 D20⋊6C4 D20⋊5C4 D4⋊Dic5 C5×D4⋊C4 D5×C4⋊C4 D5×C2×C8 C2×D4×D5 D4×D5 C4×D5 C2×Dic5 C22×D5 D4⋊C4 D10 D10 C4⋊C4 C2×C8 C2×D4 D4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 2 4 4 2 2 2 8 2 2 4 4

Matrix representation of D5×D4⋊C4 in GL6(𝔽41)

 7 1 0 0 0 0 33 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 40 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 9 0 0 0 0 0 0 0 29 29 0 0 0 0 29 12

G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,29,29,0,0,0,0,29,12] >;

D5×D4⋊C4 in GAP, Magma, Sage, TeX

D_5\times D_4\rtimes C_4
% in TeX

G:=Group("D5xD4:C4");
// GroupNames label

G:=SmallGroup(320,396);
// by ID

G=gap.SmallGroup(320,396);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^2=e^4=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,d*c*d=e*c*e^-1=c^-1,e*d*e^-1=c*d>;
// generators/relations

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