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G = M4(2).13D10order 320 = 26·5

13rd non-split extension by M4(2) of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).13D10, C8⋊C22.D5, (C5×D4).12D4, C4.179(D4×D5), C52C8.47D4, (C5×Q8).12D4, C4○D4.25D10, (C2×D4).80D10, C20.196(C2×D4), C56(D4.3D4), D4.5(C5⋊D4), D4.Dic56C2, Q8.5(C5⋊D4), D4.9D104C2, C20.D410C2, C4.12D209C2, (C2×C20).15C23, C20.53D410C2, C10.125(C4⋊D4), (D4×C10).105C22, C4.Dic5.25C22, C2.31(Dic5⋊D4), C22.14(D42D5), (C5×M4(2)).23C22, (C2×Dic10).138C22, C4.52(C2×C5⋊D4), (C2×D4.D5)⋊22C2, (C5×C8⋊C22).1C2, (C2×C4).15(C22×D5), (C2×C10).37(C4○D4), (C5×C4○D4).13C22, (C2×C52C8).171C22, SmallGroup(320,827)

Series: Derived Chief Lower central Upper central

C1C2×C20 — M4(2).13D10
C1C5C10C20C2×C20C2×Dic10D4.9D10 — M4(2).13D10
C5C10C2×C20 — M4(2).13D10
C1C2C2×C4C8⋊C22

Generators and relations for M4(2).13D10
 G = < a,b,c,d | a8=b2=c10=1, d2=a2, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 350 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×5], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8 [×2], C23, C10, C10 [×3], C2×C8 [×2], M4(2), M4(2) [×3], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20 [×2], C20, C2×C10, C2×C10 [×3], C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8 [×2], C52C8 [×2], C40, Dic10 [×2], C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4 [×3], C5×Q8, C22×C10, D4.3D4, C2×C52C8, C2×C52C8, C4.Dic5 [×2], C4.Dic5, D4.D5 [×3], C5⋊Q16, C5×M4(2), C5×D8, C5×SD16, C2×Dic10, D4×C10, C5×C4○D4, C20.53D4, C4.12D20, C20.D4, C2×D4.D5, D4.Dic5, D4.9D10, C5×C8⋊C22, M4(2).13D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4.3D4, D4×D5, D42D5, C2×C5⋊D4, Dic5⋊D4, M4(2).13D10

Smallest permutation representation of M4(2).13D10
On 80 points
Generators in S80
(1 51 56 6 33 18 13 38)(2 39 14 19 34 7 57 52)(3 53 58 8 35 20 15 40)(4 31 16 11 36 9 59 54)(5 55 60 10 37 12 17 32)(21 73 78 26 63 46 41 68)(22 69 42 47 64 27 79 74)(23 75 80 28 65 48 43 70)(24 61 44 49 66 29 71 76)(25 77 72 30 67 50 45 62)
(1 33)(3 35)(5 37)(7 39)(9 31)(11 54)(13 56)(15 58)(17 60)(19 52)(22 64)(24 66)(26 68)(28 70)(30 62)(42 79)(44 71)(46 73)(48 75)(50 77)
(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 29 56 76 33 61 13 49)(2 75 14 28 34 48 57 70)(3 27 58 74 35 69 15 47)(4 73 16 26 36 46 59 68)(5 25 60 72 37 67 17 45)(6 71 18 24 38 44 51 66)(7 23 52 80 39 65 19 43)(8 79 20 22 40 42 53 64)(9 21 54 78 31 63 11 41)(10 77 12 30 32 50 55 62)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222244445588888881010101010···1020202020202040404040
size112482244022810102020204022448···84444888888

38 irreducible representations

dim1111111122222222224448
type++++++++++++++++--
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4C5⋊D4D4.3D4D4×D5D42D5M4(2).13D10
kernelM4(2).13D10C20.53D4C4.12D20C20.D4C2×D4.D5D4.Dic5D4.9D10C5×C8⋊C22C52C8C5×D4C5×Q8C8⋊C22C2×C10M4(2)C2×D4C4○D4D4Q8C5C4C22C1
# reps1111111121122222442222

Matrix representation of M4(2).13D10 in GL6(𝔽41)

010000
100000
0000400
000011
001200
0004000
,
4000000
0400000
0040000
0004000
000010
000001
,
3870000
7380000
000010
000001
001000
000100
,
32300000
1190000
00003030
00002611
0003000
0015000

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,40,0,0,40,1,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,7,0,0,0,0,7,38,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[32,11,0,0,0,0,30,9,0,0,0,0,0,0,0,0,0,15,0,0,0,0,30,0,0,0,30,26,0,0,0,0,30,11,0,0] >;

M4(2).13D10 in GAP, Magma, Sage, TeX

M_4(2)._{13}D_{10}
% in TeX

G:=Group("M4(2).13D10");
// GroupNames label

G:=SmallGroup(320,827);
// by ID

G=gap.SmallGroup(320,827);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,219,1123,297,136,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^2,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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