metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44⋊7D4, C22⋊1D44, C23.24D22, (C2×C22)⋊5D4, D22⋊C4⋊3C2, (C2×D44)⋊6C2, C44⋊C4⋊9C2, C11⋊3(C4⋊D4), C4⋊3(C11⋊D4), (C22×C44)⋊6C2, C22.43(C2×D4), (C2×C4).85D22, C2.17(C2×D44), (C22×C4)⋊4D11, C22.19(C4○D4), (C2×C22).48C23, (C2×C44).94C22, C2.19(D44⋊5C2), (C22×C22).40C22, C22.56(C22×D11), (C2×Dic11).16C22, (C22×D11).10C22, (C2×C11⋊D4)⋊3C2, C2.7(C2×C11⋊D4), SmallGroup(352,125)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44⋊7D4
G = < a,b,c | a44=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 618 in 94 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C22, C4⋊D4, Dic11, C44, C44, D22, C2×C22, C2×C22, C2×C22, D44, C2×Dic11, C11⋊D4, C2×C44, C2×C44, C22×D11, C22×C22, C44⋊C4, D22⋊C4, C2×D44, C2×C11⋊D4, C22×C44, C44⋊7D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4⋊D4, D22, D44, C11⋊D4, C22×D11, C2×D44, D44⋊5C2, C2×C11⋊D4, C44⋊7D4
►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 45 143 111)(2 88 144 110)(3 87 145 109)(4 86 146 108)(5 85 147 107)(6 84 148 106)(7 83 149 105)(8 82 150 104)(9 81 151 103)(10 80 152 102)(11 79 153 101)(12 78 154 100)(13 77 155 99)(14 76 156 98)(15 75 157 97)(16 74 158 96)(17 73 159 95)(18 72 160 94)(19 71 161 93)(20 70 162 92)(21 69 163 91)(22 68 164 90)(23 67 165 89)(24 66 166 132)(25 65 167 131)(26 64 168 130)(27 63 169 129)(28 62 170 128)(29 61 171 127)(30 60 172 126)(31 59 173 125)(32 58 174 124)(33 57 175 123)(34 56 176 122)(35 55 133 121)(36 54 134 120)(37 53 135 119)(38 52 136 118)(39 51 137 117)(40 50 138 116)(41 49 139 115)(42 48 140 114)(43 47 141 113)(44 46 142 112)
(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)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 132)(69 131)(70 130)(71 129)(72 128)(73 127)(74 126)(75 125)(76 124)(77 123)(78 122)(79 121)(80 120)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(133 153)(134 152)(135 151)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(154 176)(155 175)(156 174)(157 173)(158 172)(159 171)(160 170)(161 169)(162 168)(163 167)(164 166)
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), (1,45,143,111)(2,88,144,110)(3,87,145,109)(4,86,146,108)(5,85,147,107)(6,84,148,106)(7,83,149,105)(8,82,150,104)(9,81,151,103)(10,80,152,102)(11,79,153,101)(12,78,154,100)(13,77,155,99)(14,76,156,98)(15,75,157,97)(16,74,158,96)(17,73,159,95)(18,72,160,94)(19,71,161,93)(20,70,162,92)(21,69,163,91)(22,68,164,90)(23,67,165,89)(24,66,166,132)(25,65,167,131)(26,64,168,130)(27,63,169,129)(28,62,170,128)(29,61,171,127)(30,60,172,126)(31,59,173,125)(32,58,174,124)(33,57,175,123)(34,56,176,122)(35,55,133,121)(36,54,134,120)(37,53,135,119)(38,52,136,118)(39,51,137,117)(40,50,138,116)(41,49,139,115)(42,48,140,114)(43,47,141,113)(44,46,142,112), (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)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(133,153)(134,152)(135,151)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(154,176)(155,175)(156,174)(157,173)(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)>;
