metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D44, C23.16D22, D22⋊C4⋊7C2, C44⋊C4⋊5C2, (C2×C22).4D4, C22.6(C2×D4), (C2×C4).9D22, C2.8(C2×D44), C22⋊C4⋊6D11, (C2×C44).3C22, C22.23(C4○D4), (C2×C22).27C23, (C22×Dic11)⋊2C2, C11⋊2(C22.D4), C2.10(D4⋊2D11), (C22×C22).16C22, (C22×D11).5C22, C22.45(C22×D11), (C2×Dic11).28C22, (C11×C22⋊C4)⋊4C2, (C2×C11⋊D4).5C2, SmallGroup(352,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D44
G = < a,b,c,d | a2=b2=c44=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 466 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C22, C22, C22.D4, Dic11, C44, D22, C2×C22, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×C22, C44⋊C4, D22⋊C4, C11×C22⋊C4, C22×Dic11, C2×C11⋊D4, C22.D44
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C22.D4, D22, D44, C22×D11, C2×D44, D4⋊2D11, C22.D44
►On 176 points
Generators in S176
(1 159)(2 73)(3 161)(4 75)(5 163)(6 77)(7 165)(8 79)(9 167)(10 81)(11 169)(12 83)(13 171)(14 85)(15 173)(16 87)(17 175)(18 45)(19 133)(20 47)(21 135)(22 49)(23 137)(24 51)(25 139)(26 53)(27 141)(28 55)(29 143)(30 57)(31 145)(32 59)(33 147)(34 61)(35 149)(36 63)(37 151)(38 65)(39 153)(40 67)(41 155)(42 69)(43 157)(44 71)(46 93)(48 95)(50 97)(52 99)(54 101)(56 103)(58 105)(60 107)(62 109)(64 111)(66 113)(68 115)(70 117)(72 119)(74 121)(76 123)(78 125)(80 127)(82 129)(84 131)(86 89)(88 91)(90 174)(92 176)(94 134)(96 136)(98 138)(100 140)(102 142)(104 144)(106 146)(108 148)(110 150)(112 152)(114 154)(116 156)(118 158)(120 160)(122 162)(124 164)(126 166)(128 168)(130 170)(132 172)
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 176)(46 133)(47 134)(48 135)(49 136)(50 137)(51 138)(52 139)(53 140)(54 141)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 169)(83 170)(84 171)(85 172)(86 173)(87 174)(88 175)
(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)
(1 118 119 44)(2 43 120 117)(3 116 121 42)(4 41 122 115)(5 114 123 40)(6 39 124 113)(7 112 125 38)(8 37 126 111)(9 110 127 36)(10 35 128 109)(11 108 129 34)(12 33 130 107)(13 106 131 32)(14 31 132 105)(15 104 89 30)(16 29 90 103)(17 102 91 28)(18 27 92 101)(19 100 93 26)(20 25 94 99)(21 98 95 24)(22 23 96 97)(45 141 176 54)(46 53 133 140)(47 139 134 52)(48 51 135 138)(49 137 136 50)(55 175 142 88)(56 87 143 174)(57 173 144 86)(58 85 145 172)(59 171 146 84)(60 83 147 170)(61 169 148 82)(62 81 149 168)(63 167 150 80)(64 79 151 166)(65 165 152 78)(66 77 153 164)(67 163 154 76)(68 75 155 162)(69 161 156 74)(70 73 157 160)(71 159 158 72)
G:=sub<Sym(176)| (1,159)(2,73)(3,161)(4,75)(5,163)(6,77)(7,165)(8,79)(9,167)(10,81)(11,169)(12,83)(13,171)(14,85)(15,173)(16,87)(17,175)(18,45)(19,133)(20,47)(21,135)(22,49)(23,137)(24,51)(25,139)(26,53)(27,141)(28,55)(29,143)(30,57)(31,145)(32,59)(33,147)(34,61)(35,149)(36,63)(37,151)(38,65)(39,153)(40,67)(41,155)(42,69)(43,157)(44,71)(46,93)(48,95)(50,97)(52,99)(54,101)(56,103)(58,105)(60,107)(62,109)(64,111)(66,113)(68,115)(70,117)(72,119)(74,121