metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D88⋊5C2, Q16⋊3D11, D22.3D4, C8.10D22, Q8.5D22, C88.8C22, C44.10C23, D44.5C22, Dic11.14D4, (C8×D11)⋊3C2, C11⋊4(C4○D8), Q8⋊D11⋊4C2, (C11×Q16)⋊3C2, C22.36(C2×D4), C2.24(D4×D11), C11⋊C8.8C22, D44⋊C2⋊3C2, C4.10(C22×D11), (Q8×C11).5C22, (C4×D11).12C22, SmallGroup(352,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D88⋊5C2
G = < a,b,c | a88=b2=c2=1, bab=a-1, cac=a65, cbc=a20b >
Subgroups: 442 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, C11, C2×C8, D8, SD16, Q16, C4○D4, D11, C22, C4○D8, Dic11, C44, C44, D22, D22, C11⋊C8, C88, C4×D11, C4×D11, D44, D44, Q8×C11, C8×D11, D88, Q8⋊D11, C11×Q16, D44⋊C2, D88⋊5C2
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C4○D8, D22, C22×D11, D4×D11, D88⋊5C2
►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 88)(2 87)(3 86)(4 85)(5 84)(6 83)(7 82)(8 81)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(127 176)(128 175)(129 174)(130 173)(131 172)(132 171)(133 170)(134 169)(135 168)(136 167)(137 166)(138 165)(139 164)(140 163)(141 162)(142 161)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)
(1 174)(2 151)(3 128)(4 105)(5 170)(6 147)(7 124)(8 101)(9 166)(10 143)(11 120)(12 97)(13 162)(14 139)(15 116)(16 93)(17 158)(18 135)(19 112)(20 89)(21 154)(22 131)(23 108)(24 173)(25 150)(26 127)(27 104)(28 169)(29 146)(30 123)(31 100)(32 165)(33 142)(34 119)(35 96)(36 161)(37 138)(38 115)(39 92)(40 157)(41 134)(42 111)(43 176)(44 153)(45 130)(46 107)(47 172)(48 149)(49 126)(50 103)(51 168)(52 145)(53 122)(54 99)(55 164)(56 141)(57 118)(58 95)(59 160)(60 137)(61 114)(62 91)(63 156)(64 133)(65 110)(66 175)(67 152)(68 129)(69 106)(70 171)(71 148)(72 125)(73 102)(74 167)(75 144)(76 121)(77 98)(78 163)(79 140)(80 117)(81 94)(82 159)(83 136)(84 113)(85 90)(86 155)(87 132)(88 109)
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), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152), (1,174)(2,151)(3,128)(4,105)(5,170)(6,147)(7,124)(8,101)(9,166)(10,143)(11,120)(12,97)(13,162)(14,139)(15,116)(16,93)(17,158)(18,135)(19,112)(20,89)(21,154)(22,131)(23,108)(24,173)(25,150)(26,127)(27,104)(28,169)(29,146)(30,123)(31,100)(32,165)(33,142)(34,119)(35,96)(36,161)(37,138)(38,115)(39,92)(40,157)(41,134)(42,111)(43,176)(44,153)(45,130)(46,107)(47,172)(48,149)(49,126)(50,103)(51,168)(52,145)(53,122)(54,99)(55,164)(56,141)(57,118)(58,95)(59,160)(60,137)(61,114)(62,91)(63,156)(64,133)(65,110)(66,175)(67,152)(68,129)(69,106)(70,171)(71,148)(72,125)(73,102)(74,167)(75,144)(76,121)(77,98)(78,163)(79,140)(80,117)(81,94)(82,159)(83,136)(84,113)(85,90)(86,155)(87,132)(88,109)>;
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), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152), (1,174)(2,151)(3,128)(4,105)(5,170)(6,147)(7,124)(8,101)(9,166)(10,143)(11,120)(12,97)(13,162)(14,139)(15,116)(16,93)(17,158)(18,135)(19,112)(20,89)(21,154)(22,131)(23,108)(24,173)(25,150)(26,127)(27,104)(28,169)(29,146)(30,123)(31,100)(32,165)(33,142)(34,119)(35,96)(36,161)(37,138)(38,115)(39,92)(40,157)(41,134)(42,111)(43,176)(44,153)(45,130)(46,107)(47,172)(48,149)(49,126)(50,103)(51,168)(52,145)(53,122)(54,99)(55,164)(56,141)(57,118)(58,95)(59,160)(60,137)(61,114)(62,91)(63,156)(64,133)(65,110)(66,175)(67,152)(68,129)(69,106)(70,171)(71,148)(72,125)(73,102)(74,167)(75,144)(76,121)(77,98)(78,163)(79,140)(80,117)(81,94)(82,159)(83,136)(84,113)(85,90)(86,155)(87,132)(88,109) );
