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G = C8×C13⋊C4order 416 = 25·13

Direct product of C8 and C13⋊C4

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

Aliases: C8×C13⋊C4, C1043C4, C26.1C42, C13⋊C83C4, C131(C4×C8), D13.(C2×C8), C132C87C4, C52.15(C2×C4), D13⋊C8.3C2, (C8×D13).9C2, D26.5(C2×C4), Dic13.7(C2×C4), (C4×D13).31C22, C2.1(C4×C13⋊C4), (C4×C13⋊C4).3C2, (C2×C13⋊C4).2C4, C4.16(C2×C13⋊C4), SmallGroup(416,66)

Series: Derived Chief Lower central Upper central

C1C13 — C8×C13⋊C4
C1C13C26D26C4×D13C4×C13⋊C4 — C8×C13⋊C4
C13 — C8×C13⋊C4
C1C8

Generators and relations for C8×C13⋊C4
 G = < a,b,c | a8=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

13C2
13C2
13C4
13C4
13C22
13C4
13C4
13C4
13C8
13C8
13C2×C4
13C2×C4
13C2×C4
13C8
13C2×C8
13C42
13C2×C8
13C4×C8

Smallest permutation representation of C8×C13⋊C4
On 104 points
Generators in S104
(1 92 40 66 14 79 27 53)(2 93 41 67 15 80 28 54)(3 94 42 68 16 81 29 55)(4 95 43 69 17 82 30 56)(5 96 44 70 18 83 31 57)(6 97 45 71 19 84 32 58)(7 98 46 72 20 85 33 59)(8 99 47 73 21 86 34 60)(9 100 48 74 22 87 35 61)(10 101 49 75 23 88 36 62)(11 102 50 76 24 89 37 63)(12 103 51 77 25 90 38 64)(13 104 52 78 26 91 39 65)
(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)(2 22 13 19)(3 17 12 24)(4 25 11 16)(5 20 10 21)(6 15 9 26)(7 23 8 18)(27 40)(28 48 39 45)(29 43 38 50)(30 51 37 42)(31 46 36 47)(32 41 35 52)(33 49 34 44)(53 66)(54 74 65 71)(55 69 64 76)(56 77 63 68)(57 72 62 73)(58 67 61 78)(59 75 60 70)(79 92)(80 100 91 97)(81 95 90 102)(82 103 89 94)(83 98 88 99)(84 93 87 104)(85 101 86 96)

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C···4L8A8B8C8D8E···8P13A13B13C26A26B26C52A···52F104A···104L
order1222444···488888···813131326262652···52104···104
size1113131113···13111113···134444444···44···4

56 irreducible representations

dim1111111114444
type++++++
imageC1C2C2C2C4C4C4C4C8C13⋊C4C2×C13⋊C4C4×C13⋊C4C8×C13⋊C4
kernelC8×C13⋊C4C8×D13D13⋊C8C4×C13⋊C4C132C8C104C13⋊C8C2×C13⋊C4C13⋊C4C8C4C2C1
# reps111122441633612

Matrix representation of C8×C13⋊C4 in GL4(𝔽313) generated by

5000
0500
0050
0005
,
31231231228
100213
010100
001284
,
1010183214
2850201201
100312214183
211029311
G:=sub<GL(4,GF(313))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[312,1,0,0,312,0,1,0,312,0,0,1,28,213,100,284],[101,285,100,211,0,0,312,0,183,201,214,29,214,201,183,311] >;

C8×C13⋊C4 in GAP, Magma, Sage, TeX

C_8\times C_{13}\rtimes C_4
% in TeX

G:=Group("C8xC13:C4");
// GroupNames label

G:=SmallGroup(416,66);
// by ID

G=gap.SmallGroup(416,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,69,9221,3473]);
// Polycyclic

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

Export

Subgroup lattice of C8×C13⋊C4 in TeX

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