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## G = Q8×C24order 192 = 26·3

### Direct product of C24 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C24
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C2×C24 — C3×C4⋊C8 — Q8×C24
 Lower central C1 — C2 — Q8×C24
 Upper central C1 — C2×C24 — Q8×C24

Generators and relations for Q8×C24
G = < a,b,c | a24=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 114 in 102 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, C12, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C24, C24, C2×C12, C2×C12, C3×Q8, C4×C8, C4⋊C8, C4×Q8, C4×C12, C3×C4⋊C4, C2×C24, C2×C24, C6×Q8, C8×Q8, C4×C24, C3×C4⋊C8, Q8×C12, Q8×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, Q8, C23, C12, C2×C6, C2×C8, C22×C4, C2×Q8, C4○D4, C24, C2×C12, C3×Q8, C22×C6, C4×Q8, C22×C8, C8○D4, C2×C24, C22×C12, C6×Q8, C3×C4○D4, C8×Q8, Q8×C12, C22×C24, C3×C8○D4, Q8×C24

Smallest permutation representation of Q8×C24
Regular action on 192 points
Generators in S192
(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)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)
(1 43 132 81)(2 44 133 82)(3 45 134 83)(4 46 135 84)(5 47 136 85)(6 48 137 86)(7 25 138 87)(8 26 139 88)(9 27 140 89)(10 28 141 90)(11 29 142 91)(12 30 143 92)(13 31 144 93)(14 32 121 94)(15 33 122 95)(16 34 123 96)(17 35 124 73)(18 36 125 74)(19 37 126 75)(20 38 127 76)(21 39 128 77)(22 40 129 78)(23 41 130 79)(24 42 131 80)(49 158 108 190)(50 159 109 191)(51 160 110 192)(52 161 111 169)(53 162 112 170)(54 163 113 171)(55 164 114 172)(56 165 115 173)(57 166 116 174)(58 167 117 175)(59 168 118 176)(60 145 119 177)(61 146 120 178)(62 147 97 179)(63 148 98 180)(64 149 99 181)(65 150 100 182)(66 151 101 183)(67 152 102 184)(68 153 103 185)(69 154 104 186)(70 155 105 187)(71 156 106 188)(72 157 107 189)
(1 162 132 170)(2 163 133 171)(3 164 134 172)(4 165 135 173)(5 166 136 174)(6 167 137 175)(7 168 138 176)(8 145 139 177)(9 146 140 178)(10 147 141 179)(11 148 142 180)(12 149 143 181)(13 150 144 182)(14 151 121 183)(15 152 122 184)(16 153 123 185)(17 154 124 186)(18 155 125 187)(19 156 126 188)(20 157 127 189)(21 158 128 190)(22 159 129 191)(23 160 130 192)(24 161 131 169)(25 59 87 118)(26 60 88 119)(27 61 89 120)(28 62 90 97)(29 63 91 98)(30 64 92 99)(31 65 93 100)(32 66 94 101)(33 67 95 102)(34 68 96 103)(35 69 73 104)(36 70 74 105)(37 71 75 106)(38 72 76 107)(39 49 77 108)(40 50 78 109)(41 51 79 110)(42 52 80 111)(43 53 81 112)(44 54 82 113)(45 55 83 114)(46 56 84 115)(47 57 85 116)(48 58 86 117)

G:=sub<Sym(192)| (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,43,132,81)(2,44,133,82)(3,45,134,83)(4,46,135,84)(5,47,136,85)(6,48,137,86)(7,25,138,87)(8,26,139,88)(9,27,140,89)(10,28,141,90)(11,29,142,91)(12,30,143,92)(13,31,144,93)(14,32,121,94)(15,33,122,95)(16,34,123,96)(17,35,124,73)(18,36,125,74)(19,37,126,75)(20,38,127,76)(21,39,128,77)(22,40,129,78)(23,41,130,79)(24,42,131,80)(49,158,108,190)(50,159,109,191)(51,160,110,192)(52,161,111,169)(53,162,112,170)(54,163,113,171)(55,164,114,172)(56,165,115,173)(57,166,116,174)(58,167,117,175)(59,168,118,176)(60,145,119,177)(61,146,120,178)(62,147,97,179)(63,148,98,180)(64,149,99,181)(65,150,100,182)(66,151,101,183)(67,152,102,184)(68,153,103,185)(69,154,104,186)(70,155,105,187)(71,156,106,188)(72,157,107,189), (1,162,132,170)(2,163,133,171)(3,164,134,172)(4,165,135,173)(5,166,136,174)(6,167,137,175)(7,168,138,176)(8,145,139,177)(9,146,140,178)(10,147,141,179)(11,148,142,180)(12,149,143,181)(13,150,144,182)(14,151,121,183)(15,152,122,184)(16,153,123,185)(17,154,124,186)(18,155,125,187)(19,156,126,188)(20,157,127,189)(21,158,128,190)(22,159,129,191)(23,160,130,192)(24,161,131,169)(25,59,87,118)(26,60,88,119)(27,61,89,120)(28,62,90,97)(29,63,91,98)(30,64,92,99)(31,65,93,100)(32,66,94,101)(33,67,95,102)(34,68,96,103)(35,69,73,104)(36,70,74,105)(37,71,75,106)(38,72,76,107)(39,49,77,108)(40,50,78,109)(41,51,79,110)(42,52,80,111)(43,53,81,112)(44,54,82,113)(45,55,83,114)(46,56,84,115)(47,57,85,116)(48,58,86,117)>;

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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,43,132,81)(2,44,133,82)(3,45,134,83)(4,46,135,84)(5,47,136,85)(6,48,137,86)(7,25,138,87)(8,26,139,88)(9,27,140,89)(10,28,141,90)(11,29,142,91)(12,30,143,92)(13,31,144,93)(14,32,121,94)(15,33,122,95)(16,34,123,96)(17,35,124,73)(18,36,125,74)(19,37,126,75)(20,38,127,76)(21,39,128,77)(22,40,129,78)(23,41,130,79)(24,42,131,80)(49,158,108,190)(50,159,109,191)(51,160,110,192)(52,161,111,169)(53,162,112,170)(54,163,113,171)(55,164,114,172)(56,165,115,173)(57,166,116,174)(58,167,117,175)(59,168,118,176)(60,145,119,177)(61,146,120,178)(62,147,97,179)(63,148,98,180)(64,149,99,181)(65,150,100,182)(66,151,101,183)(67,152,102,184)(68,153,103,185)(69,154,104,186)(70,155,105,187)(71,156,106,188)(72,157,107,189), (1,162,132,170)(2,163,133,171)(3,164,134,172)(4,165,135,173)(5,166,136,174)(6,167,137,175)(7,168,138,176)(8,145,139,177)(9,146,140,178)(10,147,141,179)(11,148,142,180)(12,149,143,181)(13,150,144,182)(14,151,121,183)(15,152,122,184)(16,153,123,185)(17,154,124,186)(18,155,125,187)(19,156,126,188)(20,157,127,189)(21,158,128,190)(22,159,129,191)(23,160,130,192)(24,161,131,169)(25,59,87,118)(26,60,88,119)(27,61,89,120)(28,62,90,97)(29,63,91,98)(30,64,92,99)(31,65,93,100)(32,66,94,101)(33,67,95,102)(34,68,96,103)(35,69,73,104)(36,70,74,105)(37,71,75,106)(38,72,76,107)(39,49,77,108)(40,50,78,109)(41,51,79,110)(42,52,80,111)(43,53,81,112)(44,54,82,113)(45,55,83,114)(46,56,84,115)(47,57,85,116)(48,58,86,117) );

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),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)], [(1,43,132,81),(2,44,133,82),(3,45,134,83),(4,46,135,84),(5,47,136,85),(6,48,137,86),(7,25,138,87),(8,26,139,88),(9,27,140,89),(10,28,141,90),(11,29,142,91),(12,30,143,92),(13,31,144,93),(14,32,121,94),(15,33,122,95),(16,34,123,96),(17,35,124,73),(18,36,125,74),(19,37,126,75),(20,38,127,76),(21,39,128,77),(22,40,129,78),(23,41,130,79),(24,42,131,80),(49,158,108,190),(50,159,109,191),(51,160,110,192),(52,161,111,169),(53,162,112,170),(54,163,113,171),(55,164,114,172),(56,165,115,173),(57,166,116,174),(58,167,117,175),(59,168,118,176),(60,145,119,177),(61,146,120,178),(62,147,97,179),(63,148,98,180),(64,149,99,181),(65,150,100,182),(66,151,101,183),(67,152,102,184),(68,153,103,185),(69,154,104,186),(70,155,105,187),(71,156,106,188),(72,157,107,189)], [(1,162,132,170),(2,163,133,171),(3,164,134,172),(4,165,135,173),(5,166,136,174),(6,167,137,175),(7,168,138,176),(8,145,139,177),(9,146,140,178),(10,147,141,179),(11,148,142,180),(12,149,143,181),(13,150,144,182),(14,151,121,183),(15,152,122,184),(16,153,123,185),(17,154,124,186),(18,155,125,187),(19,156,126,188),(20,157,127,189),(21,158,128,190),(22,159,129,191),(23,160,130,192),(24,161,131,169),(25,59,87,118),(26,60,88,119),(27,61,89,120),(28,62,90,97),(29,63,91,98),(30,64,92,99),(31,65,93,100),(32,66,94,101),(33,67,95,102),(34,68,96,103),(35,69,73,104),(36,70,74,105),(37,71,75,106),(38,72,76,107),(39,49,77,108),(40,50,78,109),(41,51,79,110),(42,52,80,111),(43,53,81,112),(44,54,82,113),(45,55,83,114),(46,56,84,115),(47,57,85,116),(48,58,86,117)]])

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E ··· 4P 6A ··· 6F 8A ··· 8H 8I ··· 8T 12A ··· 12H 12I ··· 12AF 24A ··· 24P 24Q ··· 24AN order 1 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 Q8 C4○D4 C3×Q8 C8○D4 C3×C4○D4 C3×C8○D4 kernel Q8×C24 C4×C24 C3×C4⋊C8 Q8×C12 C8×Q8 C3×C4⋊C4 C6×Q8 C4×C8 C4⋊C8 C4×Q8 C3×Q8 C4⋊C4 C2×Q8 Q8 C24 C12 C8 C6 C4 C2 # reps 1 3 3 1 2 6 2 6 6 2 16 12 4 32 2 2 4 4 4 8

Matrix representation of Q8×C24 in GL3(𝔽73) generated by

 10 0 0 0 9 0 0 0 9
,
 72 0 0 0 0 1 0 72 0
,
 72 0 0 0 43 62 0 62 30
G:=sub<GL(3,GF(73))| [10,0,0,0,9,0,0,0,9],[72,0,0,0,0,72,0,1,0],[72,0,0,0,43,62,0,62,30] >;

Q8×C24 in GAP, Magma, Sage, TeX

Q_8\times C_{24}
% in TeX

G:=Group("Q8xC24");
// GroupNames label

G:=SmallGroup(192,878);
// by ID

G=gap.SmallGroup(192,878);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,394,124]);
// Polycyclic

G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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