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)], [(1,88),(2,87),(3,86),(4,85),(5,84),(6,83),(7,82),(8,81),(9,80),(10,79),(11,78),(12,77),(13,76),(14,75),(15,74),(16,73),(17,72),(18,71),(19,70),(20,69),(21,68),(22,67),(23,66),(24,65),(25,64),(26,63),(27,62),(28,61),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45),(89,126),(90,125),(91,124),(92,123),(93,122),(94,121),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(127,176),(128,175),(129,174),(130,173),(131,172),(132,171),(133,170),(134,169),(135,168),(136,167),(137,166),(138,165),(139,164),(140,163),(141,162),(142,161),(143,160),(144,159),(145,158),(146,157),(147,156),(148,155),(149,154),(150,153),(151,152)], [(1,174),(2,151),(3,128),(4,105),(5,170),(6,147),(7,124),(8,101),(9,166),(10,143),(11,120),(12,97),(13,162),(14,139),(15,116),(16,93),(17,158),(18,135),(19,112),(20,89),(21,154),(22,131),(23,108),(24,173),(25,150),(26,127),(27,104),(28,169),(29,146),(30,123),(31,100),(32,165),(33,142),(34,119),(35,96),(36,161),(37,138),(38,115),(39,92),(40,157),(41,134),(42,111),(43,176),(44,153),(45,130),(46,107),(47,172),(48,149),(49,126),(50,103),(51,168),(52,145),(53,122),(54,99),(55,164),(56,141),(57,118),(58,95),(59,160),(60,137),(61,114),(62,91),(63,156),(64,133),(65,110),(66,175),(67,152),(68,129),(69,106),(70,171),(71,148),(72,125),(73,102),(74,167),(75,144),(76,121),(77,98),(78,163),(79,140),(80,117),(81,94),(82,159),(83,136),(84,113),(85,90),(86,155),(87,132),(88,109)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44E | 44F | ··· | 44O | 88A | ··· | 88J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 22 | 44 | 44 | 2 | 4 | 4 | 11 | 11 | 2 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D11 | C4○D8 | D22 | D22 | D4×D11 | D88⋊5C2 |
| kernel | D88⋊5C2 | C8×D11 | D88 | Q8⋊D11 | C11×Q16 | D44⋊C2 | Dic11 | D22 | Q16 | C11 | C8 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 5 | 4 | 5 | 10 | 5 | 10 |
Matrix representation of D88⋊5C2 ►in GL4(𝔽89) generated by
| 78 | 13 | 0 | 0 |
| 60 | 18 | 0 | 0 |
| 0 | 0 | 64 | 67 |
| 0 | 0 | 85 | 0 |
| 53 | 2 | 0 | 0 |
| 20 | 36 | 0 | 0 |
| 0 | 0 | 64 | 67 |
| 0 | 0 | 85 | 25 |
| 33 | 1 | 0 | 0 |
| 69 | 56 | 0 | 0 |
| 0 | 0 | 55 | 79 |
| 0 | 0 | 71 | 34 |
G:=sub<GL(4,GF(89))| [78,60,0,0,13,18,0,0,0,0,64,85,0,0,67,0],[53,20,0,0,2,36,0,0,0,0,64,85,0,0,67,25],[33,69,0,0,1,56,0,0,0,0,55,71,0,0,79,34] >;
D88⋊5C2 in GAP, Magma, Sage, TeX
D_{88}\rtimes_5C_2 % in TeX
G:=Group("D88:5C2"); // GroupNames label
G:=SmallGroup(352,114);
// by ID
G=gap.SmallGroup(352,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,362,116,86,297,159,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^88=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^65,c*b*c=a^20*b>;
// generators/relations