metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)⋊4F5, C20.4(C4⋊C4), (C4×D5).6Q8, C4.19(C4⋊F5), C4.Dic5⋊3C4, C22⋊F5.2C4, C22.9(C4×F5), (C4×D5).111D4, (C5×M4(2))⋊4C4, (C2×C10).4C42, D10.23(C4⋊C4), Dic5.7(C4⋊C4), (D5×M4(2)).6C2, C4.29(C22⋊F5), C20.29(C22⋊C4), C5⋊3(M4(2)⋊4C4), D10.7(C22⋊C4), Dic5.34(C22⋊C4), C2.17(D10.3Q8), C10.16(C2.C42), D10.C23.2C2, (C2×C5⋊C8)⋊2C4, (C2×C4).13(C2×F5), (C2×C4.F5).2C2, (C2×C20).38(C2×C4), (C2×C4×D5).189C22, (C2×Dic5).44(C2×C4), (C22×D5).37(C2×C4), SmallGroup(320,240)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊4F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a5, ac=ca, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c3 >
Subgroups: 370 in 90 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×7], C23, D5 [×2], C10, C10, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×3], M4(2), M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10, C2×C10, C42⋊C2, C2×M4(2) [×2], C5⋊2C8, C40, C5⋊C8 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×2], C22×D5, M4(2)⋊4C4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C4.F5 [×2], C4×F5, C4⋊F5, C2×C5⋊C8 [×2], C22⋊F5 [×2], C2×C4×D5, D5×M4(2), C2×C4.F5, D10.C23, M4(2)⋊4F5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C2×F5, M4(2)⋊4C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, M4(2)⋊4F5
►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 48)(2 45)(3 42)(4 47)(5 44)(6 41)(7 46)(8 43)(9 52)(10 49)(11 54)(12 51)(13 56)(14 53)(15 50)(16 55)(17 33)(18 38)(19 35)(20 40)(21 37)(22 34)(23 39)(24 36)(25 77)(26 74)(27 79)(28 76)(29 73)(30 78)(31 75)(32 80)(57 65)(58 70)(59 67)(60 72)(61 69)(62 66)(63 71)(64 68)
(1 55 29 40 66)(2 56 30 33 67)(3 49 31 34 68)(4 50 32 35 69)(5 51 25 36 70)(6 52 26 37 71)(7 53 27 38 72)(8 54 28 39 65)(9 74 21 63 41)(10 75 22 64 42)(11 76 23 57 43)(12 77 24 58 44)(13 78 17 59 45)(14 79 18 60 46)(15 80 19 61 47)(16 73 20 62 48)
(1 3)(2 47 6 43)(4 45 8 41)(5 7)(9 32 59 39)(10 77 64 24)(11 30 61 37)(12 75 58 22)(13 28 63 35)(14 73 60 20)(15 26 57 33)(16 79 62 18)(17 54 74 69)(19 52 76 67)(21 50 78 65)(23 56 80 71)(25 72 36 53)(27 70 38 51)(29 68 40 49)(31 66 34 55)(42 44)(46 48)
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), (1,48)(2,45)(3,42)(4,47)(5,44)(6,41)(7,46)(8,43)(9,52)(10,49)(11,54)(12,51)(13,56)(14,53)(15,50)(16,55)(17,33)(18,38)(19,35)(20,40)(21,37)(22,34)(23,39)(24,36)(25,77)(26,74)(27,79)(28,76)(29,73)(30,78)(31,75)(32,80)(57,65)(58,70)(59,67)(60,72)(61,69)(62,66)(63,71)(64,68), (1,55,29,40,66)(2,56,30,33,67)(3,49,31,34,68)(4,50,32,35,69)(5,51,25,36,70)(6,52,26,37,71)(7,53,27,38,72)(8,54,28,39,65)(9,74,21,63,41)(10,75,22,64,42)(11,76,23,57,43)(12,77,24,58,44)(13,78,17,59,45)(14,79,18,60,46)(15,80,19,61,47)(16,73,20,62,48), (1,3)(2,47,6,43)(4,45,8,41)(5,7)(9,32,59,39)(10,77,64,24)(11,30,61,37)(12,75,58,22)(13,28,63,35)(14,73,60,20)(15,26,57,33)(16,79,62,18)(17,54,74,69)(19,52,76,67)(21,50,78,65)(23,56,80,71)(25,72,36,53)(27,70,38,51)(29,68,40,49)(31,66,34,55)(42,44)(46,48)>;
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), (1,48)(2,45)(3,42)(4,47)(5,44)(6,41)(7,46)(8,43)(9,52)(10,49)(11,54)(12,51)(13,56)(14,53)(15,50)(16,55)(17,33)(18,38)(19,35)(20,40)(21,37)(22,34)(23,39)(24,36)(25,77)(26,74)(27,79)(28,76)(29,73)(30,78)(31,75)(32,80)(57,65)(58,70)(59,67)(60,72)(61,69)(62,66)(63,71)(64,68), (1,55,29,40,66)(2,56,30,33,67)(3,49,31,34,68)(4,50,32,35,69)(5,51,25,36,70)(6,52,26,37,71)(7,53,27,38,72)(8,54,28,39,65)(9,74,21,63,41)(10,75,22,64,42)(11,76,23,57,43)(12,77,24,58,44)(13,78,17,59,45)(14,79,18,60,46)(15,80,19,61,47)(16,73,20,62,48), (1,3)(2,47,6,43)(4,45,8,41)(5,7)(9,32,59,39)(10,77,64,24)(11,30,61,37)(12,75,58,22)(13,28,63,35)(14,73,60,20)(15,26,57,33)(16,79,62,18)(17,54,74,69)(19,52,76,67)(21,50,78,65)(23,56,80,71)(25,72,36,53)(27,70,38,51)(29,68,40,49)(31,66,34,55)(42,44)(46,48) );
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)], [(1,48),(2,45),(3,42),(4,47),(5,44),(6,41),(7,46),(8,43),(9,52),(10,49),(11,54),(12,51),(13,56),(14,53),(15,50),(16,55),(17,33),(18,38),(19,35),(20,40),(21,37),(22,34),(23,39),(24,36),(25,77),(26,74),(27,79),(28,76),(29,73),(30,78),(31,75),(32,80),(57,65),(58,70),(59,67),(60,72),(61,69),(62,66),(63,71),(64,68)], [(1,55,29,40,66),(2,56,30,33,67),(3,49,31,34,68),(4,50,32,35,69),(5,51,25,36,70),(6,52,26,37,71),(7,53,27,38,72),(8,54,28,39,65),(9,74,21,63,41),(10,75,22,64,42),(11,76,23,57,43),(12,77,24,58,44),(13,78,17,59,45),(14,79,18,60,46),(15,80,19,61,47),(16,73,20,62,48)], [(1,3),(2,47,6,43),(4,45,8,41),(5,7),(9,32,59,39),(10,77,64,24),(11,30,61,37),(12,75,58,22),(13,28,63,35),(14,73,60,20),(15,26,57,33),(16,79,62,18),(17,54,74,69),(19,52,76,67),(21,50,78,65),(23,56,80,71),(25,72,36,53),(27,70,38,51),(29,68,40,49),(31,66,34,55),(42,44),(46,48)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5 | 8A | 8B | 8C | ··· | 8H | 10A | 10B | 20A | 20B | 20C | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 5 | 5 | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 20 | ··· | 20 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | - | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | F5 | C2×F5 | M4(2)⋊4C4 | C4⋊F5 | C22⋊F5 | C4×F5 | M4(2)⋊4F5 |
| kernel | M4(2)⋊4F5 | D5×M4(2) | C2×C4.F5 | D10.C23 | C4.Dic5 | C5×M4(2) | C2×C5⋊C8 | C22⋊F5 | C4×D5 | C4×D5 | M4(2) | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of M4(2)⋊4F5 ►in GL8(𝔽41)
| 40 | 0 | 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 18 | 9 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 | 27 | 27 |
| 0 | 0 | 0 | 0 | 14 | 7 | 14 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 7 | 14 |
| 0 | 0 | 0 | 0 | 27 | 27 | 0 | 34 |
| 1 | 0 | 32 | 32 | 0 | 0 | 0 | 0 |
| 39 | 40 | 9 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 32 | 40 | 0 | 0 | 0 | 0 | 0 |
| 18 | 9 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(8,GF(41))| [40,0,18,0,0,0,0,0,0,0,9,32,0,0,0,0,4,1,1,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,0,34,14,0,27,0,0,0,0,0,7,14,27,0,0,0,0,27,14,7,0,0,0,0,0,27,0,14,34],[1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,40,0,0,0,0,32,9,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[32,18,0,0,0,0,0,0,32,9,0,0,0,0,0,0,40,1,9,0,0,0,0,0,0,1,0,32,0,0,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1] >;
M4(2)⋊4F5 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_4F_5
% in TeX
G:=Group("M4(2):4F5"); // GroupNames label
G:=SmallGroup(320,240);
// by ID
G=gap.SmallGroup(320,240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,136,851,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^5*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations