metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C44⋊1C4, C4⋊Dic11, C2.1D44, C22.4D4, C22.2Q8, C2.2Dic22, C22.5D22, C11⋊2(C4⋊C4), C22.8(C2×C4), (C2×C44).3C2, (C2×C4).3D11, (C2×C22).5C22, C2.4(C2×Dic11), (C2×Dic11).2C2, SmallGroup(176,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44⋊C4
G = < a,b | a44=b4=1, bab-1=a-1 >
Smallest permutation representation of C44⋊C4
►Regular action 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)
(1 130 170 48)(2 129 171 47)(3 128 172 46)(4 127 173 45)(5 126 174 88)(6 125 175 87)(7 124 176 86)(8 123 133 85)(9 122 134 84)(10 121 135 83)(11 120 136 82)(12 119 137 81)(13 118 138 80)(14 117 139 79)(15 116 140 78)(16 115 141 77)(17 114 142 76)(18 113 143 75)(19 112 144 74)(20 111 145 73)(21 110 146 72)(22 109 147 71)(23 108 148 70)(24 107 149 69)(25 106 150 68)(26 105 151 67)(27 104 152 66)(28 103 153 65)(29 102 154 64)(30 101 155 63)(31 100 156 62)(32 99 157 61)(33 98 158 60)(34 97 159 59)(35 96 160 58)(36 95 161 57)(37 94 162 56)(38 93 163 55)(39 92 164 54)(40 91 165 53)(41 90 166 52)(42 89 167 51)(43 132 168 50)(44 131 169 49)
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), (1,130,170,48)(2,129,171,47)(3,128,172,46)(4,127,173,45)(5,126,174,88)(6,125,175,87)(7,124,176,86)(8,123,133,85)(9,122,134,84)(10,121,135,83)(11,120,136,82)(12,119,137,81)(13,118,138,80)(14,117,139,79)(15,116,140,78)(16,115,141,77)(17,114,142,76)(18,113,143,75)(19,112,144,74)(20,111,145,73)(21,110,146,72)(22,109,147,71)(23,108,148,70)(24,107,149,69)(25,106,150,68)(26,105,151,67)(27,104,152,66)(28,103,153,65)(29,102,154,64)(30,101,155,63)(31,100,156,62)(32,99,157,61)(33,98,158,60)(34,97,159,59)(35,96,160,58)(36,95,161,57)(37,94,162,56)(38,93,163,55)(39,92,164,54)(40,91,165,53)(41,90,166,52)(42,89,167,51)(43,132,168,50)(44,131,169,49)>;
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), (1,130,170,48)(2,129,171,47)(3,128,172,46)(4,127,173,45)(5,126,174,88)(6,125,175,87)(7,124,176,86)(8,123,133,85)(9,122,134,84)(10,121,135,83)(11,120,136,82)(12,119,137,81)(13,118,138,80)(14,117,139,79)(15,116,140,78)(16,115,141,77)(17,114,142,76)(18,113,143,75)(19,112,144,74)(20,111,145,73)(21,110,146,72)(22,109,147,71)(23,108,148,70)(24,107,149,69)(25,106,150,68)(26,105,151,67)(27,104,152,66)(28,103,153,65)(29,102,154,64)(30,101,155,63)(31,100,156,62)(32,99,157,61)(33,98,158,60)(34,97,159,59)(35,96,160,58)(36,95,161,57)(37,94,162,56)(38,93,163,55)(39,92,164,54)(40,91,165,53)(41,90,166,52)(42,89,167,51)(43,132,168,50)(44,131,169,49) );
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)], [(1,130,170,48),(2,129,171,47),(3,128,172,46),(4,127,173,45),(5,126,174,88),(6,125,175,87),(7,124,176,86),(8,123,133,85),(9,122,134,84),(10,121,135,83),(11,120,136,82),(12,119,137,81),(13,118,138,80),(14,117,139,79),(15,116,140,78),(16,115,141,77),(17,114,142,76),(18,113,143,75),(19,112,144,74),(20,111,145,73),(21,110,146,72),(22,109,147,71),(23,108,148,70),(24,107,149,69),(25,106,150,68),(26,105,151,67),(27,104,152,66),(28,103,153,65),(29,102,154,64),(30,101,155,63),(31,100,156,62),(32,99,157,61),(33,98,158,60),(34,97,159,59),(35,96,160,58),(36,95,161,57),(37,94,162,56),(38,93,163,55),(39,92,164,54),(40,91,165,53),(41,90,166,52),(42,89,167,51),(43,132,168,50),(44,131,169,49)]])
C44⋊C4 is a maximal subgroup of
C44.Q8 C4.Dic22 C44.44D4 C44.4Q8 C44.5Q8 C2.D88 D4⋊Dic11 Q8⋊Dic11 C4×Dic22 C44⋊2Q8 C44.6Q8 C4×D44 C22⋊Dic22 C23.D22 D22.D4 C22.D44 C44⋊Q8 Dic11.Q8 C44.3Q8 C4⋊C4×D11 C4⋊C4⋊7D11 D22⋊2Q8 C4⋊C4⋊D11 C44.48D4 C23.21D22 C44⋊7D4 D4×Dic11 C44⋊2D4 Q8×Dic11 D22⋊3Q8
C44⋊C4 is a maximal quotient of
C44⋊C8 C44.4Q8 C44.5Q8 C88.C4 C22.C42
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D11 | Dic11 | D22 | Dic22 | D44 |
| kernel | C44⋊C4 | C2×Dic11 | C2×C44 | C44 | C22 | C22 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 5 | 10 | 5 | 10 | 10 |
Matrix representation of C44⋊C4 ►in GL4(𝔽89) generated by
| 34 | 87 | 0 | 0 |
| 2 | 13 | 0 | 0 |
| 0 | 0 | 14 | 73 |
| 0 | 0 | 2 | 55 |
| 23 | 63 | 0 | 0 |
| 82 | 66 | 0 | 0 |
| 0 | 0 | 8 | 41 |
| 0 | 0 | 18 | 81 |
G:=sub<GL(4,GF(89))| [34,2,0,0,87,13,0,0,0,0,14,2,0,0,73,55],[23,82,0,0,63,66,0,0,0,0,8,18,0,0,41,81] >;
C44⋊C4 in GAP, Magma, Sage, TeX
C_{44}\rtimes C_4 % in TeX
G:=Group("C44:C4"); // GroupNames label
G:=SmallGroup(176,12);
// by ID
G=gap.SmallGroup(176,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,20,101,46,4004]);
// Polycyclic
G:=Group<a,b|a^44=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export