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## G = C3×C4.10D8order 192 = 26·3

### Direct product of C3 and C4.10D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C4.10D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C12 — C3×C4⋊C8 — C3×C4.10D8
 Lower central C1 — C22 — C2×C4 — C3×C4.10D8
 Upper central C1 — C2×C6 — C4×C12 — C3×C4.10D8

Generators and relations for C3×C4.10D8
G = < a,b,c,d | a3=b4=c8=1, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=bc-1 >

Subgroups: 114 in 64 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C4⋊C8, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, C4.10D8, C3×C4⋊C8, C3×C4⋊Q8, C3×C4.10D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, Q16, C2×C12, C3×D4, C4.10D4, D4⋊C4, Q8⋊C4, C3×C22⋊C4, C3×D8, C3×SD16, C3×Q16, C4.10D8, C3×C4.10D4, C3×D4⋊C4, C3×Q8⋊C4, C3×C4.10D8

Smallest permutation representation of C3×C4.10D8
Regular action on 192 points
Generators in S192
(1 167 159)(2 168 160)(3 161 153)(4 162 154)(5 163 155)(6 164 156)(7 165 157)(8 166 158)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(13 29 21)(14 30 22)(15 31 23)(16 32 24)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(37 53 45)(38 54 46)(39 55 47)(40 56 48)(57 73 65)(58 74 66)(59 75 67)(60 76 68)(61 77 69)(62 78 70)(63 79 71)(64 80 72)(81 97 89)(82 98 90)(83 99 91)(84 100 92)(85 101 93)(86 102 94)(87 103 95)(88 104 96)(105 121 113)(106 122 114)(107 123 115)(108 124 116)(109 125 117)(110 126 118)(111 127 119)(112 128 120)(129 145 137)(130 146 138)(131 147 139)(132 148 140)(133 149 141)(134 150 142)(135 151 143)(136 152 144)(169 185 177)(170 186 178)(171 187 179)(172 188 180)(173 189 181)(174 190 182)(175 191 183)(176 192 184)
(1 57 11 109)(2 110 12 58)(3 59 13 111)(4 112 14 60)(5 61 15 105)(6 106 16 62)(7 63 9 107)(8 108 10 64)(17 115 157 71)(18 72 158 116)(19 117 159 65)(20 66 160 118)(21 119 153 67)(22 68 154 120)(23 113 155 69)(24 70 156 114)(25 123 165 79)(26 80 166 124)(27 125 167 73)(28 74 168 126)(29 127 161 75)(30 76 162 128)(31 121 163 77)(32 78 164 122)(33 83 173 135)(34 136 174 84)(35 85 175 129)(36 130 176 86)(37 87 169 131)(38 132 170 88)(39 81 171 133)(40 134 172 82)(41 91 181 143)(42 144 182 92)(43 93 183 137)(44 138 184 94)(45 95 177 139)(46 140 178 96)(47 89 179 141)(48 142 180 90)(49 99 189 151)(50 152 190 100)(51 101 191 145)(52 146 192 102)(53 103 185 147)(54 148 186 104)(55 97 187 149)(56 150 188 98)
(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)
(1 132 109 38 11 88 57 170)(2 37 58 131 12 169 110 87)(3 130 111 36 13 86 59 176)(4 35 60 129 14 175 112 85)(5 136 105 34 15 84 61 174)(6 33 62 135 16 173 106 83)(7 134 107 40 9 82 63 172)(8 39 64 133 10 171 108 81)(17 90 71 180 157 142 115 48)(18 179 116 89 158 47 72 141)(19 96 65 178 159 140 117 46)(20 177 118 95 160 45 66 139)(21 94 67 184 153 138 119 44)(22 183 120 93 154 43 68 137)(23 92 69 182 155 144 113 42)(24 181 114 91 156 41 70 143)(25 98 79 188 165 150 123 56)(26 187 124 97 166 55 80 149)(27 104 73 186 167 148 125 54)(28 185 126 103 168 53 74 147)(29 102 75 192 161 146 127 52)(30 191 128 101 162 51 76 145)(31 100 77 190 163 152 121 50)(32 189 122 99 164 49 78 151)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 8A ··· 8H 12A ··· 12H 12I 12J 12K 12L 12M 12N 24A ··· 24P order 1 2 2 2 3 3 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 1 1 2 2 2 2 4 8 8 1 ··· 1 4 ··· 4 2 ··· 2 4 4 8 8 8 8 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + - - image C1 C2 C2 C3 C4 C6 C6 C12 D4 D8 SD16 Q16 C3×D4 C3×D8 C3×SD16 C3×Q16 C4.10D4 C3×C4.10D4 kernel C3×C4.10D8 C3×C4⋊C8 C3×C4⋊Q8 C4.10D8 C3×C4⋊C4 C4⋊C8 C4⋊Q8 C4⋊C4 C2×C12 C12 C12 C12 C2×C4 C4 C4 C4 C6 C2 # reps 1 2 1 2 4 4 2 8 2 2 4 2 4 4 8 4 1 2

Matrix representation of C3×C4.10D8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 0 72 0 0 1 0 0 0 0 0 72 0 0 0 0 72
,
 27 32 0 0 32 46 0 0 0 0 0 61 0 0 6 12
,
 67 6 0 0 67 67 0 0 0 0 41 60 0 0 62 32
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[0,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[27,32,0,0,32,46,0,0,0,0,0,6,0,0,61,12],[67,67,0,0,6,67,0,0,0,0,41,62,0,0,60,32] >;

C3×C4.10D8 in GAP, Magma, Sage, TeX

C_3\times C_4._{10}D_8
% in TeX

G:=Group("C3xC4.10D8");
// GroupNames label

G:=SmallGroup(192,138);
// by ID

G=gap.SmallGroup(192,138);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,1683,1522,248,2951,242]);
// Polycyclic

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

׿
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