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G = Q8×C19order 152 = 23·19

Direct product of C19 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C19, C4.C38, C76.3C2, C38.7C22, C2.2(C2×C38), SmallGroup(152,10)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C19
C1C2C38C76 — Q8×C19
C1C2 — Q8×C19
C1C38 — Q8×C19

Generators and relations for Q8×C19
 G = < a,b,c | a19=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C19
Regular action on 152 points
Generators in S152
(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)
(1 41 60 128)(2 42 61 129)(3 43 62 130)(4 44 63 131)(5 45 64 132)(6 46 65 133)(7 47 66 115)(8 48 67 116)(9 49 68 117)(10 50 69 118)(11 51 70 119)(12 52 71 120)(13 53 72 121)(14 54 73 122)(15 55 74 123)(16 56 75 124)(17 57 76 125)(18 39 58 126)(19 40 59 127)(20 144 109 88)(21 145 110 89)(22 146 111 90)(23 147 112 91)(24 148 113 92)(25 149 114 93)(26 150 96 94)(27 151 97 95)(28 152 98 77)(29 134 99 78)(30 135 100 79)(31 136 101 80)(32 137 102 81)(33 138 103 82)(34 139 104 83)(35 140 105 84)(36 141 106 85)(37 142 107 86)(38 143 108 87)
(1 148 60 92)(2 149 61 93)(3 150 62 94)(4 151 63 95)(5 152 64 77)(6 134 65 78)(7 135 66 79)(8 136 67 80)(9 137 68 81)(10 138 69 82)(11 139 70 83)(12 140 71 84)(13 141 72 85)(14 142 73 86)(15 143 74 87)(16 144 75 88)(17 145 76 89)(18 146 58 90)(19 147 59 91)(20 124 109 56)(21 125 110 57)(22 126 111 39)(23 127 112 40)(24 128 113 41)(25 129 114 42)(26 130 96 43)(27 131 97 44)(28 132 98 45)(29 133 99 46)(30 115 100 47)(31 116 101 48)(32 117 102 49)(33 118 103 50)(34 119 104 51)(35 120 105 52)(36 121 106 53)(37 122 107 54)(38 123 108 55)

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

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

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

Q8×C19 is a maximal subgroup of   Q8⋊D19  C19⋊Q16  D76⋊C2  C38.A4

95 conjugacy classes

class 1  2 4A4B4C19A···19R38A···38R76A···76BB
order1244419···1938···3876···76
size112221···11···12···2

95 irreducible representations

dim111122
type++-
imageC1C2C19C38Q8Q8×C19
kernelQ8×C19C76Q8C4C19C1
# reps131854118

Matrix representation of Q8×C19 in GL2(𝔽229) generated by

170
017
,
57227
22172
,
819
136148
G:=sub<GL(2,GF(229))| [17,0,0,17],[57,22,227,172],[81,136,9,148] >;

Q8×C19 in GAP, Magma, Sage, TeX

Q_8\times C_{19}
% in TeX

G:=Group("Q8xC19");
// GroupNames label

G:=SmallGroup(152,10);
// by ID

G=gap.SmallGroup(152,10);
# by ID

G:=PCGroup([4,-2,-2,-19,-2,304,625,309]);
// Polycyclic

G:=Group<a,b,c|a^19=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C19 in TeX

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