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G = C7×C13⋊C4order 364 = 22·7·13

Direct product of C7 and C13⋊C4

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

Aliases: C7×C13⋊C4, C13⋊C28, C912C4, D13.C14, (C7×D13).2C2, SmallGroup(364,5)

Series: Derived Chief Lower central Upper central

C1C13 — C7×C13⋊C4
C1C13D13C7×D13 — C7×C13⋊C4
C13 — C7×C13⋊C4
C1C7

Generators and relations for C7×C13⋊C4
 G = < a,b,c | a7=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

13C2
13C4
13C14
13C28

Smallest permutation representation of C7×C13⋊C4
On 91 points
Generators in S91
(1 79 66 53 40 27 14)(2 80 67 54 41 28 15)(3 81 68 55 42 29 16)(4 82 69 56 43 30 17)(5 83 70 57 44 31 18)(6 84 71 58 45 32 19)(7 85 72 59 46 33 20)(8 86 73 60 47 34 21)(9 87 74 61 48 35 22)(10 88 75 62 49 36 23)(11 89 76 63 50 37 24)(12 90 77 64 51 38 25)(13 91 78 65 52 39 26)
(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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)(54 61 65 58)(55 56 64 63)(57 59 62 60)(67 74 78 71)(68 69 77 76)(70 72 75 73)(80 87 91 84)(81 82 90 89)(83 85 88 86)

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

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

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

49 conjugacy classes

class 1  2 4A4B7A···7F13A13B13C14A···14F28A···28L91A···91R
order12447···713131314···1428···2891···91
size11313131···144413···1313···134···4

49 irreducible representations

dim11111144
type+++
imageC1C2C4C7C14C28C13⋊C4C7×C13⋊C4
kernelC7×C13⋊C4C7×D13C91C13⋊C4D13C13C7C1
# reps1126612318

Matrix representation of C7×C13⋊C4 in GL4(𝔽1093) generated by

243000
024300
002430
000243
,
5695237561092
100907
0101091
001338
,
5241861092906
233817617673
234708799905
057286246
G:=sub<GL(4,GF(1093))| [243,0,0,0,0,243,0,0,0,0,243,0,0,0,0,243],[569,1,0,0,523,0,1,0,756,0,0,1,1092,907,1091,338],[524,233,234,0,186,817,708,572,1092,617,799,862,906,673,905,46] >;

C7×C13⋊C4 in GAP, Magma, Sage, TeX

C_7\times C_{13}\rtimes C_4
% in TeX

G:=Group("C7xC13:C4");
// GroupNames label

G:=SmallGroup(364,5);
// by ID

G=gap.SmallGroup(364,5);
# by ID

G:=PCGroup([4,-2,-7,-2,-13,56,3587,395]);
// Polycyclic

G:=Group<a,b,c|a^7=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 C7×C13⋊C4 in TeX

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