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## G = D5×C4⋊1D4order 320 = 26·5

### Direct product of D5 and C4⋊1D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×C4⋊1D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×D4×D5 — D5×C4⋊1D4
 Lower central C5 — C2×C10 — D5×C4⋊1D4
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for D5×C41D4
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, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 2286 in 498 conjugacy classes, 131 normal (12 characteristic)
C1, C2 [×3], C2 [×12], C4 [×6], C4 [×6], C22, C22 [×46], C5, C2×C4 [×3], C2×C4 [×15], D4 [×48], C23 [×4], C23 [×29], D5 [×4], D5 [×4], C10 [×3], C10 [×4], C42, C42 [×3], C22×C4 [×3], C2×D4 [×6], C2×D4 [×42], C24 [×4], Dic5 [×6], C20 [×6], D10 [×6], D10 [×28], C2×C10, C2×C10 [×12], C2×C42, C41D4, C41D4 [×7], C22×D4 [×6], C4×D5 [×12], D20 [×12], C2×Dic5 [×3], C5⋊D4 [×24], C2×C20 [×3], C5×D4 [×12], C22×D5, C22×D5 [×4], C22×D5 [×24], C22×C10 [×4], C2×C41D4, C4×Dic5 [×3], C4×C20, C2×C4×D5 [×3], C2×D20 [×6], D4×D5 [×24], C2×C5⋊D4 [×12], D4×C10 [×6], C23×D5 [×4], D5×C42, C204D4, C20⋊D4 [×6], C5×C41D4, C2×D4×D5 [×6], D5×C41D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D5, C2×D4 [×18], C24, D10 [×7], C41D4 [×4], C22×D4 [×3], C22×D5 [×7], C2×C41D4, D4×D5 [×6], C23×D5, C2×D4×D5 [×3], D5×C41D4

Smallest permutation representation of D5×C41D4
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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)
(1 54 19 49)(2 55 20 50)(3 51 16 46)(4 52 17 47)(5 53 18 48)(6 56 11 41)(7 57 12 42)(8 58 13 43)(9 59 14 44)(10 60 15 45)(21 76 36 61)(22 77 37 62)(23 78 38 63)(24 79 39 64)(25 80 40 65)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)
(1 64 14 74)(2 65 15 75)(3 61 11 71)(4 62 12 72)(5 63 13 73)(6 66 16 76)(7 67 17 77)(8 68 18 78)(9 69 19 79)(10 70 20 80)(21 41 31 51)(22 42 32 52)(23 43 33 53)(24 44 34 54)(25 45 35 55)(26 46 36 56)(27 47 37 57)(28 48 38 58)(29 49 39 59)(30 50 40 60)
(1 54)(2 55)(3 51)(4 52)(5 53)(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)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 D10 D10 D4×D5 kernel D5×C4⋊1D4 D5×C42 C20⋊4D4 C20⋊D4 C5×C4⋊1D4 C2×D4×D5 C4×D5 C4⋊1D4 C42 C2×D4 C4 # reps 1 1 1 6 1 6 12 2 2 12 12

Matrix representation of D5×C41D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 33 7 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 40 0 0 0 0 0 33 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 39 0 0 0 0 1 1
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 1 1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,1] >;

D5×C41D4 in GAP, Magma, Sage, TeX

D_5\times C_4\rtimes_1D_4
% in TeX

G:=Group("D5xC4:1D4");
// GroupNames label

G:=SmallGroup(320,1386);
// by ID

G=gap.SmallGroup(320,1386);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,570,185,12550]);
// 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,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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