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G = C52.4C4order 208 = 24·13

1st non-split extension by C52 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C52.4C4, C4.Dic13, C4.15D26, C134M4(2), C22.Dic13, C52.15C22, C132C85C2, (C2×C26).5C4, (C2×C52).5C2, (C2×C4).2D13, C26.14(C2×C4), C2.3(C2×Dic13), SmallGroup(208,10)

Series: Derived Chief Lower central Upper central

C1C26 — C52.4C4
C1C13C26C52C132C8 — C52.4C4
C13C26 — C52.4C4
C1C4C2×C4

Generators and relations for C52.4C4
 G = < a,b | a52=1, b4=a26, bab-1=a-1 >

2C2
2C26
13C8
13C8
13M4(2)

Smallest permutation representation of C52.4C4
On 104 points
Generators in S104
(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 99 100 101 102 103 104)
(1 95 14 82 27 69 40 56)(2 94 15 81 28 68 41 55)(3 93 16 80 29 67 42 54)(4 92 17 79 30 66 43 53)(5 91 18 78 31 65 44 104)(6 90 19 77 32 64 45 103)(7 89 20 76 33 63 46 102)(8 88 21 75 34 62 47 101)(9 87 22 74 35 61 48 100)(10 86 23 73 36 60 49 99)(11 85 24 72 37 59 50 98)(12 84 25 71 38 58 51 97)(13 83 26 70 39 57 52 96)

G:=sub<Sym(104)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,95,14,82,27,69,40,56)(2,94,15,81,28,68,41,55)(3,93,16,80,29,67,42,54)(4,92,17,79,30,66,43,53)(5,91,18,78,31,65,44,104)(6,90,19,77,32,64,45,103)(7,89,20,76,33,63,46,102)(8,88,21,75,34,62,47,101)(9,87,22,74,35,61,48,100)(10,86,23,73,36,60,49,99)(11,85,24,72,37,59,50,98)(12,84,25,71,38,58,51,97)(13,83,26,70,39,57,52,96)>;

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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,95,14,82,27,69,40,56)(2,94,15,81,28,68,41,55)(3,93,16,80,29,67,42,54)(4,92,17,79,30,66,43,53)(5,91,18,78,31,65,44,104)(6,90,19,77,32,64,45,103)(7,89,20,76,33,63,46,102)(8,88,21,75,34,62,47,101)(9,87,22,74,35,61,48,100)(10,86,23,73,36,60,49,99)(11,85,24,72,37,59,50,98)(12,84,25,71,38,58,51,97)(13,83,26,70,39,57,52,96) );

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),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,95,14,82,27,69,40,56),(2,94,15,81,28,68,41,55),(3,93,16,80,29,67,42,54),(4,92,17,79,30,66,43,53),(5,91,18,78,31,65,44,104),(6,90,19,77,32,64,45,103),(7,89,20,76,33,63,46,102),(8,88,21,75,34,62,47,101),(9,87,22,74,35,61,48,100),(10,86,23,73,36,60,49,99),(11,85,24,72,37,59,50,98),(12,84,25,71,38,58,51,97),(13,83,26,70,39,57,52,96)])

C52.4C4 is a maximal subgroup of
D524C4  C104.6C4  C52.53D4  C52.46D4  C4.12D52  C52.D4  C52.10D4  C52.56D4  D52.3C4  M4(2)×D13  D526C22  Q8.D26  D4.Dic13  D4⋊D26  D4.9D26
C52.4C4 is a maximal quotient of
C26.7C42  C523C8  C52.55D4

58 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D13A···13F26A···26R52A···52X
order122444888813···1326···2652···52
size112112262626262···22···22···2

58 irreducible representations

dim11111222222
type++++-+-
imageC1C2C2C4C4M4(2)D13Dic13D26Dic13C52.4C4
kernelC52.4C4C132C8C2×C52C52C2×C26C13C2×C4C4C4C22C1
# reps121222666624

Matrix representation of C52.4C4 in GL2(𝔽313) generated by

1620
74114
,
3230
205281
G:=sub<GL(2,GF(313))| [162,74,0,114],[32,205,30,281] >;

C52.4C4 in GAP, Magma, Sage, TeX

C_{52}._4C_4
% in TeX

G:=Group("C52.4C4");
// GroupNames label

G:=SmallGroup(208,10);
// by ID

G=gap.SmallGroup(208,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,42,4804]);
// Polycyclic

G:=Group<a,b|a^52=1,b^4=a^26,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C52.4C4 in TeX

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