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## G = D5×M5(2)  order 320 = 26·5

### Direct product of D5 and M5(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5×M5(2)
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D5×C2×C8 — D5×M5(2)
 Lower central C5 — C10 — D5×M5(2)
 Upper central C1 — C8 — M5(2)

Generators and relations for D5×M5(2)
G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >

Subgroups: 238 in 90 conjugacy classes, 53 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C16, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×C16, M5(2), M5(2), C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×M5(2), C52C16, C80, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, D5×C16, C80⋊C2, C20.4C8, C5×M5(2), D5×C2×C8, D5×M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, D10, M5(2), C22×C8, C4×D5, C22×D5, C2×M5(2), C8×D5, C2×C4×D5, D5×C2×C8, D5×M5(2)

Smallest permutation representation of D5×M5(2)
On 80 points
Generators in S80
(1 20 64 74 37)(2 21 49 75 38)(3 22 50 76 39)(4 23 51 77 40)(5 24 52 78 41)(6 25 53 79 42)(7 26 54 80 43)(8 27 55 65 44)(9 28 56 66 45)(10 29 57 67 46)(11 30 58 68 47)(12 31 59 69 48)(13 32 60 70 33)(14 17 61 71 34)(15 18 62 72 35)(16 19 63 73 36)
(1 45)(2 46)(3 47)(4 48)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 79)(18 80)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 10A 10B 10C 10D 16A ··· 16H 16I ··· 16P 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 16 ··· 16 16 ··· 16 20 20 20 20 20 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 5 5 10 1 1 2 5 5 10 2 2 1 1 1 1 2 2 5 5 5 5 10 10 2 2 4 4 2 ··· 2 10 ··· 10 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D5 D10 D10 M5(2) C4×D5 C4×D5 C8×D5 C8×D5 D5×M5(2) kernel D5×M5(2) D5×C16 C80⋊C2 C20.4C8 C5×M5(2) D5×C2×C8 C8×D5 C2×C5⋊2C8 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 M5(2) C16 C2×C8 D5 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 8 4 4 2 4 2 8 4 4 8 8 8

Matrix representation of D5×M5(2) in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 189
,
 240 0 0 0 0 240 0 0 0 0 0 1 0 0 1 0
,
 1 54 0 0 111 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 116 240 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,189],[240,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0],[1,111,0,0,54,240,0,0,0,0,1,0,0,0,0,1],[1,116,0,0,0,240,0,0,0,0,1,0,0,0,0,1] >;

D5×M5(2) in GAP, Magma, Sage, TeX

D_5\times M_5(2)
% in TeX

G:=Group("D5xM5(2)");
// GroupNames label

G:=SmallGroup(320,533);
// by ID

G=gap.SmallGroup(320,533);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,58,80,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^9>;
// generators/relations

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