metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D22⋊3Q8, C44.22D4, (C2×Q8)⋊3D11, (Q8×C22)⋊3C2, C2.9(Q8×D11), D22⋊C4.6C2, C44⋊C4⋊15C2, (C2×C4).21D22, C22.57(C2×D4), C11⋊5(C22⋊Q8), C22.17(C2×Q8), Dic11⋊C4⋊16C2, C22.36(C4○D4), C4.18(C11⋊D4), (C2×C22).58C23, (C2×C44).64C22, C2.8(D44⋊C2), C22.64(C22×D11), (C2×Dic11).21C22, (C22×D11).27C22, (C2×C4×D11).5C2, C2.21(C2×C11⋊D4), SmallGroup(352,141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D22⋊3Q8
G = < a,b,c,d | a22=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a11b, bd=db, dcd-1=c-1 >
Subgroups: 410 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, D11, C22, C22⋊Q8, Dic11, C44, C44, D22, D22, C2×C22, C4×D11, C2×Dic11, C2×Dic11, C2×C44, C2×C44, Q8×C11, C22×D11, Dic11⋊C4, C44⋊C4, D22⋊C4, C2×C4×D11, Q8×C22, D22⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D11, C22⋊Q8, D22, C11⋊D4, C22×D11, Q8×D11, D44⋊C2, C2×C11⋊D4, D22⋊3Q8
►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)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 132)(16 131)(17 130)(18 129)(19 128)(20 127)(21 126)(22 125)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 96)(44 95)(45 157)(46 156)(47 155)(48 176)(49 175)(50 174)(51 173)(52 172)(53 171)(54 170)(55 169)(56 168)(57 167)(58 166)(59 165)(60 164)(61 163)(62 162)(63 161)(64 160)(65 159)(66 158)(67 141)(68 140)(69 139)(70 138)(71 137)(72 136)(73 135)(74 134)(75 133)(76 154)(77 153)(78 152)(79 151)(80 150)(81 149)(82 148)(83 147)(84 146)(85 145)(86 144)(87 143)(88 142)
(1 27 125 102)(2 28 126 103)(3 29 127 104)(4 30 128 105)(5 31 129 106)(6 32 130 107)(7 33 131 108)(8 34 132 109)(9 35 111 110)(10 36 112 89)(11 37 113 90)(12 38 114 91)(13 39 115 92)(14 40 116 93)(15 41 117 94)(16 42 118 95)(17 43 119 96)(18 44 120 97)(19 23 121 98)(20 24 122 99)(21 25 123 100)(22 26 124 101)(45 147 174 67)(46 148 175 68)(47 149 176 69)(48 150 155 70)(49 151 156 71)(50 152 157 72)(51 153 158 73)(52 154 159 74)(53 133 160 75)(54 134 161 76)(55 135 162 77)(56 136 163 78)(57 137 164 79)(58 138 165 80)(59 139 166 81)(60 140 167 82)(61 141 168 83)(62 142 169 84)(63 143 170 85)(64 144 171 86)(65 145 172 87)(66 146 173 88)
(1 166 125 59)(2 167 126 60)(3 168 127 61)(4 169 128 62)(5 170 129 63)(6 171 130 64)(7 172 131 65)(8 173 132 66)(9 174 111 45)(10 175 112 46)(11 176 113 47)(12 155 114 48)(13 156 115 49)(14 157 116 50)(15 158 117 51)(16 159 118 52)(17 160 119 53)(18 161 120 54)(19 162 121 55)(20 163 122 56)(21 164 123 57)(22 165 124 58)(23 135 98 77)(24 136 99 78)(25 137 100 79)(26 138 101 80)(27 139 102 81)(28 140 103 82)(29 141 104 83)(30 142 105 84)(31 143 106 85)(32 144 107 86)(33 145 108 87)(34 146 109 88)(35 147 110 67)(36 148 89 68)(37 149 90 69)(38 150 91 70)(39 151 92 71)(40 152 93 72)(41 153 94 73)(42 154 95 74)(43 133 96 75)(44 134 97 76)
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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,157)(46,156)(47,155)(48,176)(49,175)(50,174)(51,173)(52,172)(53,171)(54,170)(55,169)(56,168)(57,167)(58,166)(59,165)(60,164)(61,163)(62,162)(63,161)(64,160)(65,159)(66,158)(67,141)(68,140)(69,139)(70,138)(71,137)(72,136)(73,135)(74,134)(75,133)(76,154)(77,153)(78,152)(79,151)(80,150)(81,149)(82,148)(83,147)(84,146)(85,145)(86,144)(87,143)(88,142), (1,27,125,102)(2,28,126,103)(3,29,127,104)(4,30,128,105)(5,31,129,106)(6,32,130,107)(7,33,131,108)(8,34,132,109)(9,35,111,110)(10,36,112,89)(11,37,113,90)(12,38,114,91)(13,39,115,92)(14,40,116,93)(15,41,117,94)(16,42,118,95)(17,43,119,96)(18,44,120,97)(19,23,121,98)(20,24,122,99)(21,25,123,100)(22,26,124,101)(45,147,174,67)(46,148,175,68)(47,149,176,69)(48,150,155,70)(49,151,156,71)(50,152,157,72)(51,153,158,73)(52,154,159,74)(53,133,160,75)(54,134,161,76)(55,135,162,77)(56,136,163,78)(57,137,164,79)(58,138,165,80)(59,139,166,81)(60,140,167,82)(61,141,168,83)(62,142,169,84)(63,143,170,85)(64,144,171,86)(65,145,172,87)(66,146,173,88), (1,166,125,59)(2,167,126,60)(3,168,127,61)(4,169,128,62)(5,170,129,63)(6,171,130,64)(7,172,131,65)(8,173,132,66)(9,174,111,45)(10,175,112,46)(11,176,113,47)(12,155,114,48)(13,156,115,49)(14,157,116,50)(15,158,117,51)(16,159,118,52)(17,160,119,53)(18,161,120,54)(19,162,121,55)(20,163,122,56)(21,164,123,57)(22,165,124,58)(23,135,98,77)(24,136,99,78)(25,137,100,79)(26,138,101,80)(27,139,102,81)(28,140,103,82)(29,141,104,83)(30,142,105,84)(31,143,106,85)(32,144,107,86)(33,145,108,87)(34,146,109,88)(35,147,110,67)(36,148,89,68)(37,149,90,69)(38,150,91,70)(39,151,92,71)(40,152,93,72)(41,153,94,73)(42,154,95,74)(43,133,96,75)(44,134,97,76)>;
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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,157)(46,156)(47,155)(48,176)(49,175)(50,174)(51,173)(52,172)(53,171)(54,170)(55,169)(56,168)(57,167)(58,166)(59,165)(60,164)(61,163)(62,162)(63,161)(64,160)(65,159)(66,158)(67,141)(68,140)(69,139)(70,138)(71,137)(72,136)(73,135)(74,134)(75,133)(76,154)(77,153)(78,152)(79,151)(80,150)(81,149)(82,148)(83,147)(84,146)(85,145)(86,144)(87,143)(88,142), (1,27,125,102)(2,28,126,103)(3,29,127,104)(4,30,128,105)(5,31,129,106)(6,32,130,107)(7,33,131,108)(8,34,132,109)(9,35,111,110)(10,36,112,89)(11,37,113,90)(12,38,114,91)(13,39,115,92)(14,40,116,93)(15,41,117,94)(16,42,118,95)(17,43,119,96)(18,44,120,97)(19,23,121,98)(20,24,122,99)(21,25,123,100)(22,26,124,101)(45,147,174,67)(46,148,175,68)(47,149,176,69)(48,150,155,70)(49,151,156,71)(50,152,157,72)(51,153,158,73)(52,154,159,74)(53,133,160,75)(54,134,161,76)(55,135,162,77)(56,136,163,78)(57,137,164,79)(58,138,165,80)(59,139,166,81)(60,140,167,82)(61,141,168,83)(62,142,169,84)(63,143,170,85)(64,144,171,86)(65,145,172,87)(66,146,173,88), (1,166,125,59)(2,167,126,60)(3,168,127,61)(4,169,128,62)(5,170,129,63)(6,171,130,64)(7,172,131,65)(8,173,132,66)(9,174,111,45)(10,175,112,46)(11,176,113,47)(12,155,114,48)(13,156,115,49)(14,157,116,50)(15,158,117,51)(16,159,118,52)(17,160,119,53)(18,161,120,54)(19,162,121,55)(20,163,122,56)(21,164,123,57)(22,165,124,58)(23,135,98,77)(24,136,99,78)(25,137,100,79)(26,138,101,80)(27,139,102,81)(28,140,103,82)(29,141,104,83)(30,142,105,84)(31,143,106,85)(32,144,107,86)(33,145,108,87)(34,146,109,88)(35,147,110,67)(36,148,89,68)(37,149,90,69)(38,150,91,70)(39,151,92,71)(40,152,93,72)(41,153,94,73)(42,154,95,74)(43,133,96,75)(44,134,97,76) );
