metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).25D10, C8.10(C4×D5), C40.69(C2×C4), C8.C4⋊3D5, C8⋊D5.3C4, (C2×C8).68D10, C4.211(D4×D5), C40.6C4⋊7C2, (C4×D5).106D4, C20.370(C2×D4), C22.4(Q8×D5), (C2×Dic5).4Q8, D10.18(C4⋊C4), (C22×D5).3Q8, C20.53D4⋊8C2, (C2×C40).40C22, (D5×M4(2)).9C2, C5⋊2(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), C5⋊2C8.3(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×C5⋊2C8).77C22, (C2×C4).412(C22×D5), SmallGroup(320,520)
Series: Derived ►Chief ►Lower central ►Upper central
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], C5⋊2C8 [×2], C5⋊2C8 [×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×C5⋊2C8, 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
►On 80 points
(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 | 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