)(76,123)(78,125)(80,127)(82,129)(84,131)(86,89)(88,91)(90,174)(92,176)(94,134)(96,136)(98,138)(100,140)(102,142)(104,144)(106,146)(108,148)(110,150)(112,152)(114,154)(116,156)(118,158)(120,160)(122,162)(124,164)(126,166)(128,168)(130,170)(132,172), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,176)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,140)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,169)(83,170)(84,171)(85,172)(86,173)(87,174)(88,175), (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), (1,118,119,44)(2,43,120,117)(3,116,121,42)(4,41,122,115)(5,114,123,40)(6,39,124,113)(7,112,125,38)(8,37,126,111)(9,110,127,36)(10,35,128,109)(11,108,129,34)(12,33,130,107)(13,106,131,32)(14,31,132,105)(15,104,89,30)(16,29,90,103)(17,102,91,28)(18,27,92,101)(19,100,93,26)(20,25,94,99)(21,98,95,24)(22,23,96,97)(45,141,176,54)(46,53,133,140)(47,139,134,52)(48,51,135,138)(49,137,136,50)(55,175,142,88)(56,87,143,174)(57,173,144,86)(58,85,145,172)(59,171,146,84)(60,83,147,170)(61,169,148,82)(62,81,149,168)(63,167,150,80)(64,79,151,166)(65,165,152,78)(66,77,153,164)(67,163,154,76)(68,75,155,162)(69,161,156,74)(70,73,157,160)(71,159,158,72)>;
G:=Group( (1,159)(2,73)(3,161)(4,75)(5,163)(6,77)(7,165)(8,79)(9,167)(10,81)(11,169)(12,83)(13,171)(14,85)(15,173)(16,87)(17,175)(18,45)(19,133)(20,47)(21,135)(22,49)(23,137)(24,51)(25,139)(26,53)(27,141)(28,55)(29,143)(30,57)(31,145)(32,59)(33,147)(34,61)(35,149)(36,63)(37,151)(38,65)(39,153)(40,67)(41,155)(42,69)(43,157)(44,71)(46,93)(48,95)(50,97)(52,99)(54,101)(56,103)(58,105)(60,107)(62,109)(64,111)(66,113)(68,115)(70,117)(72,119)(74,121)(76,123)(78,125)(80,127)(82,129)(84,131)(86,89)(88,91)(90,174)(92,176)(94,134)(96,136)(98,138)(100,140)(102,142)(104,144)(106,146)(108,148)(110,150)(112,152)(114,154)(116,156)(118,158)(120,160)(122,162)(124,164)(126,166)(128,168)(130,170)(132,172), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,176)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,140)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,169)(83,170)(84,171)(85,172)(86,173)(87,174)(88,175), (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), (1,118,119,44)(2,43,120,117)(3,116,121,42)(4,41,122,115)(5,114,123,40)(6,39,124,113)(7,112,125,38)(8,37,126,111)(9,110,127,36)(10,35,128,109)(11,108,129,34)(12,33,130,107)(13,106,131,32)(14,31,132,105)(15,104,89,30)(16,29,90,103)(17,102,91,28)(18,27,92,101)(19,100,93,26)(20,25,94,99)(21,98,95,24)(22,23,96,97)(45,141,176,54)(46,53,133,140)(47,139,134,52)(48,51,135,138)(49,137,136,50)(55,175,142,88)(56,87,143,174)(57,173,144,86)(58,85,145,172)(59,171,146,84)(60,83,147,170)(61,169,148,82)(62,81,149,168)(63,167,150,80)(64,79,151,166)(65,165,152,78)(66,77,153,164)(67,163,154,76)(68,75,155,162)(69,161,156,74)(70,73,157,160)(71,159,158,72) );
