metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.23D4, (Q8×C22)⋊4C2, (C2×Q8)⋊4D11, D22⋊C4⋊16C2, (C2×D44).9C2, C22.58(C2×D4), (C2×C4).57D22, (C4×Dic11)⋊7C2, C11⋊4(C4.4D4), C22.37(C4○D4), C4.11(C11⋊D4), (C2×C44).40C22, (C2×C22).59C23, C2.9(D44⋊C2), C22.65(C22×D11), (C2×Dic11).41C22, (C22×D11).13C22, C2.22(C2×C11⋊D4), SmallGroup(352,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.23D4
G = < a,b,c | a44=b4=c2=1, bab-1=a21, cac=a-1, cbc=a22b-1 >
Subgroups: 506 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C11, C42, C22⋊C4, C2×D4, C2×Q8, D11, C22, C22, C4.4D4, Dic11, C44, C44, D22, C2×C22, D44, C2×Dic11, C2×C44, C2×C44, Q8×C11, C22×D11, C4×Dic11, D22⋊C4, C2×D44, Q8×C22, C44.23D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4.4D4, D22, C11⋊D4, C22×D11, D44⋊C2, C2×C11⋊D4, C44.23D4
►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 82 132 154)(2 59 89 175)(3 80 90 152)(4 57 91 173)(5 78 92 150)(6 55 93 171)(7 76 94 148)(8 53 95 169)(9 74 96 146)(10 51 97 167)(11 72 98 144)(12 49 99 165)(13 70 100 142)(14 47 101 163)(15 68 102 140)(16 45 103 161)(17 66 104 138)(18 87 105 159)(19 64 106 136)(20 85 107 157)(21 62 108 134)(22 83 109 155)(23 60 110 176)(24 81 111 153)(25 58 112 174)(26 79 113 151)(27 56 114 172)(28 77 115 149)(29 54 116 170)(30 75 117 147)(31 52 118 168)(32 73 119 145)(33 50 120 166)(34 71 121 143)(35 48 122 164)(36 69 123 141)(37 46 124 162)(38 67 125 139)(39 88 126 160)(40 65 127 137)(41 86 128 158)(42 63 129 135)(43 84 130 156)(44 61 131 133)
(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(45 169)(46 168)(47 167)(48 166)(49 165)(50 164)(51 163)(52 162)(53 161)(54 160)(55 159)(56 158)(57 157)(58 156)(59 155)(60 154)(61 153)(62 152)(63 151)(64 150)(65 149)(66 148)(67 147)(68 146)(69 145)(70 144)(71 143)(72 142)(73 141)(74 140)(75 139)(76 138)(77 137)(78 136)(79 135)(80 134)(81 133)(82 176)(83 175)(84 174)(85 173)(86 172)(87 171)(88 170)(89 131)(90 130)(91 129)(92 128)(93 127)(94 126)(95 125)(96 124)(97 123)(98 122)(99 121)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)
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), (1,82,132,154)(2,59,89,175)(3,80,90,152)(4,57,91,173)(5,78,92,150)(6,55,93,171)(7,76,94,148)(8,53,95,169)(9,74,96,146)(10,51,97,167)(11,72,98,144)(12,49,99,165)(13,70,100,142)(14,47,101,163)(15,68,102,140)(16,45,103,161)(17,66,104,138)(18,87,105,159)(19,64,106,136)(20,85,107,157)(21,62,108,134)(22,83,109,155)(23,60,110,176)(24,81,111,153)(25,58,112,174)(26,79,113,151)(27,56,114,172)(28,77,115,149)(29,54,116,170)(30,75,117,147)(31,52,118,168)(32,73,119,145)(33,50,120,166)(34,71,121,143)(35,48,122,164)(36,69,123,141)(37,46,124,162)(38,67,125,139)(39,88,126,160)(40,65,127,137)(41,86,128,158)(42,63,129,135)(43,84,130,156)(44,61,131,133), (2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(45,169)(46,168)(47,167)(48,166)(49,165)(50,164)(51,163)(52,162)(53,161)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,153)(62,152)(63,151)(64,150)(65,149)(66,148)(67,147)(68,146)(69,145)(70,144)(71,143)(72,142)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(81,133)(82,176)(83,175)(84,174)(85,173)(86,172)(87,171)(88,170)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)>;
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), (1,82,132,154)(2,59,89,175)(3,80,90,152)(4,57,91,173)(5,78,92,150)(6,55,93,171)(7,76,94,148)(8,53,95,169)(9,74,96,146)(10,51,97,167)(11,72,98,144)(12,49,99,165)(13,70,100,142)(14,47,101,163)(15,68,102,140)(16,45,103,161)(17,66,104,138)(18,87,105,159)(19,64,106,136)(20,85,107,157)(21,62,108,134)(22,83,109,155)(23,60,110,176)(24,81,111,153)(25,58,112,174)(26,79,113,151)(27,56,114,172)(28,77,115,149)(29,54,116,170)(30,75,117,147)(31,52,118,168)(32,73,119,145)(33,50,120,166)(34,71,121,143)(35,48,122,164)(36,69,123,141)(37,46,124,162)(38,67,125,139)(39,88,126,160)(40,65,127,137)(41,86,128,158)(42,63,129,135)(43,84,130,156)(44,61,131,133), (2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(45,169)(46,168)(47,167)(48,166)(49,165)(50,164)(51,163)(52,162)(53,161)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,153)(62,152)(63,151)(64,150)(65,149)(66,148)(67,147)(68,146)(69,145)(70,144)(71,143)(72,142)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(81,133)(82,176)(83,175)(84,174)(85,173)(86,172)(87,171)(88,170)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111) );
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)], [(1,82,132,154),(2,59,89,175),(3,80,90,152),(4,57,91,173),(5,78,92,150),(6,55,93,171),(7,76,94,148),(8,53,95,169),(9,74,96,146),(10,51,97,167),(11,72,98,144),(12,49,99,165),(13,70,100,142),(14,47,101,163),(15,68,102,140),(16,45,103,161),(17,66,104,138),(18,87,105,159),(19,64,106,136),(20,85,107,157),(21,62,108,134),(22,83,109,155),(23,60,110,176),(24,81,111,153),(25,58,112,174),(26,79,113,151),(27,56,114,172),(28,77,115,149),(29,54,116,170),(30,75,117,147),(31,52,118,168),(32,73,119,145),(33,50,120,166),(34,71,121,143),(35,48,122,164),(36,69,123,141),(37,46,124,162),(38,67,125,139),(39,88,126,160),(40,65,127,137),(41,86,128,158),(42,63,129,135),(43,84,130,156),(44,61,131,133)], [(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(45,169),(46,168),(47,167),(48,166),(49,165),(50,164),(51,163),(52,162),(53,161),(54,160),(55,159),(56,158),(57,157),(58,156),(59,155),(60,154),(61,153),(62,152),(63,151),(64,150),(65,149),(66,148),(67,147),(68,146),(69,145),(70,144),(71,143),(72,142),(73,141),(74,140),(75,139),(76,138),(77,137),(78,136),(79,135),(80,134),(81,133),(82,176),(83,175),(84,174),(85,173),(86,172),(87,171),(88,170),(89,131),(90,130),(91,129),(92,128),(93,127),(94,126),(95,125),(96,124),(97,123),(98,122),(99,121),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 44 | 44 | 2 | 2 | 4 | 4 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D11 | D22 | C11⋊D4 | D44⋊C2 |
| kernel | C44.23D4 | C4×Dic11 | D22⋊C4 | C2×D44 | Q8×C22 | C44 | C22 | C2×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 5 | 15 | 20 | 10 |
Matrix representation of C44.23D4 ►in GL4(𝔽89) generated by
| 3 | 8 | 0 | 0 |
| 81 | 68 | 0 | 0 |
| 0 | 0 | 31 | 2 |
| 0 | 0 | 53 | 58 |
| 42 | 28 | 0 | 0 |
| 80 | 47 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 55 |
| 1 | 0 | 0 | 0 |
| 86 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 58 | 88 |
G:=sub<GL(4,GF(89))| [3,81,0,0,8,68,0,0,0,0,31,53,0,0,2,58],[42,80,0,0,28,47,0,0,0,0,55,0,0,0,0,55],[1,86,0,0,0,88,0,0,0,0,1,58,0,0,0,88] >;
C44.23D4 in GAP, Magma, Sage, TeX
C_{44}._{23}D_4 % in TeX
G:=Group("C44.23D4"); // GroupNames label
G:=SmallGroup(352,142);
// by ID
G=gap.SmallGroup(352,142);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,218,188,86,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=b^4=c^2=1,b*a*b^-1=a^21,c*a*c=a^-1,c*b*c=a^22*b^-1>;
// generators/relations