Copied to
clipboard

## 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 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C16 [×2], C16 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16 [×2], M5(2), M5(2) [×3], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×M5(2), C52C16 [×2], C80 [×2], C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, D5×C16 [×2], C80⋊C2 [×2], C20.4C8, C5×M5(2), D5×C2×C8, D5×M5(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], C22×C4, D10 [×3], M5(2) [×2], C22×C8, C4×D5 [×2], C22×D5, C2×M5(2), C8×D5 [×2], C2×C4×D5, D5×C2×C8, D5×M5(2)

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

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

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

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

׿
×
𝔽