metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D44.C4, C8.12D22, Dic22.C4, M4(2)⋊5D11, C88.12C22, C44.39C23, C11⋊D4.C4, (C8×D11)⋊8C2, C11⋊2(C8○D4), C88⋊C2⋊6C2, C4.5(C4×D11), C44.13(C2×C4), D22.2(C2×C4), (C2×C4).46D22, C11⋊C8.12C22, C22.1(C4×D11), D44⋊5C2.3C2, (C11×M4(2))⋊4C2, C22.16(C22×C4), (C2×C44).26C22, Dic11.4(C2×C4), C4.39(C22×D11), (C4×D11).16C22, (C2×C11⋊C8)⋊3C2, C2.17(C2×C4×D11), (C2×C22).6(C2×C4), SmallGroup(352,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D44.C4
G = < a,b,c | a44=b2=1, c4=a22, bab=a-1, cac-1=a23, cbc-1=a22b >
Subgroups: 298 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C11, C2×C8, M4(2), M4(2), C4○D4, D11, C22, C22, C8○D4, Dic11, C44, D22, C2×C22, C11⋊C8, C88, Dic22, C4×D11, D44, C11⋊D4, C2×C44, C8×D11, C88⋊C2, C2×C11⋊C8, C11×M4(2), D44⋊5C2, D44.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D11, C8○D4, D22, C4×D11, C22×D11, C2×C4×D11, D44.C4
►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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(89 97)(90 96)(91 95)(92 94)(98 132)(99 131)(100 130)(101 129)(102 128)(103 127)(104 126)(105 125)(106 124)(107 123)(108 122)(109 121)(110 120)(111 119)(112 118)(113 117)(114 116)(133 135)(136 176)(137 175)(138 174)(139 173)(140 172)(141 171)(142 170)(143 169)(144 168)(145 167)(146 166)(147 165)(148 164)(149 163)(150 162)(151 161)(152 160)(153 159)(154 158)(155 157)
(1 99 173 80 23 121 151 58)(2 122 174 59 24 100 152 81)(3 101 175 82 25 123 153 60)(4 124 176 61 26 102 154 83)(5 103 133 84 27 125 155 62)(6 126 134 63 28 104 156 85)(7 105 135 86 29 127 157 64)(8 128 136 65 30 106 158 87)(9 107 137 88 31 129 159 66)(10 130 138 67 32 108 160 45)(11 109 139 46 33 131 161 68)(12 132 140 69 34 110 162 47)(13 111 141 48 35 89 163 70)(14 90 142 71 36 112 164 49)(15 113 143 50 37 91 165 72)(16 92 144 73 38 114 166 51)(17 115 145 52 39 93 167 74)(18 94 146 75 40 116 168 53)(19 117 147 54 41 95 169 76)(20 96 148 77 42 118 170 55)(21 119 149 56 43 97 171 78)(22 98 150 79 44 120 172 57)
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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(89,97)(90,96)(91,95)(92,94)(98,132)(99,131)(100,130)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122)(109,121)(110,120)(111,119)(112,118)(113,117)(114,116)(133,135)(136,176)(137,175)(138,174)(139,173)(140,172)(141,171)(142,170)(143,169)(144,168)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)(151,161)(152,160)(153,159)(154,158)(155,157), (1,99,173,80,23,121,151,58)(2,122,174,59,24,100,152,81)(3,101,175,82,25,123,153,60)(4,124,176,61,26,102,154,83)(5,103,133,84,27,125,155,62)(6,126,134,63,28,104,156,85)(7,105,135,86,29,127,157,64)(8,128,136,65,30,106,158,87)(9,107,137,88,31,129,159,66)(10,130,138,67,32,108,160,45)(11,109,139,46,33,131,161,68)(12,132,140,69,34,110,162,47)(13,111,141,48,35,89,163,70)(14,90,142,71,36,112,164,49)(15,113,143,50,37,91,165,72)(16,92,144,73,38,114,166,51)(17,115,145,52,39,93,167,74)(18,94,146,75,40,116,168,53)(19,117,147,54,41,95,169,76)(20,96,148,77,42,118,170,55)(21,119,149,56,43,97,171,78)(22,98,150,79,44,120,172,57)>;
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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(89,97)(90,96)(91,95)(92,94)(98,132)(99,131)(100,130)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122)(109,121)(110,120)(111,119)(112,118)(113,117)(114,116)(133,135)(136,176)(137,175)(138,174)(139,173)(140,172)(141,171)(142,170)(143,169)(144,168)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)(151,161)(152,160)(153,159)(154,158)(155,157), (1,99,173,80,23,121,151,58)(2,122,174,59,24,100,152,81)(3,101,175,82,25,123,153,60)(4,124,176,61,26,102,154,83)(5,103,133,84,27,125,155,62)(6,126,134,63,28,104,156,85)(7,105,135,86,29,127,157,64)(8,128,136,65,30,106,158,87)(9,107,137,88,31,129,159,66)(10,130,138,67,32,108,160,45)(11,109,139,46,33,131,161,68)(12,132,140,69,34,110,162,47)(13,111,141,48,35,89,163,70)(14,90,142,71,36,112,164,49)(15,113,143,50,37,91,165,72)(16,92,144,73,38,114,166,51)(17,115,145,52,39,93,167,74)(18,94,146,75,40,116,168,53)(19,117,147,54,41,95,169,76)(20,96,148,77,42,118,170,55)(21,119,149,56,43,97,171,78)(22,98,150,79,44,120,172,57) );
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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(60,88),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(89,97),(90,96),(91,95),(92,94),(98,132),(99,131),(100,130),(101,129),(102,128),(103,127),(104,126),(105,125),(106,124),(107,123),(108,122),(109,121),(110,120),(111,119),(112,118),(113,117),(114,116),(133,135),(136,176),(137,175),(138,174),(139,173),(140,172),(141,171),(142,170),(143,169),(144,168),(145,167),(146,166),(147,165),(148,164),(149,163),(150,162),(151,161),(152,160),(153,159),(154,158),(155,157)], [(1,99,173,80,23,121,151,58),(2,122,174,59,24,100,152,81),(3,101,175,82,25,123,153,60),(4,124,176,61,26,102,154,83),(5,103,133,84,27,125,155,62),(6,126,134,63,28,104,156,85),(7,105,135,86,29,127,157,64),(8,128,136,65,30,106,158,87),(9,107,137,88,31,129,159,66),(10,130,138,67,32,108,160,45),(11,109,139,46,33,131,161,68),(12,132,140,69,34,110,162,47),(13,111,141,48,35,89,163,70),(14,90,142,71,36,112,164,49),(15,113,143,50,37,91,165,72),(16,92,144,73,38,114,166,51),(17,115,145,52,39,93,167,74),(18,94,146,75,40,116,168,53),(19,117,147,54,41,95,169,76),(20,96,148,77,42,118,170,55),(21,119,149,56,43,97,171,78),(22,98,150,79,44,120,172,57)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44J | 44K | ··· | 44O | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 22 | 22 | 1 | 1 | 2 | 22 | 22 | 2 | 2 | 2 | 2 | 11 | 11 | 11 | 11 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D11 | C8○D4 | D22 | D22 | C4×D11 | C4×D11 | D44.C4 |
| kernel | D44.C4 | C8×D11 | C88⋊C2 | C2×C11⋊C8 | C11×M4(2) | D44⋊5C2 | Dic22 | D44 | C11⋊D4 | M4(2) | C11 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 5 | 4 | 10 | 5 | 10 | 10 | 10 |
Matrix representation of D44.C4 ►in GL4(𝔽89) generated by
| 7 | 33 | 0 | 0 |
| 62 | 0 | 0 | 0 |
| 0 | 0 | 40 | 74 |
| 0 | 0 | 83 | 49 |
| 41 | 36 | 0 | 0 |
| 72 | 48 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 54 | 1 |
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 35 | 87 |
| 0 | 0 | 51 | 54 |
G:=sub<GL(4,GF(89))| [7,62,0,0,33,0,0,0,0,0,40,83,0,0,74,49],[41,72,0,0,36,48,0,0,0,0,88,54,0,0,0,1],[34,0,0,0,0,34,0,0,0,0,35,51,0,0,87,54] >;
D44.C4 in GAP, Magma, Sage, TeX
D_{44}.C_4 % in TeX
G:=Group("D44.C4"); // GroupNames label
G:=SmallGroup(352,102);
// by ID
G=gap.SmallGroup(352,102);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,188,50,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=b^2=1,c^4=a^22,b*a*b=a^-1,c*a*c^-1=a^23,c*b*c^-1=a^22*b>;
// generators/relations