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## G = D5×C4⋊D4order 320 = 26·5

### Direct product of D5 and C4⋊D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C4⋊D4
G = < a,b,c,d,e | a5=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1822 in 426 conjugacy classes, 121 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×40], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×22], D4 [×24], C23, C23 [×2], C23 [×24], D5 [×4], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×11], C2×D4, C2×D4 [×2], C2×D4 [×21], C24 [×3], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×8], D10 [×24], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C23×C4, C22×D4 [×3], C4×D5 [×4], C4×D5 [×10], D20 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5 [×2], C22×D5 [×6], C22×D5 [×16], C22×C10, C22×C10 [×2], C2×C4⋊D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×4], C2×C4×D5 [×2], C2×C4×D5 [×4], C2×D20, C2×D20 [×2], D4×D5 [×12], C22×Dic5, C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], C23×D5, C23×D5 [×2], D5×C22⋊C4 [×2], D10⋊D4 [×2], D5×C4⋊C4, C4⋊D20, C207D4, C202D4, Dic5⋊D4 [×2], C5×C4⋊D4, D5×C22×C4, C2×D4×D5, C2×D4×D5 [×2], D5×C4⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C4○D4 [×2], C24, D10 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C22×D5 [×7], C2×C4⋊D4, D4×D5 [×4], C23×D5, C2×D4×D5 [×2], D5×C4○D4, D5×C4⋊D4

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 20A ··· 20H 20I 20J 20K 20L 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 10 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 2 2 4 4 5 5 5 5 10 10 20 20 2 2 2 2 4 4 10 10 10 10 20 20 2 2 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4 8 8 8 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 D10 D4×D5 D4×D5 D5×C4○D4 kernel D5×C4⋊D4 D5×C22⋊C4 D10⋊D4 D5×C4⋊C4 C4⋊D20 C20⋊7D4 C20⋊2D4 Dic5⋊D4 C5×C4⋊D4 D5×C22×C4 C2×D4×D5 C4×D5 C22×D5 C4⋊D4 D10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C22 C2 # reps 1 2 2 1 1 1 1 2 1 1 3 4 4 2 4 4 2 2 6 4 4 4

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

 0 1 0 0 0 0 40 6 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
,
 0 1 0 0 0 0 1 0 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 0 40 0 0 0 0 0 0 1 4 0 0 0 0 20 40 0 0 0 0 0 0 1 4 0 0 0 0 20 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 20 40 0 0 0 0 0 0 32 5 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 21 1 0 0 0 0 0 0 40 0 0 0 0 0 21 1

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,20,0,0,0,0,4,40,0,0,0,0,0,0,1,20,0,0,0,0,4,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,5,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,21,0,0,0,0,0,1,0,0,0,0,0,0,40,21,0,0,0,0,0,1] >;

D5×C4⋊D4 in GAP, Magma, Sage, TeX

D_5\times C_4\rtimes D_4
% in TeX

G:=Group("D5xC4:D4");
// GroupNames label

G:=SmallGroup(320,1276);
// by ID

G=gap.SmallGroup(320,1276);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,794,297,12550]);
// Polycyclic

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

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