direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C22, C24.C22, C23⋊2C44, C2.1(D4×C22), (C22×C44)⋊3C2, C22⋊2(C2×C44), (C22×C22)⋊3C4, (C22×C4)⋊1C22, C22.64(C2×D4), (C2×C22).50D4, (C2×C44)⋊11C22, (C23×C22).1C2, C2.1(C22×C44), C23.5(C2×C22), (C2×C22).70C23, C22.29(C22×C4), C22.12(D4×C11), C22.4(C22×C22), (C22×C22).24C22, (C2×C22)⋊7(C2×C4), (C2×C4)⋊3(C2×C22), SmallGroup(352,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C22
G = < a,b,c,d | a22=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 188 in 132 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C11, C22⋊C4, C22×C4, C24, C22, C22, C22, C2×C22⋊C4, C44, C2×C22, C2×C22, C2×C22, C2×C44, C2×C44, C22×C22, C22×C22, C22×C22, C11×C22⋊C4, C22×C44, C23×C22, C22⋊C4×C22
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C11, C22⋊C4, C22×C4, C2×D4, C22, C2×C22⋊C4, C44, C2×C22, C2×C44, D4×C11, C22×C22, C11×C22⋊C4, C22×C44, D4×C22, C22⋊C4×C22
►On 176 points
Generators in S176
(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 42)(2 43)(3 44)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(45 153)(46 154)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 170)(68 171)(69 172)(70 173)(71 174)(72 175)(73 176)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)(82 163)(83 164)(84 165)(85 166)(86 167)(87 168)(88 169)(89 132)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 111)(22 112)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 86)(46 87)(47 88)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(133 169)(134 170)(135 171)(136 172)(137 173)(138 174)(139 175)(140 176)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)
(1 60 42 160)(2 61 43 161)(3 62 44 162)(4 63 23 163)(5 64 24 164)(6 65 25 165)(7 66 26 166)(8 45 27 167)(9 46 28 168)(10 47 29 169)(11 48 30 170)(12 49 31 171)(13 50 32 172)(14 51 33 173)(15 52 34 174)(16 53 35 175)(17 54 36 176)(18 55 37 155)(19 56 38 156)(20 57 39 157)(21 58 40 158)(22 59 41 159)(67 102 134 123)(68 103 135 124)(69 104 136 125)(70 105 137 126)(71 106 138 127)(72 107 139 128)(73 108 140 129)(74 109 141 130)(75 110 142 131)(76 89 143 132)(77 90 144 111)(78 91 145 112)(79 92 146 113)(80 93 147 114)(81 94 148 115)(82 95 149 116)(83 96 150 117)(84 97 151 118)(85 98 152 119)(86 99 153 120)(87 100 154 121)(88 101 133 122)
G:=sub<Sym(176)| (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), (1,42)(2,43)(3,44)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(45,153)(46,154)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,132)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,111)(22,112)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,86)(46,87)(47,88)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168), (1,60,42,160)(2,61,43,161)(3,62,44,162)(4,63,23,163)(5,64,24,164)(6,65,25,165)(7,66,26,166)(8,45,27,167)(9,46,28,168)(10,47,29,169)(11,48,30,170)(12,49,31,171)(13,50,32,172)(14,51,33,173)(15,52,34,174)(16,53,35,175)(17,54,36,176)(18,55,37,155)(19,56,38,156)(20,57,39,157)(21,58,40,158)(22,59,41,159)(67,102,134,123)(68,103,135,124)(69,104,136,125)(70,105,137,126)(71,106,138,127)(72,107,139,128)(73,108,140,129)(74,109,141,130)(75,110,142,131)(76,89,143,132)(77,90,144,111)(78,91,145,112)(79,92,146,113)(80,93,147,114)(81,94,148,115)(82,95,149,116)(83,96,150,117)(84,97,151,118)(85,98,152,119)(86,99,153,120)(87,100,154,121)(88,101,133,122)>;
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), (1,42)(2,43)(3,44)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(45,153)(46,154)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,132)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,111)(22,112)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,86)(46,87)(47,88)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168), (1,60,42,160)(2,61,43,161)(3,62,44,162)(4,63,23,163)(5,64,24,164)(6,65,25,165)(7,66,26,166)(8,45,27,167)(9,46,28,168)(10,47,29,169)(11,48,30,170)(12,49,31,171)(13,50,32,172)(14,51,33,173)(15,52,34,174)(16,53,35,175)(17,54,36,176)(18,55,37,155)(19,56,38,156)(20,57,39,157)(21,58,40,158)(22,59,41,159)(67,102,134,123)(68,103,135,124)(69,104,136,125)(70,105,137,126)(71,106,138,127)(72,107,139,128)(73,108,140,129)(74,109,141,130)(75,110,142,131)(76,89,143,132)(77,90,144,111)(78,91,145,112)(79,92,146,113)(80,93,147,114)(81,94,148,115)(82,95,149,116)(83,96,150,117)(84,97,151,118)(85,98,152,119)(86,99,153,120)(87,100,154,121)(88,101,133,122) );
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)], [(1,42),(2,43),(3,44),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(45,153),(46,154),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(61,147),(62,148),(63,149),(64,150),(65,151),(66,152),(67,170),(68,171),(69,172),(70,173),(71,174),(72,175),(73,176),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,161),(81,162),(82,163),(83,164),(84,165),(85,166),(86,167),(87,168),(88,169),(89,132),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,118),(98,119),(99,120),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,131)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,127),(16,128),(17,129),(18,130),(19,131),(20,132),(21,111),(22,112),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,86),(46,87),(47,88),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(133,169),(134,170),(135,171),(136,172),(137,173),(138,174),(139,175),(140,176),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168)], [(1,60,42,160),(2,61,43,161),(3,62,44,162),(4,63,23,163),(5,64,24,164),(6,65,25,165),(7,66,26,166),(8,45,27,167),(9,46,28,168),(10,47,29,169),(11,48,30,170),(12,49,31,171),(13,50,32,172),(14,51,33,173),(15,52,34,174),(16,53,35,175),(17,54,36,176),(18,55,37,155),(19,56,38,156),(20,57,39,157),(21,58,40,158),(22,59,41,159),(67,102,134,123),(68,103,135,124),(69,104,136,125),(70,105,137,126),(71,106,138,127),(72,107,139,128),(73,108,140,129),(74,109,141,130),(75,110,142,131),(76,89,143,132),(77,90,144,111),(78,91,145,112),(79,92,146,113),(80,93,147,114),(81,94,148,115),(82,95,149,116),(83,96,150,117),(84,97,151,118),(85,98,152,119),(86,99,153,120),(87,100,154,121),(88,101,133,122)]])
220 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 11A | ··· | 11J | 22A | ··· | 22BR | 22BS | ··· | 22DF | 44A | ··· | 44CB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
220 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C11 | C22 | C22 | C22 | C44 | D4 | D4×C11 |
| kernel | C22⋊C4×C22 | C11×C22⋊C4 | C22×C44 | C23×C22 | C22×C22 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C2×C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 10 | 40 | 20 | 10 | 80 | 4 | 40 |
Matrix representation of C22⋊C4×C22 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 73 | 0 |
| 0 | 0 | 0 | 73 |
| 1 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 55 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 34 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 55 | 2 |
| 0 | 0 | 0 | 34 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,73,0,0,0,0,73],[1,0,0,0,0,88,0,0,0,0,88,55,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[34,0,0,0,0,88,0,0,0,0,55,0,0,0,2,34] >;
C22⋊C4×C22 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{22} % in TeX
G:=Group("C2^2:C4xC22"); // GroupNames label
G:=SmallGroup(352,150);
// by ID
G=gap.SmallGroup(352,150);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1056,1081]);
// Polycyclic
G:=Group<a,b,c,d|a^22=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations