C19:C24 - GroupNames

(1 123 68 60 99 129 91 107 31 83 45 115 23 53 76 37 137 145 15)(2 24 130 146 46 61 38 32 124 54 92 16 116 100 138 84 69 77 108)(3 117 62 78 93 147 85 125 25 101 39 109 17 47 70 55 131 139 9)(4 18 148 140 40 79 56 26 118 48 86 10 110 94 132 102 63 71 126)(5 111 80 72 87 141 103 119 19 95 33 127 11 41 64 49 149 133 27)(6 12 142 134 34 73 50 20 112 42 104 28 128 88 150 96 57 65 120)(7 105 74 66 81 135 97 113 13 89 51 121 29 35 58 43 143 151 21)(8 30 136 152 52 67 44 14 106 36 98 22 122 82 144 90 75 59 114)

(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)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)

G:=sub<Sym(152)| (1,123,68,60,99,129,91,107,31,83,45,115,23,53,76,37,137,145,15)(2,24,130,146,46,61,38,32,124,54,92,16,116,100,138,84,69,77,108)(3,117,62,78,93,147,85,125,25,101,39,109,17,47,70,55,131,139,9)(4,18,148,140,40,79,56,26,118,48,86,10,110,94,132,102,63,71,126)(5,111,80,72,87,141,103,119,19,95,33,127,11,41,64,49,149,133,27)(6,12,142,134,34,73,50,20,112,42,104,28,128,88,150,96,57,65,120)(7,105,74,66,81,135,97,113,13,89,51,121,29,35,58,43,143,151,21)(8,30,136,152,52,67,44,14,106,36,98,22,122,82,144,90,75,59,114), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)>;

G:=Group( (1,123,68,60,99,129,91,107,31,83,45,115,23,53,76,37,137,145,15)(2,24,130,146,46,61,38,32,124,54,92,16,116,100,138,84,69,77,108)(3,117,62,78,93,147,85,125,25,101,39,109,17,47,70,55,131,139,9)(4,18,148,140,40,79,56,26,118,48,86,10,110,94,132,102,63,71,126)(5,111,80,72,87,141,103,119,19,95,33,127,11,41,64,49,149,133,27)(6,12,142,134,34,73,50,20,112,42,104,28,128,88,150,96,57,65,120)(7,105,74,66,81,135,97,113,13,89,51,121,29,35,58,43,143,151,21)(8,30,136,152,52,67,44,14,106,36,98,22,122,82,144,90,75,59,114), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152) );

G=PermutationGroup([[(1,123,68,60,99,129,91,107,31,83,45,115,23,53,76,37,137,145,15),(2,24,130,146,46,61,38,32,124,54,92,16,116,100,138,84,69,77,108),(3,117,62,78,93,147,85,125,25,101,39,109,17,47,70,55,131,139,9),(4,18,148,140,40,79,56,26,118,48,86,10,110,94,132,102,63,71,126),(5,111,80,72,87,141,103,119,19,95,33,127,11,41,64,49,149,133,27),(6,12,142,134,34,73,50,20,112,42,104,28,128,88,150,96,57,65,120),(7,105,74,66,81,135,97,113,13,89,51,121,29,35,58,43,143,151,21),(8,30,136,152,52,67,44,14,106,36,98,22,122,82,144,90,75,59,114)], [(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),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)]])

36 conjugacy classes

class	1	2	3A	3B	4A	4B	6A	6B	8A	8B	8C	8D	12A	12B	12C	12D	19A	19B	19C	24A	···	24H	38A	38B	38C	76A	···	76F
order	1	2	3	3	4	4	6	6	8	8	8	8	12	12	12	12	19	19	19	24	···	24	38	38	38	76	···	76
size	1	1	19	19	1	1	19	19	19	19	19	19	19	19	19	19	6	6	6	19	···	19	6	6	6	6	···	6

36 irreducible representations

dim	1	1	1	1	1	1	1	1	6	6	6
type	+	+							+	-
image	C₁	C₂	C₃	C₄	C₆	C₈	C₁₂	C₂₄	C₁₉⋊C₆	C₁₉⋊C₁₂	C₁₉⋊C₂₄
kernel	C₁₉⋊C₂₄	C₄×C₁₉⋊C₃	C₁₉⋊C₈	C₂×C₁₉⋊C₃	C₇₆	C₁₉⋊C₃	C₃₈	C₁₉	C₄	C₂	C₁
# reps	1	1	2	2	2	4	4	8	3	3	6

Matrix representation of C₁₉⋊C₂₄ ►in GL₆(𝔽₄₅₇)

0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
456	46	171	267	171	46

159	415	52	413	71	436
279	3	335	32	193	185
21	386	44	405	42	109
266	380	128	453	286	40
117	190	353	379	236	416
19	21	388	111	19	19

G:=sub<GL(6,GF(457))| [0,0,0,0,0,456,1,0,0,0,0,46,0,1,0,0,0,171,0,0,1,0,0,267,0,0,0,1,0,171,0,0,0,0,1,46],[159,279,21,266,117,19,415,3,386,380,190,21,52,335,44,128,353,388,413,32,405,453,379,111,71,193,42,286,236,19,436,185,109,40,416,19] >;

C₁₉⋊C₂₄ in GAP, Magma, Sage, TeX

C_{19}\rtimes C_{24}

% in TeX

G:=Group("C19:C24");

// GroupNames label

G:=SmallGroup(456,1);

// by ID

G=gap.SmallGroup(456,1);

# by ID

G:=PCGroup([5,-2,-3,-2,-2,-19,30,42,10804,2109]);

// Polycyclic

G:=Group<a,b|a^19=b^24=1,b*a*b^-1=a^12>;

// generators/relations

Export

Subgroup lattice of C₁₉⋊C₂₄ in TeX

159	415	52	413	71	436
279	3	335	32	193	185
21	386	44	405	42	109
266	380	128	453	286	40
117	190	353	379	236	416
19	21	388	111	19	19

159	415	52	413	71	436
279	3	335	32	193	185
21	386	44	405	42	109
266	380	128	453	286	40
117	190	353	379	236	416
19	21	388	111	19	19

G = C19⋊C24 order 456 = 23·3·19

The semidirect product of C19 and C24 acting via C24/C4=C6

G = C₁₉⋊C₂₄ order 456 = 2³·3·19

The semidirect product of C₁₉ and C₂₄ acting via C₂₄/C₄=C₆

159	415	52	413	71	436
279	3	335	32	193	185
21	386	44	405	42	109
266	380	128	453	286	40
117	190	353	379	236	416
19	21	388	111	19	19