G=PermutationGroup([[(1,159),(2,73),(3,161),(4,75),(5,163),(6,77),(7,165),(8,79),(9,167),(10,81),(11,169),(12,83),(13,171),(14,85),(15,173),(16,87),(17,175),(18,45),(19,133),(20,47),(21,135),(22,49),(23,137),(24,51),(25,139),(26,53),(27,141),(28,55),(29,143),(30,57),(31,145),(32,59),(33,147),(34,61),(35,149),(36,63),(37,151),(38,65),(39,153),(40,67),(41,155),(42,69),(43,157),(44,71),(46,93),(48,95),(50,97),(52,99),(54,101),(56,103),(58,105),(60,107),(62,109),(64,111),(66,113),(68,115),(70,117),(72,119),(74,121),(76,123),(78,125),(80,127),(82,129),(84,131),(86,89),(88,91),(90,174),(92,176),(94,134),(96,136),(98,138),(100,140),(102,142),(104,144),(106,146),(108,148),(110,150),(112,152),(114,154),(116,156),(118,158),(120,160),(122,162),(124,164),(126,166),(128,168),(130,170),(132,172)], [(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,176),(46,133),(47,134),(48,135),(49,136),(50,137),(51,138),(52,139),(53,140),(54,141),(55,142),(56,143),(57,144),(58,145),(59,146),(60,147),(61,148),(62,149),(63,150),(64,151),(65,152),(66,153),(67,154),(68,155),(69,156),(70,157),(71,158),(72,159),(73,160),(74,161),(75,162),(76,163),(77,164),(78,165),(79,166),(80,167),(81,168),(82,169),(83,170),(84,171),(85,172),(86,173),(87,174),(88,175)], [(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)], [(1,118,119,44),(2,43,120,117),(3,116,121,42),(4,41,122,115),(5,114,123,40),(6,39,124,113),(7,112,125,38),(8,37,126,111),(9,110,127,36),(10,35,128,109),(11,108,129,34),(12,33,130,107),(13,106,131,32),(14,31,132,105),(15,104,89,30),(16,29,90,103),(17,102,91,28),(18,27,92,101),(19,100,93,26),(20,25,94,99),(21,98,95,24),(22,23,96,97),(45,141,176,54),(46,53,133,140),(47,139,134,52),(48,51,135,138),(49,137,136,50),(55,175,142,88),(56,87,143,174),(57,173,144,86),(58,85,145,172),(59,171,146,84),(60,83,147,170),(61,169,148,82),(62,81,149,168),(63,167,150,80),(64,79,151,166),(65,165,152,78),(66,77,153,164),(67,163,154,76),(68,75,155,162),(69,161,156,74),(70,73,157,160),(71,159,158,72)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 44 | 4 | 4 | 22 | 22 | 22 | 22 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D11 | D22 | D22 | D44 | D4⋊2D11 |
| kernel | C22.D44 | C44⋊C4 | D22⋊C4 | C11×C22⋊C4 | C22×Dic11 | C2×C11⋊D4 | C2×C22 | C22 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 5 | 10 | 5 | 20 | 10 |
Matrix representation of C22.D44 ►in GL4(𝔽89) generated by
| 0 | 55 | 0 | 0 |
| 34 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 44 | 54 |
| 0 | 0 | 37 | 7 |
| 0 | 1 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 65 | 9 |
| 0 | 0 | 35 | 24 |
G:=sub<GL(4,GF(89))| [0,34,0,0,55,0,0,0,0,0,1,0,0,0,0,1],[88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,44,37,0,0,54,7],[0,88,0,0,1,0,0,0,0,0,65,35,0,0,9,24] >;
C22.D44 in GAP, Magma, Sage, TeX
C_2^2.D_{44} % in TeX
G:=Group("C2^2.D44"); // GroupNames label
G:=SmallGroup(352,81);
// by ID
G=gap.SmallGroup(352,81);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,218,188,122,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^44=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations