direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C22, C44.55C23, C22.18C24, (C2×D4)⋊7C22, D4⋊3(C2×C22), (C2×Q8)⋊6C22, Q8⋊3(C2×C22), (D4×C22)⋊16C2, (C22×C4)⋊6C22, (Q8×C22)⋊13C2, (C2×C44)⋊16C22, (C22×C44)⋊13C2, (C2×C22).6C23, C4.8(C22×C22), C2.3(C23×C22), (D4×C11)⋊12C22, C23.11(C2×C22), (Q8×C11)⋊11C22, C22.1(C22×C22), (C22×C22).30C22, (C2×C4)⋊5(C2×C22), SmallGroup(352,191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C22
G = < a,b,c,d | a22=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C11, C22×C4, C2×D4, C2×Q8, C4○D4, C22, C22, C22, C2×C4○D4, C44, C2×C22, C2×C22, C2×C22, C2×C44, C2×C44, D4×C11, Q8×C11, C22×C22, C22×C44, D4×C22, Q8×C22, C11×C4○D4, C4○D4×C22
Quotients: C1, C2, C22, C23, C11, C4○D4, C24, C22, C2×C4○D4, C2×C22, C22×C22, C11×C4○D4, C23×C22, C4○D4×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)
(1 130 169 100)(2 131 170 101)(3 132 171 102)(4 111 172 103)(5 112 173 104)(6 113 174 105)(7 114 175 106)(8 115 176 107)(9 116 155 108)(10 117 156 109)(11 118 157 110)(12 119 158 89)(13 120 159 90)(14 121 160 91)(15 122 161 92)(16 123 162 93)(17 124 163 94)(18 125 164 95)(19 126 165 96)(20 127 166 97)(21 128 167 98)(22 129 168 99)(23 136 57 73)(24 137 58 74)(25 138 59 75)(26 139 60 76)(27 140 61 77)(28 141 62 78)(29 142 63 79)(30 143 64 80)(31 144 65 81)(32 145 66 82)(33 146 45 83)(34 147 46 84)(35 148 47 85)(36 149 48 86)(37 150 49 87)(38 151 50 88)(39 152 51 67)(40 153 52 68)(41 154 53 69)(42 133 54 70)(43 134 55 71)(44 135 56 72)
(1 70 169 133)(2 71 170 134)(3 72 171 135)(4 73 172 136)(5 74 173 137)(6 75 174 138)(7 76 175 139)(8 77 176 140)(9 78 155 141)(10 79 156 142)(11 80 157 143)(12 81 158 144)(13 82 159 145)(14 83 160 146)(15 84 161 147)(16 85 162 148)(17 86 163 149)(18 87 164 150)(19 88 165 151)(20 67 166 152)(21 68 167 153)(22 69 168 154)(23 103 57 111)(24 104 58 112)(25 105 59 113)(26 106 60 114)(27 107 61 115)(28 108 62 116)(29 109 63 117)(30 110 64 118)(31 89 65 119)(32 90 66 120)(33 91 45 121)(34 92 46 122)(35 93 47 123)(36 94 48 124)(37 95 49 125)(38 96 50 126)(39 97 51 127)(40 98 52 128)(41 99 53 129)(42 100 54 130)(43 101 55 131)(44 102 56 132)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(133 158)(134 159)(135 160)(136 161)(137 162)(138 163)(139 164)(140 165)(141 166)(142 167)(143 168)(144 169)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 155)(153 156)(154 157)
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), (1,130,169,100)(2,131,170,101)(3,132,171,102)(4,111,172,103)(5,112,173,104)(6,113,174,105)(7,114,175,106)(8,115,176,107)(9,116,155,108)(10,117,156,109)(11,118,157,110)(12,119,158,89)(13,120,159,90)(14,121,160,91)(15,122,161,92)(16,123,162,93)(17,124,163,94)(18,125,164,95)(19,126,165,96)(20,127,166,97)(21,128,167,98)(22,129,168,99)(23,136,57,73)(24,137,58,74)(25,138,59,75)(26,139,60,76)(27,140,61,77)(28,141,62,78)(29,142,63,79)(30,143,64,80)(31,144,65,81)(32,145,66,82)(33,146,45,83)(34,147,46,84)(35,148,47,85)(36,149,48,86)(37,150,49,87)(38,151,50,88)(39,152,51,67)(40,153,52,68)(41,154,53,69)(42,133,54,70)(43,134,55,71)(44,135,56,72), (1,70,169,133)(2,71,170,134)(3,72,171,135)(4,73,172,136)(5,74,173,137)(6,75,174,138)(7,76,175,139)(8,77,176,140)(9,78,155,141)(10,79,156,142)(11,80,157,143)(12,81,158,144)(13,82,159,145)(14,83,160,146)(15,84,161,147)(16,85,162,148)(17,86,163,149)(18,87,164,150)(19,88,165,151)(20,67,166,152)(21,68,167,153)(22,69,168,154)(23,103,57,111)(24,104,58,112)(25,105,59,113)(26,106,60,114)(27,107,61,115)(28,108,62,116)(29,109,63,117)(30,110,64,118)(31,89,65,119)(32,90,66,120)(33,91,45,121)(34,92,46,122)(35,93,47,123)(36,94,48,124)(37,95,49,125)(38,96,50,126)(39,97,51,127)(40,98,52,128)(41,99,53,129)(42,100,54,130)(43,101,55,131)(44,102,56,132), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,99)(65,100)(66,101)(133,158)(134,159)(135,160)(136,161)(137,162)(138,163)(139,164)(140,165)(141,166)(142,167)(143,168)(144,169)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,155)(153,156)(154,157)>;
