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## G = M4(2).25D10order 320 = 26·5

### 8th non-split extension by M4(2) of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — M4(2).25D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C2×C8⋊D5 — M4(2).25D10
 Lower central C5 — C10 — C20 — M4(2).25D10
 Upper central C1 — C4 — C2×C4 — C8.C4

Generators and relations for M4(2).25D10
G = < a,b,c,d | a8=b2=1, c10=a2b, d2=a6b, bab=a5, cac-1=a-1b, dad-1=a3b, bc=cb, bd=db, dcd-1=a4c9 >

Subgroups: 318 in 102 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C8.C4, C8.C4, C2×M4(2), C52C8, C52C8, C40, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, M4(2).C4, C8×D5, C8⋊D5, C8⋊D5, C2×C52C8, C4.Dic5, C2×C40, C5×M4(2), C2×C4×D5, C40.6C4, C20.53D4, C5×C8.C4, C2×C8⋊D5, D5×M4(2), M4(2).25D10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C4×D5, C22×D5, M4(2).C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, M4(2).25D10

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - - + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 Q8 Q8 D5 D10 D10 C4×D5 M4(2).C4 D4×D5 Q8×D5 M4(2).25D10 kernel M4(2).25D10 C40.6C4 C20.53D4 C5×C8.C4 C2×C8⋊D5 D5×M4(2) C8⋊D5 C4×D5 C2×Dic5 C22×D5 C8.C4 C2×C8 M4(2) C8 C5 C4 C22 C1 # reps 1 1 2 1 1 2 8 2 1 1 2 2 4 8 2 2 2 8

Matrix representation of M4(2).25D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 30 0 15 0 0 9 0 12 0 0 0 0 9 0 40 0 0 40 0 17 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40
,
 6 6 0 0 0 0 35 1 0 0 0 0 0 0 24 0 34 0 0 0 0 32 0 33 0 0 40 0 17 0 0 0 0 1 0 9
,
 6 6 0 0 0 0 1 35 0 0 0 0 0 0 24 0 11 0 0 0 0 32 0 10 0 0 40 0 17 0 0 0 0 1 0 9

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,40,0,0,30,0,9,0,0,0,0,12,0,17,0,0,15,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,24,0,40,0,0,0,0,32,0,1,0,0,34,0,17,0,0,0,0,33,0,9],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,24,0,40,0,0,0,0,32,0,1,0,0,11,0,17,0,0,0,0,10,0,9] >;

M4(2).25D10 in GAP, Magma, Sage, TeX

M_4(2)._{25}D_{10}
% in TeX

G:=Group("M4(2).25D10");
// GroupNames label

G:=SmallGroup(320,520);
// by ID

G=gap.SmallGroup(320,520);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,219,58,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^10=a^2*b,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^9>;
// generators/relations

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