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, C4.12(C4×F5), C8.19(C2×F5), C40.19(C2×C4), C4.Dic5⋊6C4, C22⋊F5.3C4, D5.2(C8○D4), (C5×M4(2))⋊6C4, (C2×C10).7C42, C4.53(C22×F5), C22.12(C4×F5), C5⋊4(C8○2M4(2)), C10.15(C2×C42), C20.93(C22×C4), D5⋊C8.20C22, (C8×D5).37C22, (C4×D5).89C23, (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
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
Generators and relations
G = < a,b,c,d | a8=b2=c5=d4=1, bab=dad-1=a5, ac=ca, bc=cb, dbd-1=a4b, dcd-1=c3 >
►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 63)(2 60)(3 57)(4 62)(5 59)(6 64)(7 61)(8 58)(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 66)(26 71)(27 68)(28 65)(29 70)(30 67)(31 72)(32 69)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)
(1 66 34 16 75)(2 67 35 9 76)(3 68 36 10 77)(4 69 37 11 78)(5 70 38 12 79)(6 71 39 13 80)(7 72 40 14 73)(8 65 33 15 74)(17 43 52 64 26)(18 44 53 57 27)(19 45 54 58 28)(20 46 55 59 29)(21 47 56 60 30)(22 48 49 61 31)(23 41 50 62 32)(24 42 51 63 25)
(1 59 5 63)(2 64 6 60)(3 61 7 57)(4 58 8 62)(9 26 39 56)(10 31 40 53)(11 28 33 50)(12 25 34 55)(13 30 35 52)(14 27 36 49)(15 32 37 54)(16 29 38 51)(17 80 47 67)(18 77 48 72)(19 74 41 69)(20 79 42 66)(21 76 43 71)(22 73 44 68)(23 78 45 65)(24 75 46 70)
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), (1,63)(2,60)(3,57)(4,62)(5,59)(6,64)(7,61)(8,58)(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)(25,66)(26,71)(27,68)(28,65)(29,70)(30,67)(31,72)(32,69)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76), (1,66,34,16,75)(2,67,35,9,76)(3,68,36,10,77)(4,69,37,11,78)(5,70,38,12,79)(6,71,39,13,80)(7,72,40,14,73)(8,65,33,15,74)(17,43,52,64,26)(18,44,53,57,27)(19,45,54,58,28)(20,46,55,59,29)(21,47,56,60,30)(22,48,49,61,31)(23,41,50,62,32)(24,42,51,63,25), (1,59,5,63)(2,64,6,60)(3,61,7,57)(4,58,8,62)(9,26,39,56)(10,31,40,53)(11,28,33,50)(12,25,34,55)(13,30,35,52)(14,27,36,49)(15,32,37,54)(16,29,38,51)(17,80,47,67)(18,77,48,72)(19,74,41,69)(20,79,42,66)(21,76,43,71)(22,73,44,68)(23,78,45,65)(24,75,46,70)>;
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), (1,63)(2,60)(3,57)(4,62)(5,59)(6,64)(7,61)(8,58)(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)(25,66)(26,71)(27,68)(28,65)(29,70)(30,67)(31,72)(32,69)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76), (1,66,34,16,75)(2,67,35,9,76)(3,68,36,10,77)(4,69,37,11,78)(5,70,38,12,79)(6,71,39,13,80)(7,72,40,14,73)(8,65,33,15,74)(17,43,52,64,26)(18,44,53,57,27)(19,45,54,58,28)(20,46,55,59,29)(21,47,56,60,30)(22,48,49,61,31)(23,41,50,62,32)(24,42,51,63,25), (1,59,5,63)(2,64,6,60)(3,61,7,57)(4,58,8,62)(9,26,39,56)(10,31,40,53)(11,28,33,50)(12,25,34,55)(13,30,35,52)(14,27,36,49)(15,32,37,54)(16,29,38,51)(17,80,47,67)(18,77,48,72)(19,74,41,69)(20,79,42,66)(21,76,43,71)(22,73,44,68)(23,78,45,65)(24,75,46,70) );
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)], [(1,63),(2,60),(3,57),(4,62),(5,59),(6,64),(7,61),(8,58),(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),(25,66),(26,71),(27,68),(28,65),(29,70),(30,67),(31,72),(32,69),(49,73),(50,78),(51,75),(52,80),(53,77),(54,74),(55,79),(56,76)], [(1,66,34,16,75),(2,67,35,9,76),(3,68,36,10,77),(4,69,37,11,78),(5,70,38,12,79),(6,71,39,13,80),(7,72,40,14,73),(8,65,33,15,74),(17,43,52,64,26),(18,44,53,57,27),(19,45,54,58,28),(20,46,55,59,29),(21,47,56,60,30),(22,48,49,61,31),(23,41,50,62,32),(24,42,51,63,25)], [(1,59,5,63),(2,64,6,60),(3,61,7,57),(4,58,8,62),(9,26,39,56),(10,31,40,53),(11,28,33,50),(12,25,34,55),(13,30,35,52),(14,27,36,49),(15,32,37,54),(16,29,38,51),(17,80,47,67),(18,77,48,72),(19,74,41,69),(20,79,42,66),(21,76,43,71),(22,73,44,68),(23,78,45,65),(24,75,46,70)])
Matrix representation ►G ⊆ 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] >;
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 |
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