metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D44, C44.28D4, C42⋊6D11, (C4×C44)⋊5C2, D22⋊C4⋊1C2, C2.6(C2×D44), C22.4(C2×D4), (C2×D44).2C2, (C2×C4).77D22, (C2×Dic22)⋊1C2, C22.5(C4○D4), C11⋊1(C4.4D4), (C2×C22).16C23, (C2×C44).74C22, C2.7(D44⋊5C2), (C2×Dic11).3C22, (C22×D11).2C22, C22.37(C22×D11), SmallGroup(352,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D44
G = < a,b,c | a4=b44=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 546 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, Dic22, D44, C2×Dic11, C2×C44, C2×C44, C22×D11, D22⋊C4, C4×C44, C2×Dic22, C2×D44, C4.D44
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4.4D4, D22, D44, C22×D11, C2×D44, D44⋊5C2, C4.D44
►On 176 points
Generators in S176
(1 161 53 128)(2 162 54 129)(3 163 55 130)(4 164 56 131)(5 165 57 132)(6 166 58 89)(7 167 59 90)(8 168 60 91)(9 169 61 92)(10 170 62 93)(11 171 63 94)(12 172 64 95)(13 173 65 96)(14 174 66 97)(15 175 67 98)(16 176 68 99)(17 133 69 100)(18 134 70 101)(19 135 71 102)(20 136 72 103)(21 137 73 104)(22 138 74 105)(23 139 75 106)(24 140 76 107)(25 141 77 108)(26 142 78 109)(27 143 79 110)(28 144 80 111)(29 145 81 112)(30 146 82 113)(31 147 83 114)(32 148 84 115)(33 149 85 116)(34 150 86 117)(35 151 87 118)(36 152 88 119)(37 153 45 120)(38 154 46 121)(39 155 47 122)(40 156 48 123)(41 157 49 124)(42 158 50 125)(43 159 51 126)(44 160 52 127)
(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 52 53 44)(2 43 54 51)(3 50 55 42)(4 41 56 49)(5 48 57 40)(6 39 58 47)(7 46 59 38)(8 37 60 45)(9 88 61 36)(10 35 62 87)(11 86 63 34)(12 33 64 85)(13 84 65 32)(14 31 66 83)(15 82 67 30)(16 29 68 81)(17 80 69 28)(18 27 70 79)(19 78 71 26)(20 25 72 77)(21 76 73 24)(22 23 74 75)(89 155 166 122)(90 121 167 154)(91 153 168 120)(92 119 169 152)(93 151 170 118)(94 117 171 150)(95 149 172 116)(96 115 173 148)(97 147 174 114)(98 113 175 146)(99 145 176 112)(100 111 133 144)(101 143 134 110)(102 109 135 142)(103 141 136 108)(104 107 137 140)(105 139 138 106)(123 165 156 132)(124 131 157 164)(125 163 158 130)(126 129 159 162)(127 161 160 128)
G:=sub<Sym(176)| (1,161,53,128)(2,162,54,129)(3,163,55,130)(4,164,56,131)(5,165,57,132)(6,166,58,89)(7,167,59,90)(8,168,60,91)(9,169,61,92)(10,170,62,93)(11,171,63,94)(12,172,64,95)(13,173,65,96)(14,174,66,97)(15,175,67,98)(16,176,68,99)(17,133,69,100)(18,134,70,101)(19,135,71,102)(20,136,72,103)(21,137,73,104)(22,138,74,105)(23,139,75,106)(24,140,76,107)(25,141,77,108)(26,142,78,109)(27,143,79,110)(28,144,80,111)(29,145,81,112)(30,146,82,113)(31,147,83,114)(32,148,84,115)(33,149,85,116)(34,150,86,117)(35,151,87,118)(36,152,88,119)(37,153,45,120)(38,154,46,121)(39,155,47,122)(40,156,48,123)(41,157,49,124)(42,158,50,125)(43,159,51,126)(44,160,52,127), (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), (1,52,53,44)(2,43,54,51)(3,50,55,42)(4,41,56,49)(5,48,57,40)(6,39,58,47)(7,46,59,38)(8,37,60,45)(9,88,61,36)(10,35,62,87)(11,86,63,34)(12,33,64,85)(13,84,65,32)(14,31,66,83)(15,82,67,30)(16,29,68,81)(17,80,69,28)(18,27,70,79)(19,78,71,26)(20,25,72,77)(21,76,73,24)(22,23,74,75)(89,155,166,122)(90,121,167,154)(91,153,168,120)(92,119,169,152)(93,151,170,118)(94,117,171,150)(95,149,172,116)(96,115,173,148)(97,147,174,114)(98,113,175,146)(99,145,176,112)(100,111,133,144)(101,143,134,110)(102,109,135,142)(103,141,136,108)(104,107,137,140)(105,139,138,106)(123,165,156,132)(124,131,157,164)(125,163,158,130)(126,129,159,162)(127,161,160,128)>;
G:=Group( (1,161,53,128)(2,162,54,129)(3,163,55,130)(4,164,56,131)(5,165,57,132)(6,166,58,89)(7,167,59,90)(8,168,60,91)(9,169,61,92)(10,170,62,93)(11,171,63,94)(12,172,64,95)(13,173,65,96)(14,174,66,97)(15,175,67,98)(16,176,68,99)(17,133,69,100)(18,134,70,101)(19,135,71,102)(20,136,72,103)(21,137,73,104)(22,138,74,105)(23,139,75,106)(24,140,76,107)(25,141,77,108)(26,142,78,109)(27,143,79,110)(28,144,80,111)(29,145,81,112)(30,146,82,113)(31,147,83,114)(32,148,84,115)(33,149,85,116)(34,150,86,117)(35,151,87,118)(36,152,88,119)(37,153,45,120)(38,154,46,121)(39,155,47,122)(40,156,48,123)(41,157,49,124)(42,158,50,125)(43,159,51,126)(44,160,52,127), (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), (1,52,53,44)(2,43,54,51)(3,50,55,42)(4,41,56,49)(5,48,57,40)(6,39,58,47)(7,46,59,38)(8,37,60,45)(9,88,61,36)(10,35,62,87)(11,86,63,34)(12,33,64,85)(13,84,65,32)(14,31,66,83)(15,82,67,30)(16,29,68,81)(17,80,69,28)(18,27,70,79)(19,78,71,26)(20,25,72,77)(21,76,73,24)(22,23,74,75)(89,155,166,122)(90,121,167,154)(91,153,168,120)(92,119,169,152)(93,151,170,118)(94,117,171,150)(95,149,172,116)(96,115,173,148)(97,147,174,114)(98,113,175,146)(99,145,176,112)(100,111,133,144)(101,143,134,110)(102,109,135,142)(103,141,136,108)(104,107,137,140)(105,139,138,106)(123,165,156,132)(124,131,157,164)(125,163,158,130)(126,129,159,162)(127,161,160,128) );
G=PermutationGroup([[(1,161,53,128),(2,162,54,129),(3,163,55,130),(4,164,56,131),(5,165,57,132),(6,166,58,89),(7,167,59,90),(8,168,60,91),(9,169,61,92),(10,170,62,93),(11,171,63,94),(12,172,64,95),(13,173,65,96),(14,174,66,97),(15,175,67,98),(16,176,68,99),(17,133,69,100),(18,134,70,101),(19,135,71,102),(20,136,72,103),(21,137,73,104),(22,138,74,105),(23,139,75,106),(24,140,76,107),(25,141,77,108),(26,142,78,109),(27,143,79,110),(28,144,80,111),(29,145,81,112),(30,146,82,113),(31,147,83,114),(32,148,84,115),(33,149,85,116),(34,150,86,117),(35,151,87,118),(36,152,88,119),(37,153,45,120),(38,154,46,121),(39,155,47,122),(40,156,48,123),(41,157,49,124),(42,158,50,125),(43,159,51,126),(44,160,52,127)], [(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)], [(1,52,53,44),(2,43,54,51),(3,50,55,42),(4,41,56,49),(5,48,57,40),(6,39,58,47),(7,46,59,38),(8,37,60,45),(9,88,61,36),(10,35,62,87),(11,86,63,34),(12,33,64,85),(13,84,65,32),(14,31,66,83),(15,82,67,30),(16,29,68,81),(17,80,69,28),(18,27,70,79),(19,78,71,26),(20,25,72,77),(21,76,73,24),(22,23,74,75),(89,155,166,122),(90,121,167,154),(91,153,168,120),(92,119,169,152),(93,151,170,118),(94,117,171,150),(95,149,172,116),(96,115,173,148),(97,147,174,114),(98,113,175,146),(99,145,176,112),(100,111,133,144),(101,143,134,110),(102,109,135,142),(103,141,136,108),(104,107,137,140),(105,139,138,106),(123,165,156,132),(124,131,157,164),(125,163,158,130),(126,129,159,162),(127,161,160,128)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 44 | 44 | 2 | ··· | 2 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D11 | D22 | D44 | D44⋊5C2 |
| kernel | C4.D44 | D22⋊C4 | C4×C44 | C2×Dic22 | C2×D44 | C44 | C22 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 5 | 15 | 20 | 40 |
Matrix representation of C4.D44 ►in GL4(𝔽89) generated by
| 62 | 75 | 0 | 0 |
| 14 | 27 | 0 | 0 |
| 0 | 0 | 1 | 42 |
| 0 | 0 | 72 | 88 |
| 7 | 2 | 0 | 0 |
| 87 | 12 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 7 | 2 | 0 | 0 |
| 64 | 82 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 45 | 55 |
G:=sub<GL(4,GF(89))| [62,14,0,0,75,27,0,0,0,0,1,72,0,0,42,88],[7,87,0,0,2,12,0,0,0,0,34,0,0,0,0,34],[7,64,0,0,2,82,0,0,0,0,34,45,0,0,0,55] >;
C4.D44 in GAP, Magma, Sage, TeX
C_4.D_{44} % in TeX
G:=Group("C4.D44"); // GroupNames label
G:=SmallGroup(352,70);
// by ID
G=gap.SmallGroup(352,70);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,55,218,86,11525]);
// Polycyclic
G:=Group<a,b,c|a^4=b^44=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations