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G = M5(2)⋊D5order 320 = 26·5

3rd semidirect product of M5(2) and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.46D4, C8.25D20, M5(2)⋊3D5, C20.20M4(2), C53(C23.C8), C22.5(C8×D5), (C2×C8).152D10, C8.46(C5⋊D4), (C5×M5(2))⋊7C2, (C2×Dic5).1C8, C40.9C410C2, (C22×D5).1C8, C4.10(C8⋊D5), C10.24(C22⋊C8), (C2×C40).220C22, C4.42(D10⋊C4), C2.10(D101C8), C20.104(C22⋊C4), (C2×C4×D5).1C4, (C2×C52C8).2C4, (C2×C10).18(C2×C8), (C2×C4).137(C4×D5), (C2×C20).225(C2×C4), (C2×C8⋊D5).14C2, SmallGroup(320,72)

Series: Derived Chief Lower central Upper central

C1C2×C10 — M5(2)⋊D5
C1C5C10C20C40C2×C40C2×C8⋊D5 — M5(2)⋊D5
C5C10C2×C10 — M5(2)⋊D5
C1C4C2×C8M5(2)

Generators and relations for M5(2)⋊D5
 G = < a,b,c,d | a16=b2=c5=d2=1, bab=a9, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 214 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4 [×2], C4, C22, C22 [×2], C5, C8 [×2], C8, C2×C4, C2×C4 [×3], C23, D5, C10, C10, C16 [×2], C2×C8, C2×C8, M4(2) [×2], C22×C4, Dic5, C20 [×2], D10 [×2], C2×C10, M5(2), M5(2), C2×M4(2), C52C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C23.C8, C52C16, C80, C8⋊D5 [×2], C2×C52C8, C2×C40, C2×C4×D5, C40.9C4, C5×M5(2), C2×C8⋊D5, M5(2)⋊D5
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C4×D5, D20, C5⋊D4, C23.C8, C8×D5, C8⋊D5, D10⋊C4, D101C8, M5(2)⋊D5

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

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

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

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

62 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B8A8B8C8D8E8F10A10B10C10D16A16B16C16D16E16F16G16H20A20B20C20D20E20F40A···40H40I40J40K40L80A···80P
order122244445588888810101010161616161616161620202020202040···404040404080···80
size1122011220222222202022444444202020202222442···244444···4

62 irreducible representations

dim1111111122222222244
type++++++++
imageC1C2C2C2C4C4C8C8D4D5M4(2)D10D20C5⋊D4C4×D5C8⋊D5C8×D5C23.C8M5(2)⋊D5
kernelM5(2)⋊D5C40.9C4C5×M5(2)C2×C8⋊D5C2×C52C8C2×C4×D5C2×Dic5C22×D5C40M5(2)C20C2×C8C8C8C2×C4C4C22C5C1
# reps1111224422224448828

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

0010
0001
1544300
1988700
,
1000
0100
002400
000240
,
0100
24018900
0001
00240189
,
0100
1000
0001
0010
G:=sub<GL(4,GF(241))| [0,0,154,198,0,0,43,87,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[0,240,0,0,1,189,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

M5(2)⋊D5 in GAP, Magma, Sage, TeX

M_{5(2})\rtimes D_5
% in TeX

G:=Group("M5(2):D5");
// GroupNames label

G:=SmallGroup(320,72);
// by ID

G=gap.SmallGroup(320,72);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,100,570,102,12550]);
// Polycyclic

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

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