metabelian, supersoluble, monomial
Aliases: D10.3F5, Dic5.3F5, C52⋊4M4(2), C5⋊3(C4.F5), C10.5(C2×F5), C52⋊4C8⋊2C2, C52⋊5C8⋊2C2, (D5×C10).4C4, C2.5(D5⋊F5), (D5×Dic5).5C2, (C5×Dic5).2C4, C5⋊1(C22.F5), C52⋊6C4.5C22, (C5×C10).12(C2×C4), SmallGroup(400,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊4M4(2)
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, dad=a-1, cbc-1=b2, bd=db, dcd=c5 >
10C2
2C5
2C5
5C4
5C22
25C4
2C10
2C10
2D5
10C10
25C2×C4
25C8
25C8
5Dic5
5C20
5Dic5
10Dic5
10Dic5
25M4(2)
10C5⋊C8
10C5⋊C8
Character table of C52⋊4M4(2)
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | |
| size | 1 | 1 | 10 | 10 | 25 | 25 | 4 | 4 | 8 | 8 | 50 | 50 | 50 | 50 | 4 | 4 | 8 | 8 | 20 | 20 | 20 | 20 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | -2 | 0 | 0 | 2i | -2i | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | -2 | 0 | 0 | -2i | 2i | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ11 | 4 | 4 | 4 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from F5 |
| ρ12 | 4 | 4 | 0 | -4 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ13 | 4 | 4 | 0 | 4 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | -4 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×F5 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -4 | 1 | 1 | -√5 | √5 | 0 | 0 | symplectic lifted from C22.F5, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -4 | 1 | 1 | √5 | -√5 | 0 | 0 | symplectic lifted from C22.F5, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -4 | 1 | 1 | 1 | 0 | 0 | √-5 | -√-5 | complex lifted from C4.F5 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -4 | 1 | 1 | 1 | 0 | 0 | -√-5 | √-5 | complex lifted from C4.F5 |
| ρ19 | 8 | 8 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | orthogonal lifted from D5⋊F5 |
| ρ20 | 8 | 8 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5⋊F5 |
| ρ21 | 8 | -8 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 2 | 2 | -3 | 2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ22 | 8 | -8 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -3 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 80 points
Generators in S80
(1 10 28 77 66)(2 78 11 67 29)(3 68 79 30 12)(4 31 69 13 80)(5 14 32 73 70)(6 74 15 71 25)(7 72 75 26 16)(8 27 65 9 76)(17 34 64 50 41)(18 51 35 42 57)(19 43 52 58 36)(20 59 44 37 53)(21 38 60 54 45)(22 55 39 46 61)(23 47 56 62 40)(24 63 48 33 49)
(1 66 77 28 10)(2 78 11 67 29)(3 12 30 79 68)(4 31 69 13 80)(5 70 73 32 14)(6 74 15 71 25)(7 16 26 75 72)(8 27 65 9 76)(17 34 64 50 41)(18 57 42 35 51)(19 43 52 58 36)(20 53 37 44 59)(21 38 60 54 45)(22 61 46 39 55)(23 47 56 62 40)(24 49 33 48 63)
(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 64)(2 61)(3 58)(4 63)(5 60)(6 57)(7 62)(8 59)(9 37)(10 34)(11 39)(12 36)(13 33)(14 38)(15 35)(16 40)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(41 77)(42 74)(43 79)(44 76)(45 73)(46 78)(47 75)(48 80)(49 69)(50 66)(51 71)(52 68)(53 65)(54 70)(55 67)(56 72)
G:=sub<Sym(80)| (1,10,28,77,66)(2,78,11,67,29)(3,68,79,30,12)(4,31,69,13,80)(5,14,32,73,70)(6,74,15,71,25)(7,72,75,26,16)(8,27,65,9,76)(17,34,64,50,41)(18,51,35,42,57)(19,43,52,58,36)(20,59,44,37,53)(21,38,60,54,45)(22,55,39,46,61)(23,47,56,62,40)(24,63,48,33,49), (1,66,77,28,10)(2,78,11,67,29)(3,12,30,79,68)(4,31,69,13,80)(5,70,73,32,14)(6,74,15,71,25)(7,16,26,75,72)(8,27,65,9,76)(17,34,64,50,41)(18,57,42,35,51)(19,43,52,58,36)(20,53,37,44,59)(21,38,60,54,45)(22,61,46,39,55)(23,47,56,62,40)(24,49,33,48,63), (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), (1,64)(2,61)(3,58)(4,63)(5,60)(6,57)(7,62)(8,59)(9,37)(10,34)(11,39)(12,36)(13,33)(14,38)(15,35)(16,40)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,69)(50,66)(51,71)(52,68)(53,65)(54,70)(55,67)(56,72)>;
G:=Group( (1,10,28,77,66)(2,78,11,67,29)(3,68,79,30,12)(4,31,69,13,80)(5,14,32,73,70)(6,74,15,71,25)(7,72,75,26,16)(8,27,65,9,76)(17,34,64,50,41)(18,51,35,42,57)(19,43,52,58,36)(20,59,44,37,53)(21,38,60,54,45)(22,55,39,46,61)(23,47,56,62,40)(24,63,48,33,49), (1,66,77,28,10)(2,78,11,67,29)(3,12,30,79,68)(4,31,69,13,80)(5,70,73,32,14)(6,74,15,71,25)(7,16,26,75,72)(8,27,65,9,76)(17,34,64,50,41)(18,57,42,35,51)(19,43,52,58,36)(20,53,37,44,59)(21,38,60,54,45)(22,61,46,39,55)(23,47,56,62,40)(24,49,33,48,63), (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), (1,64)(2,61)(3,58)(4,63)(5,60)(6,57)(7,62)(8,59)(9,37)(10,34)(11,39)(12,36)(13,33)(14,38)(15,35)(16,40)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,69)(50,66)(51,71)(52,68)(53,65)(54,70)(55,67)(56,72) );
G=PermutationGroup([(1,10,28,77,66),(2,78,11,67,29),(3,68,79,30,12),(4,31,69,13,80),(5,14,32,73,70),(6,74,15,71,25),(7,72,75,26,16),(8,27,65,9,76),(17,34,64,50,41),(18,51,35,42,57),(19,43,52,58,36),(20,59,44,37,53),(21,38,60,54,45),(22,55,39,46,61),(23,47,56,62,40),(24,63,48,33,49)], [(1,66,77,28,10),(2,78,11,67,29),(3,12,30,79,68),(4,31,69,13,80),(5,70,73,32,14),(6,74,15,71,25),(7,16,26,75,72),(8,27,65,9,76),(17,34,64,50,41),(18,57,42,35,51),(19,43,52,58,36),(20,53,37,44,59),(21,38,60,54,45),(22,61,46,39,55),(23,47,56,62,40),(24,49,33,48,63)], [(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)], [(1,64),(2,61),(3,58),(4,63),(5,60),(6,57),(7,62),(8,59),(9,37),(10,34),(11,39),(12,36),(13,33),(14,38),(15,35),(16,40),(17,28),(18,25),(19,30),(20,27),(21,32),(22,29),(23,26),(24,31),(41,77),(42,74),(43,79),(44,76),(45,73),(46,78),(47,75),(48,80),(49,69),(50,66),(51,71),(52,68),(53,65),(54,70),(55,67),(56,72)])
Matrix representation of C52⋊4M4(2) ►in GL10(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 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 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,40,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[14,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,40],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,40,0] >;
C52⋊4M4(2) in GAP, Magma, Sage, TeX
C_5^2\rtimes_4M_4(2)
% in TeX
G:=Group("C5^2:4M4(2)"); // GroupNames label
G:=SmallGroup(400,128);
// by ID
G=gap.SmallGroup(400,128);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,1444,970,496,8645,2897]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^3,d*a*d=a^-1,c*b*c^-1=b^2,b*d=d*b,d*c*d=c^5>;
// generators/relations
Export
Subgroup lattice of C52⋊4M4(2) in TeX
Character table of C52⋊4M4(2) in TeX