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G = C32⋊D7order 448 = 26·7

3rd semidirect product of C32 and D7 acting via D7/C7=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C323D7, C2246C2, D14.C16, C71M6(2), Dic7.C16, C16.20D14, C112.25C22, C7⋊C324C2, C7⋊C8.2C8, C7⋊C16.2C4, (C4×D7).2C8, (C8×D7).2C4, C2.3(D7×C16), C4.17(C8×D7), C8.37(C4×D7), C28.22(C2×C8), C14.2(C2×C16), C56.54(C2×C4), (D7×C16).2C2, SmallGroup(448,4)

Series: Derived Chief Lower central Upper central

C1C14 — C32⋊D7
C1C7C14C28C56C112D7×C16 — C32⋊D7
C7C14 — C32⋊D7
C1C16C32

Generators and relations for C32⋊D7
 G = < a,b,c | a32=b7=c2=1, ab=ba, cac=a17, cbc=b-1 >

14C2
7C22
7C4
2D7
7C2×C4
7C8
7C2×C8
7C16
7C2×C16
7C32
7M6(2)

Smallest permutation representation of C32⋊D7
On 224 points
Generators in S224
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 189 91 44 157 121 205)(2 190 92 45 158 122 206)(3 191 93 46 159 123 207)(4 192 94 47 160 124 208)(5 161 95 48 129 125 209)(6 162 96 49 130 126 210)(7 163 65 50 131 127 211)(8 164 66 51 132 128 212)(9 165 67 52 133 97 213)(10 166 68 53 134 98 214)(11 167 69 54 135 99 215)(12 168 70 55 136 100 216)(13 169 71 56 137 101 217)(14 170 72 57 138 102 218)(15 171 73 58 139 103 219)(16 172 74 59 140 104 220)(17 173 75 60 141 105 221)(18 174 76 61 142 106 222)(19 175 77 62 143 107 223)(20 176 78 63 144 108 224)(21 177 79 64 145 109 193)(22 178 80 33 146 110 194)(23 179 81 34 147 111 195)(24 180 82 35 148 112 196)(25 181 83 36 149 113 197)(26 182 84 37 150 114 198)(27 183 85 38 151 115 199)(28 184 86 39 152 116 200)(29 185 87 40 153 117 201)(30 186 88 41 154 118 202)(31 187 89 42 155 119 203)(32 188 90 43 156 120 204)
(1 205)(2 222)(3 207)(4 224)(5 209)(6 194)(7 211)(8 196)(9 213)(10 198)(11 215)(12 200)(13 217)(14 202)(15 219)(16 204)(17 221)(18 206)(19 223)(20 208)(21 193)(22 210)(23 195)(24 212)(25 197)(26 214)(27 199)(28 216)(29 201)(30 218)(31 203)(32 220)(33 49)(35 51)(37 53)(39 55)(41 57)(43 59)(45 61)(47 63)(65 131)(66 148)(67 133)(68 150)(69 135)(70 152)(71 137)(72 154)(73 139)(74 156)(75 141)(76 158)(77 143)(78 160)(79 145)(80 130)(81 147)(82 132)(83 149)(84 134)(85 151)(86 136)(87 153)(88 138)(89 155)(90 140)(91 157)(92 142)(93 159)(94 144)(95 129)(96 146)(97 165)(98 182)(99 167)(100 184)(101 169)(102 186)(103 171)(104 188)(105 173)(106 190)(107 175)(108 192)(109 177)(110 162)(111 179)(112 164)(113 181)(114 166)(115 183)(116 168)(117 185)(118 170)(119 187)(120 172)(121 189)(122 174)(123 191)(124 176)(125 161)(126 178)(127 163)(128 180)

G:=sub<Sym(224)| (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,189,91,44,157,121,205)(2,190,92,45,158,122,206)(3,191,93,46,159,123,207)(4,192,94,47,160,124,208)(5,161,95,48,129,125,209)(6,162,96,49,130,126,210)(7,163,65,50,131,127,211)(8,164,66,51,132,128,212)(9,165,67,52,133,97,213)(10,166,68,53,134,98,214)(11,167,69,54,135,99,215)(12,168,70,55,136,100,216)(13,169,71,56,137,101,217)(14,170,72,57,138,102,218)(15,171,73,58,139,103,219)(16,172,74,59,140,104,220)(17,173,75,60,141,105,221)(18,174,76,61,142,106,222)(19,175,77,62,143,107,223)(20,176,78,63,144,108,224)(21,177,79,64,145,109,193)(22,178,80,33,146,110,194)(23,179,81,34,147,111,195)(24,180,82,35,148,112,196)(25,181,83,36,149,113,197)(26,182,84,37,150,114,198)(27,183,85,38,151,115,199)(28,184,86,39,152,116,200)(29,185,87,40,153,117,201)(30,186,88,41,154,118,202)(31,187,89,42,155,119,203)(32,188,90,43,156,120,204), (1,205)(2,222)(3,207)(4,224)(5,209)(6,194)(7,211)(8,196)(9,213)(10,198)(11,215)(12,200)(13,217)(14,202)(15,219)(16,204)(17,221)(18,206)(19,223)(20,208)(21,193)(22,210)(23,195)(24,212)(25,197)(26,214)(27,199)(28,216)(29,201)(30,218)(31,203)(32,220)(33,49)(35,51)(37,53)(39,55)(41,57)(43,59)(45,61)(47,63)(65,131)(66,148)(67,133)(68,150)(69,135)(70,152)(71,137)(72,154)(73,139)(74,156)(75,141)(76,158)(77,143)(78,160)(79,145)(80,130)(81,147)(82,132)(83,149)(84,134)(85,151)(86,136)(87,153)(88,138)(89,155)(90,140)(91,157)(92,142)(93,159)(94,144)(95,129)(96,146)(97,165)(98,182)(99,167)(100,184)(101,169)(102,186)(103,171)(104,188)(105,173)(106,190)(107,175)(108,192)(109,177)(110,162)(111,179)(112,164)(113,181)(114,166)(115,183)(116,168)(117,185)(118,170)(119,187)(120,172)(121,189)(122,174)(123,191)(124,176)(125,161)(126,178)(127,163)(128,180)>;

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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,189,91,44,157,121,205)(2,190,92,45,158,122,206)(3,191,93,46,159,123,207)(4,192,94,47,160,124,208)(5,161,95,48,129,125,209)(6,162,96,49,130,126,210)(7,163,65,50,131,127,211)(8,164,66,51,132,128,212)(9,165,67,52,133,97,213)(10,166,68,53,134,98,214)(11,167,69,54,135,99,215)(12,168,70,55,136,100,216)(13,169,71,56,137,101,217)(14,170,72,57,138,102,218)(15,171,73,58,139,103,219)(16,172,74,59,140,104,220)(17,173,75,60,141,105,221)(18,174,76,61,142,106,222)(19,175,77,62,143,107,223)(20,176,78,63,144,108,224)(21,177,79,64,145,109,193)(22,178,80,33,146,110,194)(23,179,81,34,147,111,195)(24,180,82,35,148,112,196)(25,181,83,36,149,113,197)(26,182,84,37,150,114,198)(27,183,85,38,151,115,199)(28,184,86,39,152,116,200)(29,185,87,40,153,117,201)(30,186,88,41,154,118,202)(31,187,89,42,155,119,203)(32,188,90,43,156,120,204), (1,205)(2,222)(3,207)(4,224)(5,209)(6,194)(7,211)(8,196)(9,213)(10,198)(11,215)(12,200)(13,217)(14,202)(15,219)(16,204)(17,221)(18,206)(19,223)(20,208)(21,193)(22,210)(23,195)(24,212)(25,197)(26,214)(27,199)(28,216)(29,201)(30,218)(31,203)(32,220)(33,49)(35,51)(37,53)(39,55)(41,57)(43,59)(45,61)(47,63)(65,131)(66,148)(67,133)(68,150)(69,135)(70,152)(71,137)(72,154)(73,139)(74,156)(75,141)(76,158)(77,143)(78,160)(79,145)(80,130)(81,147)(82,132)(83,149)(84,134)(85,151)(86,136)(87,153)(88,138)(89,155)(90,140)(91,157)(92,142)(93,159)(94,144)(95,129)(96,146)(97,165)(98,182)(99,167)(100,184)(101,169)(102,186)(103,171)(104,188)(105,173)(106,190)(107,175)(108,192)(109,177)(110,162)(111,179)(112,164)(113,181)(114,166)(115,183)(116,168)(117,185)(118,170)(119,187)(120,172)(121,189)(122,174)(123,191)(124,176)(125,161)(126,178)(127,163)(128,180) );

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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,189,91,44,157,121,205),(2,190,92,45,158,122,206),(3,191,93,46,159,123,207),(4,192,94,47,160,124,208),(5,161,95,48,129,125,209),(6,162,96,49,130,126,210),(7,163,65,50,131,127,211),(8,164,66,51,132,128,212),(9,165,67,52,133,97,213),(10,166,68,53,134,98,214),(11,167,69,54,135,99,215),(12,168,70,55,136,100,216),(13,169,71,56,137,101,217),(14,170,72,57,138,102,218),(15,171,73,58,139,103,219),(16,172,74,59,140,104,220),(17,173,75,60,141,105,221),(18,174,76,61,142,106,222),(19,175,77,62,143,107,223),(20,176,78,63,144,108,224),(21,177,79,64,145,109,193),(22,178,80,33,146,110,194),(23,179,81,34,147,111,195),(24,180,82,35,148,112,196),(25,181,83,36,149,113,197),(26,182,84,37,150,114,198),(27,183,85,38,151,115,199),(28,184,86,39,152,116,200),(29,185,87,40,153,117,201),(30,186,88,41,154,118,202),(31,187,89,42,155,119,203),(32,188,90,43,156,120,204)], [(1,205),(2,222),(3,207),(4,224),(5,209),(6,194),(7,211),(8,196),(9,213),(10,198),(11,215),(12,200),(13,217),(14,202),(15,219),(16,204),(17,221),(18,206),(19,223),(20,208),(21,193),(22,210),(23,195),(24,212),(25,197),(26,214),(27,199),(28,216),(29,201),(30,218),(31,203),(32,220),(33,49),(35,51),(37,53),(39,55),(41,57),(43,59),(45,61),(47,63),(65,131),(66,148),(67,133),(68,150),(69,135),(70,152),(71,137),(72,154),(73,139),(74,156),(75,141),(76,158),(77,143),(78,160),(79,145),(80,130),(81,147),(82,132),(83,149),(84,134),(85,151),(86,136),(87,153),(88,138),(89,155),(90,140),(91,157),(92,142),(93,159),(94,144),(95,129),(96,146),(97,165),(98,182),(99,167),(100,184),(101,169),(102,186),(103,171),(104,188),(105,173),(106,190),(107,175),(108,192),(109,177),(110,162),(111,179),(112,164),(113,181),(114,166),(115,183),(116,168),(117,185),(118,170),(119,187),(120,172),(121,189),(122,174),(123,191),(124,176),(125,161),(126,178),(127,163),(128,180)])

136 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D8E8F14A14B14C16A···16H16I16J16K16L28A···28F32A···32H32I···32P56A···56L112A···112X224A···224AV
order12244477788888814141416···161616161628···2832···3232···3256···56112···112224···224
size11141114222111114142221···1141414142···22···214···142···22···22···2

136 irreducible representations

dim11111111112222222
type++++++
imageC1C2C2C2C4C4C8C8C16C16D7D14C4×D7M6(2)C8×D7D7×C16C32⋊D7
kernelC32⋊D7C7⋊C32C224D7×C16C7⋊C16C8×D7C7⋊C8C4×D7Dic7D14C32C16C8C7C4C2C1
# reps11112244883368122448

Matrix representation of C32⋊D7 in GL2(𝔽449) generated by

131382
67318
,
431
4480
,
01
10
G:=sub<GL(2,GF(449))| [131,67,382,318],[43,448,1,0],[0,1,1,0] >;

C32⋊D7 in GAP, Magma, Sage, TeX

C_{32}\rtimes D_7
% in TeX

G:=Group("C32:D7");
// GroupNames label

G:=SmallGroup(448,4);
// by ID

G=gap.SmallGroup(448,4);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,36,58,80,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^32=b^7=c^2=1,a*b=b*a,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C32⋊D7 in TeX

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