Copied to
clipboard

## G = C52⋊6C4order 100 = 22·52

### 2nd semidirect product of C52 and C4 acting via C4/C2=C2

Aliases: C526C4, C52Dic5, C10.3D5, C2.(C5⋊D5), (C5×C10).2C2, SmallGroup(100,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊6C4
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4
 Lower central C52 — C52⋊6C4
 Upper central C1 — C2

Generators and relations for C526C4
G = < a,b,c | a5=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b-1 >

Character table of C526C4

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 5K 5L 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L size 1 1 25 25 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 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 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -i i 1 1 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 4 ρ4 1 -1 i -i 1 1 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 4 ρ5 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ6 2 2 0 0 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 2 0 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 orthogonal lifted from D5 ρ8 2 2 0 0 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 2 0 0 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ10 2 2 0 0 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ11 2 2 0 0 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ12 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ13 2 2 0 0 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ14 2 2 0 0 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ15 2 2 0 0 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ16 2 2 0 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 orthogonal lifted from D5 ρ17 2 -2 0 0 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ18 2 -2 0 0 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ19 2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ20 2 -2 0 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 -2 -2 symplectic lifted from Dic5, Schur index 2 ρ21 2 -2 0 0 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ22 2 -2 0 0 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ23 2 -2 0 0 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ24 2 -2 0 0 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ25 2 -2 0 0 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ26 2 -2 0 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 -2 -2 symplectic lifted from Dic5, Schur index 2 ρ27 2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ28 2 -2 0 0 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2

Smallest permutation representation of C526C4
Regular action on 100 points
Generators in S100
(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)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)
(1 44 38 33 28)(2 45 39 34 29)(3 41 40 35 30)(4 42 36 31 26)(5 43 37 32 27)(6 100 25 16 11)(7 96 21 17 12)(8 97 22 18 13)(9 98 23 19 14)(10 99 24 20 15)(46 52 56 61 66)(47 53 57 62 67)(48 54 58 63 68)(49 55 59 64 69)(50 51 60 65 70)(71 80 81 86 91)(72 76 82 87 92)(73 77 83 88 93)(74 78 84 89 94)(75 79 85 90 95)
(1 18 48 82)(2 17 49 81)(3 16 50 85)(4 20 46 84)(5 19 47 83)(6 65 95 35)(7 64 91 34)(8 63 92 33)(9 62 93 32)(10 61 94 31)(11 70 90 30)(12 69 86 29)(13 68 87 28)(14 67 88 27)(15 66 89 26)(21 55 80 45)(22 54 76 44)(23 53 77 43)(24 52 78 42)(25 51 79 41)(36 99 56 74)(37 98 57 73)(38 97 58 72)(39 96 59 71)(40 100 60 75)

G:=sub<Sym(100)| (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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100), (1,44,38,33,28)(2,45,39,34,29)(3,41,40,35,30)(4,42,36,31,26)(5,43,37,32,27)(6,100,25,16,11)(7,96,21,17,12)(8,97,22,18,13)(9,98,23,19,14)(10,99,24,20,15)(46,52,56,61,66)(47,53,57,62,67)(48,54,58,63,68)(49,55,59,64,69)(50,51,60,65,70)(71,80,81,86,91)(72,76,82,87,92)(73,77,83,88,93)(74,78,84,89,94)(75,79,85,90,95), (1,18,48,82)(2,17,49,81)(3,16,50,85)(4,20,46,84)(5,19,47,83)(6,65,95,35)(7,64,91,34)(8,63,92,33)(9,62,93,32)(10,61,94,31)(11,70,90,30)(12,69,86,29)(13,68,87,28)(14,67,88,27)(15,66,89,26)(21,55,80,45)(22,54,76,44)(23,53,77,43)(24,52,78,42)(25,51,79,41)(36,99,56,74)(37,98,57,73)(38,97,58,72)(39,96,59,71)(40,100,60,75)>;

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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100), (1,44,38,33,28)(2,45,39,34,29)(3,41,40,35,30)(4,42,36,31,26)(5,43,37,32,27)(6,100,25,16,11)(7,96,21,17,12)(8,97,22,18,13)(9,98,23,19,14)(10,99,24,20,15)(46,52,56,61,66)(47,53,57,62,67)(48,54,58,63,68)(49,55,59,64,69)(50,51,60,65,70)(71,80,81,86,91)(72,76,82,87,92)(73,77,83,88,93)(74,78,84,89,94)(75,79,85,90,95), (1,18,48,82)(2,17,49,81)(3,16,50,85)(4,20,46,84)(5,19,47,83)(6,65,95,35)(7,64,91,34)(8,63,92,33)(9,62,93,32)(10,61,94,31)(11,70,90,30)(12,69,86,29)(13,68,87,28)(14,67,88,27)(15,66,89,26)(21,55,80,45)(22,54,76,44)(23,53,77,43)(24,52,78,42)(25,51,79,41)(36,99,56,74)(37,98,57,73)(38,97,58,72)(39,96,59,71)(40,100,60,75) );

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),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100)], [(1,44,38,33,28),(2,45,39,34,29),(3,41,40,35,30),(4,42,36,31,26),(5,43,37,32,27),(6,100,25,16,11),(7,96,21,17,12),(8,97,22,18,13),(9,98,23,19,14),(10,99,24,20,15),(46,52,56,61,66),(47,53,57,62,67),(48,54,58,63,68),(49,55,59,64,69),(50,51,60,65,70),(71,80,81,86,91),(72,76,82,87,92),(73,77,83,88,93),(74,78,84,89,94),(75,79,85,90,95)], [(1,18,48,82),(2,17,49,81),(3,16,50,85),(4,20,46,84),(5,19,47,83),(6,65,95,35),(7,64,91,34),(8,63,92,33),(9,62,93,32),(10,61,94,31),(11,70,90,30),(12,69,86,29),(13,68,87,28),(14,67,88,27),(15,66,89,26),(21,55,80,45),(22,54,76,44),(23,53,77,43),(24,52,78,42),(25,51,79,41),(36,99,56,74),(37,98,57,73),(38,97,58,72),(39,96,59,71),(40,100,60,75)]])

C526C4 is a maximal subgroup of
C524C8  C525C8  D5×Dic5  C522D4  C522Q8  C524Q8  C4×C5⋊D5  C527D4  C522C12  C30.D5  He55C4  C50.D5  C5312C4  C53⋊C4
C526C4 is a maximal quotient of
C527C8  C30.D5  C50.D5  He56C4  C5312C4  C53⋊C4

Matrix representation of C526C4 in GL5(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 40 6 0 0 0 0 0 40 6 0 0 0 35 35
,
 1 0 0 0 0 0 0 1 0 0 0 40 6 0 0 0 0 0 0 1 0 0 0 40 6
,
 32 0 0 0 0 0 12 18 0 0 0 8 29 0 0 0 0 0 6 6 0 0 0 1 35

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,40,35,0,0,0,6,35],[1,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,1,6],[32,0,0,0,0,0,12,8,0,0,0,18,29,0,0,0,0,0,6,1,0,0,0,6,35] >;

C526C4 in GAP, Magma, Sage, TeX

C_5^2\rtimes_6C_4
% in TeX

G:=Group("C5^2:6C4");
// GroupNames label

G:=SmallGroup(100,7);
// by ID

G=gap.SmallGroup(100,7);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,8,194,1283]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

׿
×
𝔽