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G = D527C4order 416 = 25·13

4th semidirect product of D52 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D527C4, C52.54D4, Dic267C4, C22.3D52, M4(2)⋊4D13, C134C4≀C2, C4.3(C4×D13), (C2×C26).1D4, C52.27(C2×C4), (C2×C4).38D26, (C4×Dic13)⋊1C2, D525C2.2C2, C4.29(C13⋊D4), (C13×M4(2))⋊8C2, (C2×C52).15C22, C26.21(C22⋊C4), C2.11(D26⋊C4), SmallGroup(416,32)

Series: Derived Chief Lower central Upper central

C1C52 — D527C4
C1C13C26C52C2×C52D525C2 — D527C4
C13C26C52 — D527C4
C1C4C2×C4M4(2)

Generators and relations for D527C4
 G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a25, cbc-1=a11b >

2C2
52C2
26C4
26C22
26C4
26C4
2C26
4D13
2C8
13D4
13Q8
26C2×C4
26C2×C4
26D4
2Dic13
2Dic13
2D26
2Dic13
13C42
13C4○D4
2C4×D13
2C13⋊D4
2C2×Dic13
2C104
13C4≀C2

Smallest permutation representation of D527C4
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 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 104)(34 103)(35 102)(36 101)(37 100)(38 99)(39 98)(40 97)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)
(1 40 27 14)(2 13 28 39)(3 38 29 12)(4 11 30 37)(5 36 31 10)(6 9 32 35)(7 34 33 8)(15 26 41 52)(16 51 42 25)(17 24 43 50)(18 49 44 23)(19 22 45 48)(20 47 46 21)(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)

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

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

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

74 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A8B13A···13F26A···26F26G···26L52A···52L52M···52R104A···104X
order1222444444448813···1326···2626···2652···5252···52104···104
size112521122626262652442···22···24···42···24···44···4

74 irreducible representations

dim111111222222224
type+++++++++
imageC1C2C2C2C4C4D4D4D13C4≀C2D26C4×D13C13⋊D4D52D527C4
kernelD527C4C4×Dic13C13×M4(2)D525C2Dic26D52C52C2×C26M4(2)C13C2×C4C4C4C22C1
# reps1111221164612121212

Matrix representation of D527C4 in GL4(𝔽313) generated by

2514000
028800
00135227
0086293
,
312000
291100
00212243
00101101
,
28811500
0100
0086293
00135227
G:=sub<GL(4,GF(313))| [25,0,0,0,140,288,0,0,0,0,135,86,0,0,227,293],[312,291,0,0,0,1,0,0,0,0,212,101,0,0,243,101],[288,0,0,0,115,1,0,0,0,0,86,135,0,0,293,227] >;

D527C4 in GAP, Magma, Sage, TeX

D_{52}\rtimes_7C_4
% in TeX

G:=Group("D52:7C4");
// GroupNames label

G:=SmallGroup(416,32);
// by ID

G=gap.SmallGroup(416,32);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,86,579,297,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^11*b>;
// generators/relations

Export

Subgroup lattice of D527C4 in TeX

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