Copied to
clipboard

## G = C4×C3⋊Q16order 192 = 26·3

### Direct product of C4 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4×C3⋊Q16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — C2×C3⋊Q16 — C4×C3⋊Q16
 Lower central C3 — C6 — C12 — C4×C3⋊Q16
 Upper central C1 — C2×C4 — C42 — C4×Q8

Generators and relations for C4×C3⋊Q16
G = < a,b,c,d | a4=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 232 in 110 conjugacy classes, 55 normal (39 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C4 [×7], C22, C6 [×3], C8 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×4], Dic3 [×3], C12 [×2], C12 [×2], C12 [×4], C2×C6, C42, C42 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, C4×C8, Q8⋊C4 [×2], C2.D8, C4×Q8, C4×Q8, C2×Q16, C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×4], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C4×Q16, C4×C3⋊C8, C6.Q16, C6.SD16, Q82Dic3, C4×Dic6, C2×C3⋊Q16, Q8×C12, C4×C3⋊Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], Q16 [×2], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C2×Q16, C4○D8, C3⋊Q16 [×2], S3×C2×C4, C4○D12, C2×C3⋊D4, C4×Q16, C4×C3⋊D4, C2×C3⋊Q16, Q8.13D6, C4×C3⋊Q16

Smallest permutation representation of C4×C3⋊Q16
Regular action on 192 points
Generators in S192
(1 55 39 157)(2 56 40 158)(3 49 33 159)(4 50 34 160)(5 51 35 153)(6 52 36 154)(7 53 37 155)(8 54 38 156)(9 182 109 68)(10 183 110 69)(11 184 111 70)(12 177 112 71)(13 178 105 72)(14 179 106 65)(15 180 107 66)(16 181 108 67)(17 87 126 30)(18 88 127 31)(19 81 128 32)(20 82 121 25)(21 83 122 26)(22 84 123 27)(23 85 124 28)(24 86 125 29)(41 141 135 120)(42 142 136 113)(43 143 129 114)(44 144 130 115)(45 137 131 116)(46 138 132 117)(47 139 133 118)(48 140 134 119)(57 104 164 185)(58 97 165 186)(59 98 166 187)(60 99 167 188)(61 100 168 189)(62 101 161 190)(63 102 162 191)(64 103 163 192)(73 93 171 146)(74 94 172 147)(75 95 173 148)(76 96 174 149)(77 89 175 150)(78 90 176 151)(79 91 169 152)(80 92 170 145)
(1 68 187)(2 188 69)(3 70 189)(4 190 71)(5 72 191)(6 192 65)(7 66 185)(8 186 67)(9 59 55)(10 56 60)(11 61 49)(12 50 62)(13 63 51)(14 52 64)(15 57 53)(16 54 58)(17 152 114)(18 115 145)(19 146 116)(20 117 147)(21 148 118)(22 119 149)(23 150 120)(24 113 151)(25 132 172)(26 173 133)(27 134 174)(28 175 135)(29 136 176)(30 169 129)(31 130 170)(32 171 131)(33 184 100)(34 101 177)(35 178 102)(36 103 179)(37 180 104)(38 97 181)(39 182 98)(40 99 183)(41 85 77)(42 78 86)(43 87 79)(44 80 88)(45 81 73)(46 74 82)(47 83 75)(48 76 84)(89 141 124)(90 125 142)(91 143 126)(92 127 144)(93 137 128)(94 121 138)(95 139 122)(96 123 140)(105 162 153)(106 154 163)(107 164 155)(108 156 165)(109 166 157)(110 158 167)(111 168 159)(112 160 161)
(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 95 5 91)(2 94 6 90)(3 93 7 89)(4 92 8 96)(9 133 13 129)(10 132 14 136)(11 131 15 135)(12 130 16 134)(17 98 21 102)(18 97 22 101)(19 104 23 100)(20 103 24 99)(25 64 29 60)(26 63 30 59)(27 62 31 58)(28 61 32 57)(33 146 37 150)(34 145 38 149)(35 152 39 148)(36 151 40 147)(41 111 45 107)(42 110 46 106)(43 109 47 105)(44 108 48 112)(49 171 53 175)(50 170 54 174)(51 169 55 173)(52 176 56 172)(65 142 69 138)(66 141 70 137)(67 140 71 144)(68 139 72 143)(73 155 77 159)(74 154 78 158)(75 153 79 157)(76 160 80 156)(81 164 85 168)(82 163 86 167)(83 162 87 166)(84 161 88 165)(113 183 117 179)(114 182 118 178)(115 181 119 177)(116 180 120 184)(121 192 125 188)(122 191 126 187)(123 190 127 186)(124 189 128 185)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A 6B 6C 8A ··· 8H 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 1 1 2 1 1 1 1 2 2 2 2 4 4 4 4 12 12 12 12 2 2 2 6 ··· 6 2 2 2 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 Q16 C4○D4 C3⋊D4 C4×S3 C4○D8 C4○D12 C3⋊Q16 Q8.13D6 kernel C4×C3⋊Q16 C4×C3⋊C8 C6.Q16 C6.SD16 Q8⋊2Dic3 C4×Dic6 C2×C3⋊Q16 Q8×C12 C3⋊Q16 C4×Q8 C2×C12 C42 C4⋊C4 C2×Q8 C12 C12 C2×C4 Q8 C6 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 4 2 4 4 4 4 2 2

Matrix representation of C4×C3⋊Q16 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 27 0 0 0 0 27
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 16 57 0 0 16 16 0 0 0 0 12 10 0 0 22 61
,
 19 52 0 0 52 54 0 0 0 0 43 60 0 0 13 30
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[16,16,0,0,57,16,0,0,0,0,12,22,0,0,10,61],[19,52,0,0,52,54,0,0,0,0,43,13,0,0,60,30] >;

C4×C3⋊Q16 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes Q_{16}
% in TeX

G:=Group("C4xC3:Q16");
// GroupNames label

G:=SmallGroup(192,588);
// by ID

G=gap.SmallGroup(192,588);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,58,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