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G = C52.56D4order 416 = 25·13

13rd non-split extension by C52 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.56D4, Q82Dic13, D42Dic13, C135C4≀C2, (D4×C13)⋊5C4, (Q8×C13)⋊5C4, (C2×C26).3D4, C52.30(C2×C4), C4○D4.1D13, (C2×C4).41D26, C52.4C44C2, (C4×Dic13)⋊2C2, C4.3(C2×Dic13), C4.31(C13⋊D4), (C2×C52).20C22, C26.29(C22⋊C4), C22.3(C13⋊D4), C2.8(C23.D13), (C13×C4○D4).1C2, SmallGroup(416,44)

Series: Derived Chief Lower central Upper central

C1C52 — C52.56D4
C1C13C26C52C2×C52C52.4C4 — C52.56D4
C13C26C52 — C52.56D4
C1C4C2×C4C4○D4

Generators and relations for C52.56D4
 G = < a,b,c | a52=b4=1, c2=a13, bab-1=cac-1=a25, cbc-1=a13b-1 >

2C2
4C2
2C4
2C22
26C4
26C4
2C26
4C26
2D4
2C2×C4
26C2×C4
26C8
2C2×C26
2Dic13
2C52
2Dic13
13C42
13M4(2)
2C132C8
2C2×C52
2D4×C13
2C2×Dic13
13C4≀C2

Smallest permutation representation of C52.56D4
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 14 27 40)(2 39 28 13)(3 12 29 38)(4 37 30 11)(5 10 31 36)(6 35 32 9)(7 8 33 34)(15 52 41 26)(16 25 42 51)(17 50 43 24)(18 23 44 49)(19 48 45 22)(20 21 46 47)(53 65)(54 90)(55 63)(56 88)(57 61)(58 86)(60 84)(62 82)(64 80)(66 78)(67 103)(68 76)(69 101)(70 74)(71 99)(73 97)(75 95)(77 93)(79 91)(81 89)(83 87)(92 104)(94 102)(96 100)
(1 72 14 85 27 98 40 59)(2 97 15 58 28 71 41 84)(3 70 16 83 29 96 42 57)(4 95 17 56 30 69 43 82)(5 68 18 81 31 94 44 55)(6 93 19 54 32 67 45 80)(7 66 20 79 33 92 46 53)(8 91 21 104 34 65 47 78)(9 64 22 77 35 90 48 103)(10 89 23 102 36 63 49 76)(11 62 24 75 37 88 50 101)(12 87 25 100 38 61 51 74)(13 60 26 73 39 86 52 99)

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

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

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

74 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A8B13A···13F26A···26F26G···26X52A···52L52M···52AD
order1222444444448813···1326···2626···2652···5252···52
size112411242626262652522···22···24···42···24···4

74 irreducible representations

dim1111112222222224
type++++++++--
imageC1C2C2C2C4C4D4D4D13C4≀C2D26Dic13Dic13C13⋊D4C13⋊D4C52.56D4
kernelC52.56D4C52.4C4C4×Dic13C13×C4○D4D4×C13Q8×C13C52C2×C26C4○D4C13C2×C4D4Q8C4C22C1
# reps1111221164666121212

Matrix representation of C52.56D4 in GL4(𝔽313) generated by

1927800
29700
00250
00025
,
6522200
18424800
00250
00801
,
9921500
21521400
0068230
00161245
G:=sub<GL(4,GF(313))| [192,2,0,0,78,97,0,0,0,0,25,0,0,0,0,25],[65,184,0,0,222,248,0,0,0,0,25,80,0,0,0,1],[99,215,0,0,215,214,0,0,0,0,68,161,0,0,230,245] >;

C52.56D4 in GAP, Magma, Sage, TeX

C_{52}._{56}D_4
% in TeX

G:=Group("C52.56D4");
// GroupNames label

G:=SmallGroup(416,44);
// by ID

G=gap.SmallGroup(416,44);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,86,579,297,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^4=1,c^2=a^13,b*a*b^-1=c*a*c^-1=a^25,c*b*c^-1=a^13*b^-1>;
// generators/relations

Export

Subgroup lattice of C52.56D4 in TeX

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