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), (1,45,143,111)(2,88,144,110)(3,87,145,109)(4,86,146,108)(5,85,147,107)(6,84,148,106)(7,83,149,105)(8,82,150,104)(9,81,151,103)(10,80,152,102)(11,79,153,101)(12,78,154,100)(13,77,155,99)(14,76,156,98)(15,75,157,97)(16,74,158,96)(17,73,159,95)(18,72,160,94)(19,71,161,93)(20,70,162,92)(21,69,163,91)(22,68,164,90)(23,67,165,89)(24,66,166,132)(25,65,167,131)(26,64,168,130)(27,63,169,129)(28,62,170,128)(29,61,171,127)(30,60,172,126)(31,59,173,125)(32,58,174,124)(33,57,175,123)(34,56,176,122)(35,55,133,121)(36,54,134,120)(37,53,135,119)(38,52,136,118)(39,51,137,117)(40,50,138,116)(41,49,139,115)(42,48,140,114)(43,47,141,113)(44,46,142,112), (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)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(133,153)(134,152)(135,151)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(154,176)(155,175)(156,174)(157,173)(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166) );
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)], [(1,45,143,111),(2,88,144,110),(3,87,145,109),(4,86,146,108),(5,85,147,107),(6,84,148,106),(7,83,149,105),(8,82,150,104),(9,81,151,103),(10,80,152,102),(11,79,153,101),(12,78,154,100),(13,77,155,99),(14,76,156,98),(15,75,157,97),(16,74,158,96),(17,73,159,95),(18,72,160,94),(19,71,161,93),(20,70,162,92),(21,69,163,91),(22,68,164,90),(23,67,165,89),(24,66,166,132),(25,65,167,131),(26,64,168,130),(27,63,169,129),(28,62,170,128),(29,61,171,127),(30,60,172,126),(31,59,173,125),(32,58,174,124),(33,57,175,123),(34,56,176,122),(35,55,133,121),(36,54,134,120),(37,53,135,119),(38,52,136,118),(39,51,137,117),(40,50,138,116),(41,49,139,115),(42,48,140,114),(43,47,141,113),(44,46,142,112)], [(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),(45,111),(46,110),(47,109),(48,108),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,132),(69,131),(70,130),(71,129),(72,128),(73,127),(74,126),(75,125),(76,124),(77,123),(78,122),(79,121),(80,120),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(133,153),(134,152),(135,151),(136,150),(137,149),(138,148),(139,147),(140,146),(141,145),(142,144),(154,176),(155,175),(156,174),(157,173),(158,172),(159,171),(160,170),(161,169),(162,168),(163,167),(164,166)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 44 | 44 | 2 | 2 | 2 | 2 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D11 | D22 | D22 | C11⋊D4 | D44 | D44⋊5C2 |
| kernel | C44⋊7D4 | C44⋊C4 | D22⋊C4 | C2×D44 | C2×C11⋊D4 | C22×C44 | C44 | C2×C22 | C22 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 5 | 10 | 5 | 20 | 20 | 20 |
Matrix representation of C44⋊7D4 ►in GL4(𝔽89) generated by
| 50 | 65 | 0 | 0 |
| 24 | 33 | 0 | 0 |
| 0 | 0 | 86 | 25 |
| 0 | 0 | 8 | 22 |
| 42 | 28 | 0 | 0 |
| 80 | 47 | 0 | 0 |
| 0 | 0 | 27 | 82 |
| 0 | 0 | 15 | 62 |
| 1 | 0 | 0 | 0 |
| 86 | 88 | 0 | 0 |
| 0 | 0 | 27 | 82 |
| 0 | 0 | 15 | 62 |
G:=sub<GL(4,GF(89))| [50,24,0,0,65,33,0,0,0,0,86,8,0,0,25,22],[42,80,0,0,28,47,0,0,0,0,27,15,0,0,82,62],[1,86,0,0,0,88,0,0,0,0,27,15,0,0,82,62] >;
C44⋊7D4 in GAP, Magma, Sage, TeX
C_{44}\rtimes_7D_4 % in TeX
G:=Group("C44:7D4"); // GroupNames label
G:=SmallGroup(352,125);
// by ID
G=gap.SmallGroup(352,125);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,218,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations