metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)⋊5F5, C20.7C42, D10.3C42, Dic5.3C42, C4⋊F5.C4, D5⋊C8⋊4C4, (C8×F5)⋊8C2, C8⋊D5⋊3C4, C8⋊F5⋊7C2, C8.19(C2×F5), C4.12(C4×F5), C40.19(C2×C4), C4.Dic5⋊6C4, C22⋊F5.3C4, D5.2(C8○D4), (C5×M4(2))⋊6C4, (C2×C10).7C42, C22.12(C4×F5), C4.53(C22×F5), C5⋊4(C8○2M4(2)), C20.93(C22×C4), C10.15(C2×C42), D5⋊C8.20C22, (C4×D5).89C23, (C8×D5).37C22, (C4×F5).19C22, D10.35(C22×C4), (D5×M4(2)).11C2, Dic5.34(C22×C4), D10.C23.4C2, (C2×C5⋊C8)⋊7C4, C5⋊C8.2(C2×C4), C2.16(C2×C4×F5), (C2×D5⋊C8).5C2, (C2×F5).4(C2×C4), (C2×C4).77(C2×F5), (C2×C20).47(C2×C4), C5⋊2C8.21(C2×C4), (C4×D5).18(C2×C4), (C2×C4×D5).194C22, (C2×Dic5).67(C2×C4), (C22×D5).53(C2×C4), SmallGroup(320,1066)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊5F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=dad-1=a5, ac=ca, bc=cb, dbd-1=a4b, dcd-1=c3 >
Subgroups: 394 in 130 conjugacy classes, 66 normal (32 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×8], M4(2), M4(2) [×3], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C8○2M4(2), C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), D5⋊C8 [×2], D5⋊C8 [×2], C4×F5 [×2], C4⋊F5 [×2], C2×C5⋊C8 [×2], C22⋊F5 [×2], C2×C4×D5, C8×F5 [×2], C8⋊F5 [×2], D5×M4(2), C2×D5⋊C8, D10.C23, M4(2)⋊5F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], F5, C2×C42, C8○D4 [×2], C2×F5 [×3], C8○2M4(2), C4×F5 [×2], C22×F5, C2×C4×F5, M4(2)⋊5F5
►On 80 points
(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 65)(2 70)(3 67)(4 72)(5 69)(6 66)(7 71)(8 68)(9 47)(10 44)(11 41)(12 46)(13 43)(14 48)(15 45)(16 42)(17 39)(18 36)(19 33)(20 38)(21 35)(22 40)(23 37)(24 34)(25 63)(26 60)(27 57)(28 62)(29 59)(30 64)(31 61)(32 58)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)
(1 14 34 58 75)(2 15 35 59 76)(3 16 36 60 77)(4 9 37 61 78)(5 10 38 62 79)(6 11 39 63 80)(7 12 40 64 73)(8 13 33 57 74)(17 25 52 66 41)(18 26 53 67 42)(19 27 54 68 43)(20 28 55 69 44)(21 29 56 70 45)(22 30 49 71 46)(23 31 50 72 47)(24 32 51 65 48)
(1 69 5 65)(2 66 6 70)(3 71 7 67)(4 68 8 72)(9 19 74 31)(10 24 75 28)(11 21 76 25)(12 18 77 30)(13 23 78 27)(14 20 79 32)(15 17 80 29)(16 22 73 26)(33 50 61 43)(34 55 62 48)(35 52 63 45)(36 49 64 42)(37 54 57 47)(38 51 58 44)(39 56 59 41)(40 53 60 46)
G:=sub<Sym(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), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,47)(10,44)(11,41)(12,46)(13,43)(14,48)(15,45)(16,42)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76), (1,14,34,58,75)(2,15,35,59,76)(3,16,36,60,77)(4,9,37,61,78)(5,10,38,62,79)(6,11,39,63,80)(7,12,40,64,73)(8,13,33,57,74)(17,25,52,66,41)(18,26,53,67,42)(19,27,54,68,43)(20,28,55,69,44)(21,29,56,70,45)(22,30,49,71,46)(23,31,50,72,47)(24,32,51,65,48), (1,69,5,65)(2,66,6,70)(3,71,7,67)(4,68,8,72)(9,19,74,31)(10,24,75,28)(11,21,76,25)(12,18,77,30)(13,23,78,27)(14,20,79,32)(15,17,80,29)(16,22,73,26)(33,50,61,43)(34,55,62,48)(35,52,63,45)(36,49,64,42)(37,54,57,47)(38,51,58,44)(39,56,59,41)(40,53,60,46)>;
G:=Group( (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), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,47)(10,44)(11,41)(12,46)(13,43)(14,48)(15,45)(16,42)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76), (1,14,34,58,75)(2,15,35,59,76)(3,16,36,60,77)(4,9,37,61,78)(5,10,38,62,79)(6,11,39,63,80)(7,12,40,64,73)(8,13,33,57,74)(17,25,52,66,41)(18,26,53,67,42)(19,27,54,68,43)(20,28,55,69,44)(21,29,56,70,45)(22,30,49,71,46)(23,31,50,72,47)(24,32,51,65,48), (1,69,5,65)(2,66,6,70)(3,71,7,67)(4,68,8,72)(9,19,74,31)(10,24,75,28)(11,21,76,25)(12,18,77,30)(13,23,78,27)(14,20,79,32)(15,17,80,29)(16,22,73,26)(33,50,61,43)(34,55,62,48)(35,52,63,45)(36,49,64,42)(37,54,57,47)(38,51,58,44)(39,56,59,41)(40,53,60,46) );
G=PermutationGroup([(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)], [(1,65),(2,70),(3,67),(4,72),(5,69),(6,66),(7,71),(8,68),(9,47),(10,44),(11,41),(12,46),(13,43),(14,48),(15,45),(16,42),(17,39),(18,36),(19,33),(20,38),(21,35),(22,40),(23,37),(24,34),(25,63),(26,60),(27,57),(28,62),(29,59),(30,64),(31,61),(32,58),(49,73),(50,78),(51,75),(52,80),(53,77),(54,74),(55,79),(56,76)], [(1,14,34,58,75),(2,15,35,59,76),(3,16,36,60,77),(4,9,37,61,78),(5,10,38,62,79),(6,11,39,63,80),(7,12,40,64,73),(8,13,33,57,74),(17,25,52,66,41),(18,26,53,67,42),(19,27,54,68,43),(20,28,55,69,44),(21,29,56,70,45),(22,30,49,71,46),(23,31,50,72,47),(24,32,51,65,48)], [(1,69,5,65),(2,66,6,70),(3,71,7,67),(4,68,8,72),(9,19,74,31),(10,24,75,28),(11,21,76,25),(12,18,77,30),(13,23,78,27),(14,20,79,32),(15,17,80,29),(16,22,73,26),(33,50,61,43),(34,55,62,48),(35,52,63,45),(36,49,64,42),(37,54,57,47),(38,51,58,44),(39,56,59,41),(40,53,60,46)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4N | 5 | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 8M | ··· | 8T | 10A | 10B | 20A | 20B | 20C | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | ··· | 10 | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4 | C8○D4 | F5 | C2×F5 | C2×F5 | C4×F5 | C4×F5 | M4(2)⋊5F5 |
| kernel | M4(2)⋊5F5 | C8×F5 | C8⋊F5 | D5×M4(2) | C2×D5⋊C8 | D10.C23 | C8⋊D5 | C4.Dic5 | C5×M4(2) | D5⋊C8 | C4⋊F5 | C2×C5⋊C8 | C22⋊F5 | D5 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of M4(2)⋊5F5 ►in GL6(𝔽41)
| 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 27 | 0 | 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 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(6,GF(41))| [14,0,0,0,0,0,0,27,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,27,0,0,0,0,38,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[0,38,0,0,0,0,27,0,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;
M4(2)⋊5F5 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_5F_5
% in TeX
G:=Group("M4(2):5F5"); // GroupNames label
G:=SmallGroup(320,1066);
// by ID
G=gap.SmallGroup(320,1066);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,136,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=d*a*d^-1=a^5,a*c=c*a,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations