Copied to
clipboard

## G = C22⋊Dic22order 352 = 25·11

### The semidirect product of C22 and Dic22 acting via Dic22/Dic11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C22⋊Dic22
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C22×Dic11 — C22⋊Dic22
 Lower central C11 — C2×C22 — C22⋊Dic22
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22⋊Dic22
G = < a,b,c,d | a2=b2=c44=1, d2=c22, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 370 in 74 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C11, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C22, C22, C22⋊Q8, Dic11, Dic11, C44, C2×C22, C2×C22, C2×C22, Dic22, C2×Dic11, C2×Dic11, C2×C44, C22×C22, Dic11⋊C4, C44⋊C4, C23.D11, C11×C22⋊C4, C2×Dic22, C22×Dic11, C22⋊Dic22
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D11, C22⋊Q8, D22, Dic22, C22×D11, C2×Dic22, D4×D11, D42D11, C22⋊Dic22

Smallest permutation representation of C22⋊Dic22
On 176 points
Generators in S176
(1 23)(2 73)(3 25)(4 75)(5 27)(6 77)(7 29)(8 79)(9 31)(10 81)(11 33)(12 83)(13 35)(14 85)(15 37)(16 87)(17 39)(18 45)(19 41)(20 47)(21 43)(22 49)(24 51)(26 53)(28 55)(30 57)(32 59)(34 61)(36 63)(38 65)(40 67)(42 69)(44 71)(46 68)(48 70)(50 72)(52 74)(54 76)(56 78)(58 80)(60 82)(62 84)(64 86)(66 88)(89 133)(90 112)(91 135)(92 114)(93 137)(94 116)(95 139)(96 118)(97 141)(98 120)(99 143)(100 122)(101 145)(102 124)(103 147)(104 126)(105 149)(106 128)(107 151)(108 130)(109 153)(110 132)(111 155)(113 157)(115 159)(117 161)(119 163)(121 165)(123 167)(125 169)(127 171)(129 173)(131 175)(134 156)(136 158)(138 160)(140 162)(142 164)(144 166)(146 168)(148 170)(150 172)(152 174)(154 176)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 45)(41 46)(42 47)(43 48)(44 49)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)(109 175)(110 176)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 141)(120 142)(121 143)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 153)(132 154)
(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)
(1 100 23 122)(2 99 24 121)(3 98 25 120)(4 97 26 119)(5 96 27 118)(6 95 28 117)(7 94 29 116)(8 93 30 115)(9 92 31 114)(10 91 32 113)(11 90 33 112)(12 89 34 111)(13 132 35 110)(14 131 36 109)(15 130 37 108)(16 129 38 107)(17 128 39 106)(18 127 40 105)(19 126 41 104)(20 125 42 103)(21 124 43 102)(22 123 44 101)(45 171 67 149)(46 170 68 148)(47 169 69 147)(48 168 70 146)(49 167 71 145)(50 166 72 144)(51 165 73 143)(52 164 74 142)(53 163 75 141)(54 162 76 140)(55 161 77 139)(56 160 78 138)(57 159 79 137)(58 158 80 136)(59 157 81 135)(60 156 82 134)(61 155 83 133)(62 154 84 176)(63 153 85 175)(64 152 86 174)(65 151 87 173)(66 150 88 172)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 11A ··· 11E 22A ··· 22O 22P ··· 22Y 44A ··· 44T order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 1 1 2 2 4 4 22 22 22 22 44 44 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 C2 D4 Q8 C4○D4 D11 D22 D22 Dic22 D4×D11 D4⋊2D11 kernel C22⋊Dic22 Dic11⋊C4 C44⋊C4 C23.D11 C11×C22⋊C4 C2×Dic22 C22×Dic11 Dic11 C2×C22 C22 C22⋊C4 C2×C4 C23 C22 C2 C2 # reps 1 2 1 1 1 1 1 2 2 2 5 10 5 20 5 5

Matrix representation of C22⋊Dic22 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 1 0 0 0 0 88
,
 1 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
,
 10 0 0 0 47 9 0 0 0 0 0 1 0 0 1 0
,
 5 87 0 0 13 84 0 0 0 0 88 0 0 0 0 88
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[10,47,0,0,0,9,0,0,0,0,0,1,0,0,1,0],[5,13,0,0,87,84,0,0,0,0,88,0,0,0,0,88] >;

C22⋊Dic22 in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm Dic}_{22}
% in TeX

G:=Group("C2^2:Dic22");
// GroupNames label

G:=SmallGroup(352,73);
// by ID

G=gap.SmallGroup(352,73);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,218,188,50,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^44=1,d^2=c^22,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