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G = C104⋊C4order 416 = 25·13

4th semidirect product of C104 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C1044C4, C26.2C42, D13.M4(2), C13⋊C81C4, C83(C13⋊C4), C132C88C4, C131(C8⋊C4), C52.16(C2×C4), D13⋊C8.2C2, D26.6(C2×C4), (C8×D13).10C2, Dic13.8(C2×C4), (C4×D13).32C22, (C2×C13⋊C4).C4, C2.3(C4×C13⋊C4), (C4×C13⋊C4).2C2, C4.17(C2×C13⋊C4), SmallGroup(416,67)

Series: Derived Chief Lower central Upper central

C1C26 — C104⋊C4
C1C13C26D26C4×D13C4×C13⋊C4 — C104⋊C4
C13C26 — C104⋊C4
C1C4C8

Generators and relations for C104⋊C4
 G = < a,b | a104=b4=1, bab-1=a5 >

13C2
13C2
13C4
13C22
26C4
26C4
13C8
13C2×C4
13C2×C4
13C8
13C2×C4
13C8
2C13⋊C4
2C13⋊C4
13C42
13C2×C8
13C2×C8
13C8⋊C4

Smallest permutation representation of C104⋊C4
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)
(2 22 26 6)(3 43 51 11)(4 64 76 16)(5 85 101 21)(7 23 47 31)(8 44 72 36)(9 65 97 41)(10 86 18 46)(12 24 68 56)(13 45 93 61)(14 66)(15 87 39 71)(17 25 89 81)(19 67 35 91)(20 88 60 96)(28 48 52 32)(29 69 77 37)(30 90 102 42)(33 49 73 57)(34 70 98 62)(38 50 94 82)(40 92)(54 74 78 58)(55 95 103 63)(59 75 99 83)(80 100 104 84)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A8B8C···8H13A13B13C26A26B26C52A···52F104A···104L
order122244444444888···813131326262652···52104···104
size111313111313262626262226···264444444···44···4

44 irreducible representations

dim1111111124444
type++++++
imageC1C2C2C2C4C4C4C4M4(2)C13⋊C4C2×C13⋊C4C4×C13⋊C4C104⋊C4
kernelC104⋊C4C8×D13D13⋊C8C4×C13⋊C4C132C8C104C13⋊C8C2×C13⋊C4D13C8C4C2C1
# reps11112244433612

Matrix representation of C104⋊C4 in GL4(𝔽5) generated by

2303
0012
3033
4033
,
1130
3402
3313
4244
G:=sub<GL(4,GF(5))| [2,0,3,4,3,0,0,0,0,1,3,3,3,2,3,3],[1,3,3,4,1,4,3,2,3,0,1,4,0,2,3,4] >;

C104⋊C4 in GAP, Magma, Sage, TeX

C_{104}\rtimes C_4
% in TeX

G:=Group("C104:C4");
// GroupNames label

G:=SmallGroup(416,67);
// by ID

G=gap.SmallGroup(416,67);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,69,9221,3473]);
// Polycyclic

G:=Group<a,b|a^104=b^4=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C104⋊C4 in TeX

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