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## G = D4×C5⋊D4order 320 = 26·5

### Direct product of D4 and C5⋊D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C5⋊D4
G = < a,b,c,d,e | a4=b2=c5=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: 1702 in 428 conjugacy classes, 123 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×7], C22, C22 [×6], C22 [×38], C5, C2×C4 [×2], C2×C4 [×13], D4 [×4], D4 [×30], C23, C23 [×4], C23 [×23], D5 [×4], C10 [×3], C10 [×8], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×28], C24 [×2], C24 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20, D10 [×2], D10 [×16], C2×C10, C2×C10 [×6], C2×C10 [×20], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C41D4, C22×D4, C22×D4 [×3], C4×D5 [×2], D20 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×4], C5⋊D4 [×18], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C5×D4 [×6], C22×D5, C22×D5 [×2], C22×D5 [×10], C22×C10, C22×C10 [×4], C22×C10 [×10], D42, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×4], C2×C4×D5, C2×D20, D4×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4 [×10], C2×C5⋊D4 [×8], C22×C20, D4×C10 [×2], D4×C10 [×2], D4×C10 [×4], C23×D5 [×2], C23×C10 [×2], C4×C5⋊D4, C207D4, D4×Dic5, C23⋊D10 [×2], C202D4, Dic5⋊D4 [×2], C20⋊D4, C242D5 [×2], C2×D4×D5, C22×C5⋊D4 [×2], D4×C2×C10, D4×C5⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ 1+4, C5⋊D4 [×4], C22×D5 [×7], D42, D4×D5 [×2], C2×C5⋊D4 [×6], C23×D5, C2×D4×D5, D46D10, C22×C5⋊D4, D4×C5⋊D4

Smallest permutation representation of D4×C5⋊D4
On 80 points
Generators in S80
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)
(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 21 11 31)(2 25 12 35)(3 24 13 34)(4 23 14 33)(5 22 15 32)(6 26 16 36)(7 30 17 40)(8 29 18 39)(9 28 19 38)(10 27 20 37)(41 61 51 71)(42 65 52 75)(43 64 53 74)(44 63 54 73)(45 62 55 72)(46 66 56 76)(47 70 57 80)(48 69 58 79)(49 68 59 78)(50 67 60 77)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 C5⋊D4 2+ 1+4 D4×D5 D4⋊6D10 kernel D4×C5⋊D4 C4×C5⋊D4 C20⋊7D4 D4×Dic5 C23⋊D10 C20⋊2D4 Dic5⋊D4 C20⋊D4 C24⋊2D5 C2×D4×D5 C22×C5⋊D4 D4×C2×C10 C5⋊D4 C5×D4 C22×D4 C22×C4 C2×D4 C24 D4 C10 C22 C2 # reps 1 1 1 1 2 1 2 1 2 1 2 1 4 4 2 2 8 4 16 1 4 4

Matrix representation of D4×C5⋊D4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 39 0 0 1 40
,
 1 0 0 0 0 1 0 0 0 0 1 39 0 0 0 40
,
 0 40 0 0 1 34 0 0 0 0 1 0 0 0 0 1
,
 24 38 0 0 1 17 0 0 0 0 40 0 0 0 0 40
,
 1 34 0 0 0 40 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,39,40],[0,1,0,0,40,34,0,0,0,0,1,0,0,0,0,1],[24,1,0,0,38,17,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,34,40,0,0,0,0,1,0,0,0,0,1] >;

D4×C5⋊D4 in GAP, Magma, Sage, TeX

D_4\times C_5\rtimes D_4
% in TeX

G:=Group("D4xC5:D4");
// GroupNames label

G:=SmallGroup(320,1473);
// by ID

G=gap.SmallGroup(320,1473);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^5=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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