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## G = D5×2- 1+4order 320 = 26·5

### Direct product of D5 and 2- 1+4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5×2- 1+4
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — D5×2- 1+4
 Lower central C5 — C10 — D5×2- 1+4
 Upper central C1 — C2 — 2- 1+4

Generators and relations for D5×2- 1+4
G = < a,b,c,d,e,f | a5=b2=c4=d2=1, e2=f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 2190 in 794 conjugacy classes, 445 normal (8 characteristic)
C1, C2, C2 [×12], C4 [×10], C4 [×10], C22 [×5], C22 [×16], C5, C2×C4 [×15], C2×C4 [×55], D4 [×10], D4 [×30], Q8 [×10], Q8 [×30], C23 [×5], D5 [×2], D5 [×5], C10, C10 [×5], C22×C4 [×15], C2×D4 [×10], C2×Q8 [×5], C2×Q8 [×45], C4○D4 [×10], C4○D4 [×70], Dic5 [×10], C20 [×10], D10, D10 [×5], D10 [×10], C2×C10 [×5], C22×Q8 [×5], C2×C4○D4 [×10], 2- 1+4, 2- 1+4 [×15], Dic10 [×30], C4×D5 [×40], D20 [×10], C2×Dic5 [×15], C5⋊D4 [×20], C2×C20 [×15], C5×D4 [×10], C5×Q8 [×10], C22×D5 [×5], C2×2- 1+4, C2×Dic10 [×15], C2×C4×D5 [×15], C4○D20 [×30], D4×D5 [×10], D42D5 [×30], Q8×D5 [×30], Q82D5 [×10], Q8×C10 [×5], C5×C4○D4 [×10], C2×Q8×D5 [×5], Q8.10D10 [×5], D5×C4○D4 [×10], D4.10D10 [×10], C5×2- 1+4, D5×2- 1+4
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2- 1+4 [×2], C25, C22×D5 [×35], C2×2- 1+4, C23×D5 [×15], D5×C24, D5×2- 1+4

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

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

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

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

68 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I ··· 2M 4A ··· 4J 4K ··· 4T 5A 5B 10A 10B 10C ··· 10L 20A ··· 20T order 1 2 2 ··· 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 10 10 ··· 10 20 ··· 20 size 1 1 2 ··· 2 5 5 10 ··· 10 2 ··· 2 10 ··· 10 2 2 2 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D5 D10 D10 2- 1+4 D5×2- 1+4 kernel D5×2- 1+4 C2×Q8×D5 Q8.10D10 D5×C4○D4 D4.10D10 C5×2- 1+4 2- 1+4 C2×Q8 C4○D4 D5 C1 # reps 1 5 5 10 10 1 2 10 20 2 2

Matrix representation of D5×2- 1+4 in GL6(𝔽41)

 40 1 0 0 0 0 5 35 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
,
 1 0 0 0 0 0 36 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 0 40 0 0 0 0 0 0 32 9 9 39 0 0 0 0 9 0 0 0 0 9 0 0 0 0 0 0 0 9
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 40 1 1 18 0 0 40 0 0 0 0 0 32 0 9 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 32 32 2 0 0 0 0 9 0 0 0 0 9 0 0 0 0 0 40 40 32
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 9 0 0 0 32 9 9 39 0 0 9 0 0 0 0 0 0 0 0 32

G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,9,0,9,0,0,0,9,9,0,0,0,0,39,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,40,32,0,0,0,1,0,0,0,0,40,1,0,9,0,0,0,18,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,32,0,9,40,0,0,32,9,0,40,0,0,2,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,9,0,0,0,0,9,0,0,0,0,9,9,0,0,0,0,0,39,0,32] >;

D5×2- 1+4 in GAP, Magma, Sage, TeX

D_5\times 2_-^{1+4}
% in TeX

G:=Group("D5xES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1624);
// by ID

G=gap.SmallGroup(320,1624);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,297,136,851,12550]);
// Polycyclic

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

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