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G = D13.D8order 416 = 25·13

The non-split extension by D13 of D8 acting via D8/C8=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C1041C4, D13.1D8, D26.9D4, D13.1Q16, Dic13.3Q8, C81(C13⋊C4), C13⋊(C2.D8), C132C86C4, C52.9(C2×C4), C26.2(C4⋊C4), (C8×D13).3C2, C52⋊C4.4C2, C2.5(C52⋊C4), (C4×D13).26C22, C4.9(C2×C13⋊C4), SmallGroup(416,69)

Series: Derived Chief Lower central Upper central

C1C52 — D13.D8
C1C13C26D26C4×D13C52⋊C4 — D13.D8
C13C26C52 — D13.D8
C1C2C4C8

Generators and relations for D13.D8
 G = < a,b,c,d | a13=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

13C2
13C2
13C4
13C22
52C4
52C4
13C2×C4
13C8
26C2×C4
26C2×C4
4C13⋊C4
4C13⋊C4
13C2×C8
13C4⋊C4
13C4⋊C4
2C2×C13⋊C4
2C2×C13⋊C4
13C2.D8

Smallest permutation representation of D13.D8
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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 24)(15 23)(16 22)(17 21)(18 20)(25 26)(27 32)(28 31)(29 30)(33 39)(34 38)(35 37)(40 45)(41 44)(42 43)(46 52)(47 51)(48 50)(53 62)(54 61)(55 60)(56 59)(57 58)(63 65)(66 69)(67 68)(70 78)(71 77)(72 76)(73 75)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)(92 98)(93 97)(94 96)(99 104)(100 103)(101 102)
(1 86 30 68 26 102 43 58)(2 87 31 69 14 103 44 59)(3 88 32 70 15 104 45 60)(4 89 33 71 16 92 46 61)(5 90 34 72 17 93 47 62)(6 91 35 73 18 94 48 63)(7 79 36 74 19 95 49 64)(8 80 37 75 20 96 50 65)(9 81 38 76 21 97 51 53)(10 82 39 77 22 98 52 54)(11 83 27 78 23 99 40 55)(12 84 28 66 24 100 41 56)(13 85 29 67 25 101 42 57)
(1 43)(2 51 13 48)(3 46 12 40)(4 41 11 45)(5 49 10 50)(6 44 9 42)(7 52 8 47)(14 38 25 35)(15 33 24 27)(16 28 23 32)(17 36 22 37)(18 31 21 29)(19 39 20 34)(26 30)(53 57 63 59)(54 65 62 64)(55 60 61 56)(66 78 70 71)(67 73 69 76)(72 74 77 75)(79 98 80 93)(81 101 91 103)(82 96 90 95)(83 104 89 100)(84 99 88 92)(85 94 87 97)(86 102)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F8A8B8C8D13A13B13C26A26B26C52A···52F104A···104L
order1222444444888813131326262652···52104···104
size111313226525252522226264444444···44···4

38 irreducible representations

dim1111122224444
type+++-++-++
imageC1C2C2C4C4Q8D4D8Q16C13⋊C4C2×C13⋊C4C52⋊C4D13.D8
kernelD13.D8C8×D13C52⋊C4C132C8C104Dic13D26D13D13C8C4C2C1
# reps11222112233612

Matrix representation of D13.D8 in GL6(𝔽313)

100000
010000
0031323131
00282282283282
00242242242243
00312312312312
,
31200000
03120000
0031228140311
000316130
0007128171
00012422
,
253600000
2532530000
0064688660
002272232270
002684253
001818022
,
0250000
2500000
00283027232
002430252252
0028231232272
0029071311

G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,282,242,312,0,0,32,282,242,312,0,0,31,283,242,312,0,0,31,282,243,312],[312,0,0,0,0,0,0,312,0,0,0,0,0,0,312,0,0,0,0,0,281,31,71,1,0,0,40,61,281,242,0,0,311,30,71,2],[253,253,0,0,0,0,60,253,0,0,0,0,0,0,64,227,26,18,0,0,68,223,8,18,0,0,86,227,4,0,0,0,60,0,253,22],[0,25,0,0,0,0,25,0,0,0,0,0,0,0,283,243,282,29,0,0,0,0,312,0,0,0,272,252,32,71,0,0,32,252,272,311] >;

D13.D8 in GAP, Magma, Sage, TeX

D_{13}.D_8
% in TeX

G:=Group("D13.D8");
// GroupNames label

G:=SmallGroup(416,69);
// by ID

G=gap.SmallGroup(416,69);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,579,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D13.D8 in TeX

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