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)], [(1,124),(2,123),(3,122),(4,121),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,132),(16,131),(17,130),(18,129),(19,128),(20,127),(21,126),(22,125),(23,94),(24,93),(25,92),(26,91),(27,90),(28,89),(29,110),(30,109),(31,108),(32,107),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,98),(42,97),(43,96),(44,95),(45,157),(46,156),(47,155),(48,176),(49,175),(50,174),(51,173),(52,172),(53,171),(54,170),(55,169),(56,168),(57,167),(58,166),(59,165),(60,164),(61,163),(62,162),(63,161),(64,160),(65,159),(66,158),(67,141),(68,140),(69,139),(70,138),(71,137),(72,136),(73,135),(74,134),(75,133),(76,154),(77,153),(78,152),(79,151),(80,150),(81,149),(82,148),(83,147),(84,146),(85,145),(86,144),(87,143),(88,142)], [(1,27,125,102),(2,28,126,103),(3,29,127,104),(4,30,128,105),(5,31,129,106),(6,32,130,107),(7,33,131,108),(8,34,132,109),(9,35,111,110),(10,36,112,89),(11,37,113,90),(12,38,114,91),(13,39,115,92),(14,40,116,93),(15,41,117,94),(16,42,118,95),(17,43,119,96),(18,44,120,97),(19,23,121,98),(20,24,122,99),(21,25,123,100),(22,26,124,101),(45,147,174,67),(46,148,175,68),(47,149,176,69),(48,150,155,70),(49,151,156,71),(50,152,157,72),(51,153,158,73),(52,154,159,74),(53,133,160,75),(54,134,161,76),(55,135,162,77),(56,136,163,78),(57,137,164,79),(58,138,165,80),(59,139,166,81),(60,140,167,82),(61,141,168,83),(62,142,169,84),(63,143,170,85),(64,144,171,86),(65,145,172,87),(66,146,173,88)], [(1,166,125,59),(2,167,126,60),(3,168,127,61),(4,169,128,62),(5,170,129,63),(6,171,130,64),(7,172,131,65),(8,173,132,66),(9,174,111,45),(10,175,112,46),(11,176,113,47),(12,155,114,48),(13,156,115,49),(14,157,116,50),(15,158,117,51),(16,159,118,52),(17,160,119,53),(18,161,120,54),(19,162,121,55),(20,163,122,56),(21,164,123,57),(22,165,124,58),(23,135,98,77),(24,136,99,78),(25,137,100,79),(26,138,101,80),(27,139,102,81),(28,140,103,82),(29,141,104,83),(30,142,105,84),(31,143,106,85),(32,144,107,86),(33,145,108,87),(34,146,109,88),(35,147,110,67),(36,148,89,68),(37,149,90,69),(38,150,91,70),(39,151,92,71),(40,152,93,72),(41,153,94,73),(42,154,95,74),(43,133,96,75),(44,134,97,76)]])
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 | 22 | 22 | 2 | 2 | 4 | 4 | 22 | 22 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
64 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 | Q8 | C4○D4 | D11 | D22 | C11⋊D4 | Q8×D11 | D44⋊C2 |
| kernel | D22⋊3Q8 | Dic11⋊C4 | C44⋊C4 | D22⋊C4 | C2×C4×D11 | Q8×C22 | C44 | D22 | C22 | C2×Q8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 5 | 15 | 20 | 5 | 5 |
Matrix representation of D22⋊3Q8 ►in GL4(𝔽89) generated by
| 43 | 73 | 0 | 0 |
| 51 | 39 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 24 | 7 | 0 | 0 |
| 45 | 65 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 17 | 88 |
| 11 | 1 | 0 | 0 |
| 58 | 78 | 0 | 0 |
| 0 | 0 | 69 | 39 |
| 0 | 0 | 81 | 20 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 44 | 55 |
G:=sub<GL(4,GF(89))| [43,51,0,0,73,39,0,0,0,0,88,0,0,0,0,88],[24,45,0,0,7,65,0,0,0,0,1,17,0,0,0,88],[11,58,0,0,1,78,0,0,0,0,69,81,0,0,39,20],[88,0,0,0,0,88,0,0,0,0,34,44,0,0,0,55] >;
D22⋊3Q8 in GAP, Magma, Sage, TeX
D_{22}\rtimes_3Q_8 % in TeX
G:=Group("D22:3Q8"); // GroupNames label
G:=SmallGroup(352,141);
// by ID
G=gap.SmallGroup(352,141);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,188,86,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^22=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^11*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations