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G = C52⋊6C4  order 100 = 22·52

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

metabelian, supersoluble, monomial, A-group

Aliases: C52⋊6C4, C5⋊2Dic5, 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 C52⋊6C4
 G = < a,b,c | a5=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b-1 >

C1
C2
C5
C5
C5
C5
C5
C5
25C4
C10
C10
C10
C10
C10
C10
C52
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
C5×C10
C52⋊6C4

Character table of C52⋊6C4

 class 124A4B5A5B5C5D5E5F5G5H5I5J5K5L10A10B10C10D10E10F10G10H10I10J10K10L
 size 112525222222222222222222222222
ρ11111111111111111111111111111    trivial
ρ211-1-1111111111111111111111111    linear of order 2
ρ31-1-ii111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ41-1i-i111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ52200-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/222-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ62200-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ72200-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/222-1-√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+√5/222    orthogonal lifted from D5
ρ822002-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ92200-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ102200-1-√5/2-1-√5/2-1+√5/22-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ112200-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ122200-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/222-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ132200-1+√5/2-1+√5/2-1-√5/22-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ142200-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ1522002-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ162200-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/222-1+√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-√5/222    orthogonal lifted from D5
ρ172-2002-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/21-√5/2-21-√5/21+√5/21+√5/21+√5/21-√5/2-21-√5/21+√5/21-√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ182-2002-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/21+√5/2-21+√5/21-√5/21-√5/21-√5/21+√5/2-21+√5/21-√5/21+√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ192-200-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-21+√5/21-√5/21-√5/21+√5/2-21+√5/21-√5/21-√5/21+√5/21+√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ202-200-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/222-1-√5/21+√5/21+√5/21+√5/21+√5/21+√5/21-√5/21-√5/21-√5/21-√5/21-√5/2-2-2    symplectic lifted from Dic5, Schur index 2
ρ212-200-1-√5/2-1-√5/2-1+√5/22-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2-21+√5/21+√5/21-√5/2-21-√5/21-√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ222-200-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/21-√5/21+√5/2-21+√5/21-√5/21+√5/21-√5/21-√5/21+√5/2-21+√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ232-200-1+√5/2-1+√5/2-1-√5/22-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2-21-√5/21-√5/21+√5/2-21+√5/21+√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ242-200-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/21+√5/21-√5/2-21-√5/21+√5/21-√5/21+√5/21+√5/21-√5/2-21-√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ252-200-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/21-√5/21-√5/21+√5/2-21+√5/21+√5/2-21+√5/21-√5/21-√5/21+√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ262-200-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/222-1+√5/21-√5/21-√5/21-√5/21-√5/21-√5/21+√5/21+√5/21+√5/21+√5/21+√5/2-2-2    symplectic lifted from Dic5, Schur index 2
ρ272-200-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-21-√5/21+√5/21+√5/21-√5/2-21-√5/21+√5/21+√5/21-√5/21-√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ282-200-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/21+√5/21+√5/21-√5/2-21-√5/21-√5/2-21-√5/21+√5/21+√5/21-√5/21+√5/2    symplectic lifted from Dic5, Schur index 2

Smallest permutation representation of C52⋊6C4
►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)]])
 

C52⋊6C4 is a maximal subgroup of
 C52⋊4C8  C52⋊5C8  D5×Dic5  C52⋊2D4  C52⋊2Q8  C52⋊4Q8  C4×C5⋊D5  C52⋊7D4  C52⋊2C12  C30.D5  He5⋊5C4  C50.D5  C53⋊12C4  C53⋊C4
C52⋊6C4 is a maximal quotient of
 C52⋊7C8  C30.D5  C50.D5  He5⋊6C4  C53⋊12C4  C53⋊C4

Matrix representation of C52⋊6C4 ►in GL5(𝔽41)

10000
00100
040600
000406
0003535
,
10000
00100
040600
00001
000406
,
320000
0121800
082900
00066
000135

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] >;
 

C52⋊6C4 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

Subgroup lattice of C52⋊6C4 in TeX
Character table of C52⋊6C4 in TeX

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