C14xES-(3,1) - GroupNames

(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)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126)

(1 117 33 98 108 79 54 61 22)(2 118 34 85 109 80 55 62 23)(3 119 35 86 110 81 56 63 24)(4 120 36 87 111 82 43 64 25)(5 121 37 88 112 83 44 65 26)(6 122 38 89 99 84 45 66 27)(7 123 39 90 100 71 46 67 28)(8 124 40 91 101 72 47 68 15)(9 125 41 92 102 73 48 69 16)(10 126 42 93 103 74 49 70 17)(11 113 29 94 104 75 50 57 18)(12 114 30 95 105 76 51 58 19)(13 115 31 96 106 77 52 59 20)(14 116 32 97 107 78 53 60 21)

(15 40 72)(16 41 73)(17 42 74)(18 29 75)(19 30 76)(20 31 77)(21 32 78)(22 33 79)(23 34 80)(24 35 81)(25 36 82)(26 37 83)(27 38 84)(28 39 71)(57 104 113)(58 105 114)(59 106 115)(60 107 116)(61 108 117)(62 109 118)(63 110 119)(64 111 120)(65 112 121)(66 99 122)(67 100 123)(68 101 124)(69 102 125)(70 103 126)

G:=sub<Sym(126)| (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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,117,33,98,108,79,54,61,22)(2,118,34,85,109,80,55,62,23)(3,119,35,86,110,81,56,63,24)(4,120,36,87,111,82,43,64,25)(5,121,37,88,112,83,44,65,26)(6,122,38,89,99,84,45,66,27)(7,123,39,90,100,71,46,67,28)(8,124,40,91,101,72,47,68,15)(9,125,41,92,102,73,48,69,16)(10,126,42,93,103,74,49,70,17)(11,113,29,94,104,75,50,57,18)(12,114,30,95,105,76,51,58,19)(13,115,31,96,106,77,52,59,20)(14,116,32,97,107,78,53,60,21), (15,40,72)(16,41,73)(17,42,74)(18,29,75)(19,30,76)(20,31,77)(21,32,78)(22,33,79)(23,34,80)(24,35,81)(25,36,82)(26,37,83)(27,38,84)(28,39,71)(57,104,113)(58,105,114)(59,106,115)(60,107,116)(61,108,117)(62,109,118)(63,110,119)(64,111,120)(65,112,121)(66,99,122)(67,100,123)(68,101,124)(69,102,125)(70,103,126)>;

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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,117,33,98,108,79,54,61,22)(2,118,34,85,109,80,55,62,23)(3,119,35,86,110,81,56,63,24)(4,120,36,87,111,82,43,64,25)(5,121,37,88,112,83,44,65,26)(6,122,38,89,99,84,45,66,27)(7,123,39,90,100,71,46,67,28)(8,124,40,91,101,72,47,68,15)(9,125,41,92,102,73,48,69,16)(10,126,42,93,103,74,49,70,17)(11,113,29,94,104,75,50,57,18)(12,114,30,95,105,76,51,58,19)(13,115,31,96,106,77,52,59,20)(14,116,32,97,107,78,53,60,21), (15,40,72)(16,41,73)(17,42,74)(18,29,75)(19,30,76)(20,31,77)(21,32,78)(22,33,79)(23,34,80)(24,35,81)(25,36,82)(26,37,83)(27,38,84)(28,39,71)(57,104,113)(58,105,114)(59,106,115)(60,107,116)(61,108,117)(62,109,118)(63,110,119)(64,111,120)(65,112,121)(66,99,122)(67,100,123)(68,101,124)(69,102,125)(70,103,126) );

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),(99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(1,117,33,98,108,79,54,61,22),(2,118,34,85,109,80,55,62,23),(3,119,35,86,110,81,56,63,24),(4,120,36,87,111,82,43,64,25),(5,121,37,88,112,83,44,65,26),(6,122,38,89,99,84,45,66,27),(7,123,39,90,100,71,46,67,28),(8,124,40,91,101,72,47,68,15),(9,125,41,92,102,73,48,69,16),(10,126,42,93,103,74,49,70,17),(11,113,29,94,104,75,50,57,18),(12,114,30,95,105,76,51,58,19),(13,115,31,96,106,77,52,59,20),(14,116,32,97,107,78,53,60,21)], [(15,40,72),(16,41,73),(17,42,74),(18,29,75),(19,30,76),(20,31,77),(21,32,78),(22,33,79),(23,34,80),(24,35,81),(25,36,82),(26,37,83),(27,38,84),(28,39,71),(57,104,113),(58,105,114),(59,106,115),(60,107,116),(61,108,117),(62,109,118),(63,110,119),(64,111,120),(65,112,121),(66,99,122),(67,100,123),(68,101,124),(69,102,125),(70,103,126)]])

154 conjugacy classes

class	1	2	3A	3B	3C	3D	6A	6B	6C	6D	7A	···	7F	9A	···	9F	14A	···	14F	18A	···	18F	21A	···	21L	21M	···	21X	42A	···	42L	42M	···	42X	63A	···	63AJ	126A	···	126AJ
order	1	2	3	3	3	3	6	6	6	6	7	···	7	9	···	9	14	···	14	18	···	18	21	···	21	21	···	21	42	···	42	42	···	42	63	···	63	126	···	126
size	1	1	1	1	3	3	1	1	3	3	1	···	1	3	···	3	1	···	1	3	···	3	1	···	1	3	···	3	1	···	1	3	···	3	3	···	3	3	···	3

154 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	C₇	C₁₄	C₂₁	C₂₁	C₄₂	C₄₂	3_-¹⁺²	C₂×3_-¹⁺²	C₇×3_-¹⁺²	C₁₄×3_-¹⁺²
kernel	C₁₄×3_-¹⁺²	C₇×3_-¹⁺²	C₁₂₆	C₃×C₄₂	C₆₃	C₃×C₂₁	C₂×3_-¹⁺²	3_-¹⁺²	C₁₈	C₃×C₆	C₉	C₃²	C₁₄	C₇	C₂	C₁
# reps	1	1	6	2	6	2	6	6	36	12	36	12	2	2	12	12

Matrix representation of C₁₄×3_-¹⁺² ►in GL₃(𝔽₁₂₇) generated by

125	0	0
0	125	0
0	0	125

0	1	0
0	0	19
1	0	0

1	0	0
0	19	0
0	0	107

G:=sub<GL(3,GF(127))| [125,0,0,0,125,0,0,0,125],[0,0,1,1,0,0,0,19,0],[1,0,0,0,19,0,0,0,107] >;

C₁₄×3_-¹⁺² in GAP, Magma, Sage, TeX

C_{14}\times 3_-^{1+2}

% in TeX

G:=Group("C14xES-(3,1)");

// GroupNames label

G:=SmallGroup(378,46);

// by ID

G=gap.SmallGroup(378,46);

# by ID

G:=PCGroup([5,-2,-3,-3,-7,-3,636,997]);

// Polycyclic

G:=Group<a,b,c|a^14=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;

// generators/relations

Export

Subgroup lattice of C₁₄×3_-¹⁺² in TeX

G = C14×3- 1+2 order 378 = 2·33·7

Direct product of C14 and 3- 1+2

G = C₁₄×3_-¹⁺² order 378 = 2·3³·7

Direct product of C₁₄ and 3_-¹⁺²