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G = M4(2).15D10  order 320 = 26·5

15th non-split extension by M4(2) of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).15D10, (C5×D4).15D4, C4.180(D4×D5), C5⋊2C8.48D4, (C5×Q8).15D4, C8.C22⋊2D5, C4○D4.26D10, C20.199(C2×D4), C5⋊7(D4.3D4), D4.6(C5⋊D4), D4.Dic5⋊7C2, (C2×Q8).69D10, Q8.6(C5⋊D4), D4⋊D10.2C2, C20.10D4⋊9C2, (C2×C20).18C23, C20.53D4⋊11C2, C20.46D4⋊12C2, (Q8×C10).96C22, C10.126(C4⋊D4), (C2×D20).135C22, C4.Dic5.27C22, C2.32(Dic5⋊D4), C22.15(D4⋊2D5), (C5×M4(2)).25C22, (C2×Q8⋊D5)⋊22C2, C4.55(C2×C5⋊D4), (C5×C8.C22)⋊6C2, (C2×C4).18(C22×D5), (C2×C10).38(C4○D4), (C5×C4○D4).16C22, (C2×C5⋊2C8).172C22, SmallGroup(320,830)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C20 — M4(2).15D10
Chief series
C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — D4⋊D10 — M4(2).15D10
Lower central
C5 — C10 — C2×C20 — M4(2).15D10
Upper central
C1 — C2 — C2×C4 — C8.C22

Generators and relations for M4(2).15D10
 G = < a,b,c,d | a8=b2=c10=1, d2=a2, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >

Subgroups: 414 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C5⋊2C8, C5⋊2C8, C40, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, D4.3D4, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4.Dic5, D4⋊D5, Q8⋊D5, C5×M4(2), C5×SD16, C5×Q16, C2×D20, Q8×C10, C5×C4○D4, C20.53D4, C20.46D4, C20.10D4, C2×Q8⋊D5, D4.Dic5, D4⋊D10, C5×C8.C22, M4(2).15D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.3D4, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊D4, M4(2).15D10

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

(2 64)(4 66)(6 68)(8 70)(10 62)(11 47)(13 49)(15 41)(17 43)(19 45)(21 76)(23 78)(25 80)(27 72)(29 74)(31 52)(33 54)(35 56)(37 58)(39 60)

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A10B10C10D10E10F20A20B20C20D20E···20J40A40B40C40D
order1222244445588888881010101010102020202020···2040404040
size112440224822810102020204022448844448···88888

38 irreducible representations

dim1111111122222222224448
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4C5⋊D4D4.3D4D4×D5D4⋊2D5M4(2).15D10
kernelM4(2).15D10C20.53D4C20.46D4C20.10D4C2×Q8⋊D5D4.Dic5D4⋊D10C5×C8.C22C5⋊2C8C5×D4C5×Q8C8.C22C2×C10M4(2)C2×Q8C4○D4D4Q8C5C4C22C1
# reps1111111121122222442222

Matrix representation of M4(2).15D10 ►in GL8(𝔽41)

0017400000
001240000
241000000
4017000000
000000011
000000260
0000113000
0000263000
,
10000000
01000000
00100000
00010000
00001000
00000100
000000400
000000040
,
00770000
0034400000
77000000
3440000000
00000010
00000001
00001000
00000100
,
0034340000
00170000
77000000
4034000000
000000139
000000040
00001000
000014000

G:=sub<GL(8,GF(41))| [0,0,24,40,0,0,0,0,0,0,1,17,0,0,0,0,17,1,0,0,0,0,0,0,40,24,0,0,0,0,0,0,0,0,0,0,0,0,11,26,0,0,0,0,0,0,30,30,0,0,0,0,0,26,0,0,0,0,0,0,11,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,39,40,0,0] >;
 

M4(2).15D10 in GAP, Magma, Sage, TeX 

M_4(2)._{15}D_{10}
 
% in TeX
 
G:=Group("M4(2).15D10");
 
// GroupNames label
 
G:=SmallGroup(320,830);
 
// by ID
 
G=gap.SmallGroup(320,830);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,184,1123,297,136,1684,851,438,102,12550]);
 
// 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^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

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