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), (1,130,169,100)(2,131,170,101)(3,132,171,102)(4,111,172,103)(5,112,173,104)(6,113,174,105)(7,114,175,106)(8,115,176,107)(9,116,155,108)(10,117,156,109)(11,118,157,110)(12,119,158,89)(13,120,159,90)(14,121,160,91)(15,122,161,92)(16,123,162,93)(17,124,163,94)(18,125,164,95)(19,126,165,96)(20,127,166,97)(21,128,167,98)(22,129,168,99)(23,136,57,73)(24,137,58,74)(25,138,59,75)(26,139,60,76)(27,140,61,77)(28,141,62,78)(29,142,63,79)(30,143,64,80)(31,144,65,81)(32,145,66,82)(33,146,45,83)(34,147,46,84)(35,148,47,85)(36,149,48,86)(37,150,49,87)(38,151,50,88)(39,152,51,67)(40,153,52,68)(41,154,53,69)(42,133,54,70)(43,134,55,71)(44,135,56,72), (1,70,169,133)(2,71,170,134)(3,72,171,135)(4,73,172,136)(5,74,173,137)(6,75,174,138)(7,76,175,139)(8,77,176,140)(9,78,155,141)(10,79,156,142)(11,80,157,143)(12,81,158,144)(13,82,159,145)(14,83,160,146)(15,84,161,147)(16,85,162,148)(17,86,163,149)(18,87,164,150)(19,88,165,151)(20,67,166,152)(21,68,167,153)(22,69,168,154)(23,103,57,111)(24,104,58,112)(25,105,59,113)(26,106,60,114)(27,107,61,115)(28,108,62,116)(29,109,63,117)(30,110,64,118)(31,89,65,119)(32,90,66,120)(33,91,45,121)(34,92,46,122)(35,93,47,123)(36,94,48,124)(37,95,49,125)(38,96,50,126)(39,97,51,127)(40,98,52,128)(41,99,53,129)(42,100,54,130)(43,101,55,131)(44,102,56,132), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,99)(65,100)(66,101)(133,158)(134,159)(135,160)(136,161)(137,162)(138,163)(139,164)(140,165)(141,166)(142,167)(143,168)(144,169)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,155)(153,156)(154,157) );
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)], [(1,130,169,100),(2,131,170,101),(3,132,171,102),(4,111,172,103),(5,112,173,104),(6,113,174,105),(7,114,175,106),(8,115,176,107),(9,116,155,108),(10,117,156,109),(11,118,157,110),(12,119,158,89),(13,120,159,90),(14,121,160,91),(15,122,161,92),(16,123,162,93),(17,124,163,94),(18,125,164,95),(19,126,165,96),(20,127,166,97),(21,128,167,98),(22,129,168,99),(23,136,57,73),(24,137,58,74),(25,138,59,75),(26,139,60,76),(27,140,61,77),(28,141,62,78),(29,142,63,79),(30,143,64,80),(31,144,65,81),(32,145,66,82),(33,146,45,83),(34,147,46,84),(35,148,47,85),(36,149,48,86),(37,150,49,87),(38,151,50,88),(39,152,51,67),(40,153,52,68),(41,154,53,69),(42,133,54,70),(43,134,55,71),(44,135,56,72)], [(1,70,169,133),(2,71,170,134),(3,72,171,135),(4,73,172,136),(5,74,173,137),(6,75,174,138),(7,76,175,139),(8,77,176,140),(9,78,155,141),(10,79,156,142),(11,80,157,143),(12,81,158,144),(13,82,159,145),(14,83,160,146),(15,84,161,147),(16,85,162,148),(17,86,163,149),(18,87,164,150),(19,88,165,151),(20,67,166,152),(21,68,167,153),(22,69,168,154),(23,103,57,111),(24,104,58,112),(25,105,59,113),(26,106,60,114),(27,107,61,115),(28,108,62,116),(29,109,63,117),(30,110,64,118),(31,89,65,119),(32,90,66,120),(33,91,45,121),(34,92,46,122),(35,93,47,123),(36,94,48,124),(37,95,49,125),(38,96,50,126),(39,97,51,127),(40,98,52,128),(41,99,53,129),(42,100,54,130),(43,101,55,131),(44,102,56,132)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,122),(24,123),(25,124),(26,125),(27,126),(28,127),(29,128),(30,129),(31,130),(32,131),(33,132),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,118),(42,119),(43,120),(44,121),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,99),(65,100),(66,101),(133,158),(134,159),(135,160),(136,161),(137,162),(138,163),(139,164),(140,165),(141,166),(142,167),(143,168),(144,169),(145,170),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,155),(153,156),(154,157)]])
220 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22CL | 44A | ··· | 44AN | 44AO | ··· | 44CV |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
220 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | C22 | C4○D4 | C11×C4○D4 |
| kernel | C4○D4×C22 | C22×C44 | D4×C22 | Q8×C22 | C11×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C22 | C2 |
| # reps | 1 | 3 | 3 | 1 | 8 | 10 | 30 | 30 | 10 | 80 | 4 | 40 |
Matrix representation of C4○D4×C22 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 88 | 0 | 0 |
| 0 | 55 | 0 |
| 0 | 0 | 55 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 88 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 88 |
| 0 | 88 | 0 |
G:=sub<GL(3,GF(89))| [88,0,0,0,4,0,0,0,4],[88,0,0,0,55,0,0,0,55],[1,0,0,0,0,88,0,1,0],[1,0,0,0,0,88,0,88,0] >;
C4○D4×C22 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{22} % in TeX
G:=Group("C4oD4xC22"); // GroupNames label
G:=SmallGroup(352,191);
// by ID
G=gap.SmallGroup(352,191);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-11,-2,2137,806]);
// Polycyclic
G:=Group<a,b,c,d|a^22=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations