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G = C4×C25⋊C4  order 400 = 24·52

Direct product of C4 and C25⋊C4

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×C25⋊C4, C25⋊C42, C100⋊2C4, C20.7F5, Dic25⋊2C4, D50.4C22, C5.(C4×F5), D25.(C2×C4), C50.3(C2×C4), C10.8(C2×F5), (C4×D25).6C2, C2.2(C2×C25⋊C4), (C2×C25⋊C4).2C2, SmallGroup(400,30)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C25 — C4×C25⋊C4
Chief series
C1 — C5 — C25 — D25 — D50 — C2×C25⋊C4 — C4×C25⋊C4
Lower central
C25 — C4×C25⋊C4
Upper central
C1 — C4

Generators and relations for C4×C25⋊C4
 G = < a,b,c | a4=b25=c4=1, ab=ba, ac=ca, cbc-1=b18 >

C1
C2
25C2
25C2
C5
C4
25C4
25C4
25C4
25C22
25C4
25C4
C10
5D5
5D5
C25
25C2×C4
25C2×C4
25C2×C4
C20
5F5
5F5
5Dic5
5F5
5D10
5F5
D25
C50
D25
25C42
5C2×F5
5C4×D5
5C2×F5
C25⋊C4
Dic25
C25⋊C4
C25⋊C4
D50
C100
C25⋊C4
5C4×F5
C2×C25⋊C4
C4×D25
C2×C25⋊C4
C4×C25⋊C4

Smallest permutation representation of C4×C25⋊C4
►On 100 points
Generators in S100
(1 97 26 63)(2 98 27 64)(3 99 28 65)(4 100 29 66)(5 76 30 67)(6 77 31 68)(7 78 32 69)(8 79 33 70)(9 80 34 71)(10 81 35 72)(11 82 36 73)(12 83 37 74)(13 84 38 75)(14 85 39 51)(15 86 40 52)(16 87 41 53)(17 88 42 54)(18 89 43 55)(19 90 44 56)(20 91 45 57)(21 92 46 58)(22 93 47 59)(23 94 48 60)(24 95 49 61)(25 96 50 62)

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

40 conjugacy classes

class 1 2A2B2C4A4B4C···4L 5  10 20A20B25A···25E50A···50E100A···100J
order1222444···4510202025···2550···50100···100
size1125251125···2544444···44···44···4

40 irreducible representations

dim111111444444
type+++++++
imageC1C2C2C4C4C4F5C2×F5C4×F5C25⋊C4C2×C25⋊C4C4×C25⋊C4
kernelC4×C25⋊C4C4×D25C2×C25⋊C4Dic25C100C25⋊C4C20C10C5C4C2C1
# reps1122281125510

Matrix representation of C4×C25⋊C4 ►in GL5(𝔽101)

910000
01000
00100
00010
00001
,
10000
03583163
038734669
03270478
023559327
,
10000
01000
00001
00100
0100100100100

G:=sub<GL(5,GF(101))| [91,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,35,38,32,23,0,8,73,70,55,0,31,46,4,93,0,63,69,78,27],[1,0,0,0,0,0,1,0,0,100,0,0,0,1,100,0,0,0,0,100,0,0,1,0,100] >;
 

C4×C25⋊C4 in GAP, Magma, Sage, TeX 

C_4\times C_{25}\rtimes C_4
 
% in TeX
 
G:=Group("C4xC25:C4");
 
// GroupNames label
 
G:=SmallGroup(400,30);
 
// by ID
 
G=gap.SmallGroup(400,30);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,3364,2896,178,5765,2897]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^25=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^18>;
 
// generators/relations
 

Export

Subgroup lattice of C4×C25⋊C4 in TeX

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