Copied to
clipboard

G = M4(2).25D10order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).25D10, C8.10(C4×D5), C40.69(C2×C4), C8.C43D5, C8⋊D5.3C4, (C2×C8).68D10, C4.211(D4×D5), C40.6C47C2, (C4×D5).106D4, C20.370(C2×D4), C22.4(Q8×D5), (C2×Dic5).4Q8, D10.18(C4⋊C4), (C22×D5).3Q8, C20.53D48C2, (C2×C40).40C22, (D5×M4(2)).9C2, C52(M4(2).C4), Dic5.20(C4⋊C4), C20.112(C22×C4), (C2×C20).309C23, C4.Dic5.13C22, (C5×M4(2)).19C22, C4.83(C2×C4×D5), C2.18(D5×C4⋊C4), C10.40(C2×C4⋊C4), C52C8.3(C2×C4), (C4×D5).8(C2×C4), (C2×C10).2(C2×Q8), (C5×C8.C4)⋊3C2, (C2×C8⋊D5).1C2, (C2×C4×D5).47C22, (C2×C52C8).77C22, (C2×C4).412(C22×D5), SmallGroup(320,520)

Series: Derived Chief Lower central Upper central

C1C20 — M4(2).25D10
C1C5C10C20C2×C20C2×C4×D5C2×C8⋊D5 — M4(2).25D10
C5C10C20 — M4(2).25D10
C1C4C2×C4C8.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 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10, C2×C8, C2×C8 [×3], M4(2) [×2], M4(2) [×8], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10, C2×C10, C8.C4, C8.C4 [×3], C2×M4(2) [×3], C52C8 [×2], C52C8 [×2], C40 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, M4(2).C4, C8×D5 [×2], C8⋊D5 [×4], C8⋊D5 [×2], C2×C52C8, C4.Dic5 [×2], C2×C40, C5×M4(2) [×2], C2×C4×D5, C40.6C4, C20.53D4 [×2], C5×C8.C4, C2×C8⋊D5, D5×M4(2) [×2], M4(2).25D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C4×D5 [×2], 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 48 11 78 21 68 31 58)(2 79 12 69 22 59 32 49)(3 70 13 60 23 50 33 80)(4 61 14 51 24 41 34 71)(5 52 15 42 25 72 35 62)(6 43 16 73 26 63 36 53)(7 74 17 64 27 54 37 44)(8 65 18 55 28 45 38 75)(9 56 19 46 29 76 39 66)(10 47 20 77 30 67 40 57)
(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 74 71 64 61 54 51 44)(42 63 72 53 62 43 52 73)(45 70 75 60 65 50 55 80)(46 59 76 49 66 79 56 69)(47 48 77 78 67 68 57 58)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A···8F8G···8L10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order1222244444558···88···81010101020202020202040···4040···40
size11210101121010224···420···2022442222444···48···8

50 irreducible representations

dim111111122222224444
type+++++++--++++-
imageC1C2C2C2C2C2C4D4Q8Q8D5D10D10C4×D5M4(2).C4D4×D5Q8×D5M4(2).25D10
kernelM4(2).25D10C40.6C4C20.53D4C5×C8.C4C2×C8⋊D5D5×M4(2)C8⋊D5C4×D5C2×Dic5C22×D5C8.C4C2×C8M4(2)C8C5C4C22C1
# reps112112821122482228

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

3200000
0320000
00030015
0090120
0009040
00400170
,
4000000
0400000
001000
0004000
000010
0000040
,
660000
3510000
00240340
00032033
00400170
000109
,
660000
1350000
00240110
00032010
00400170
000109

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

׿
×
𝔽