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G = C28.44D4order 224 = 25·7

1st non-split extension by C28 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C28.44D4, C14.1Q16, Dic142C4, C2.1Dic28, C14.1SD16, C22.7D28, C4.7(C4×D7), (C2×C8).2D7, (C2×C56).2C2, C28.17(C2×C4), C72(Q8⋊C4), (C2×C4).67D14, (C2×C14).12D4, C4⋊Dic7.1C2, C2.7(D14⋊C4), C2.1(C56⋊C2), C4.19(C7⋊D4), C14.5(C22⋊C4), (C2×C28).80C22, (C2×Dic14).1C2, SmallGroup(224,22)

Series: Derived Chief Lower central Upper central

C1C28 — C28.44D4
C1C7C14C28C2×C28C4⋊Dic7 — C28.44D4
C7C14C28 — C28.44D4
C1C22C2×C4C2×C8

Generators and relations for C28.44D4
 G = < a,b,c | a28=b4=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a21b-1 >

14C4
14C4
28C4
2C8
7Q8
7Q8
14C2×C4
14Q8
14C2×C4
2Dic7
2Dic7
4Dic7
7C2×Q8
7C4⋊C4
2C2×Dic7
2C2×Dic7
2C56
2Dic14
7Q8⋊C4

Smallest permutation representation of C28.44D4
Regular action on 224 points
Generators in S224
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 39 117 186)(2 38 118 185)(3 37 119 184)(4 36 120 183)(5 35 121 182)(6 34 122 181)(7 33 123 180)(8 32 124 179)(9 31 125 178)(10 30 126 177)(11 29 127 176)(12 56 128 175)(13 55 129 174)(14 54 130 173)(15 53 131 172)(16 52 132 171)(17 51 133 170)(18 50 134 169)(19 49 135 196)(20 48 136 195)(21 47 137 194)(22 46 138 193)(23 45 139 192)(24 44 140 191)(25 43 113 190)(26 42 114 189)(27 41 115 188)(28 40 116 187)(57 167 224 103)(58 166 197 102)(59 165 198 101)(60 164 199 100)(61 163 200 99)(62 162 201 98)(63 161 202 97)(64 160 203 96)(65 159 204 95)(66 158 205 94)(67 157 206 93)(68 156 207 92)(69 155 208 91)(70 154 209 90)(71 153 210 89)(72 152 211 88)(73 151 212 87)(74 150 213 86)(75 149 214 85)(76 148 215 112)(77 147 216 111)(78 146 217 110)(79 145 218 109)(80 144 219 108)(81 143 220 107)(82 142 221 106)(83 141 222 105)(84 168 223 104)
(1 157 15 143)(2 156 16 142)(3 155 17 141)(4 154 18 168)(5 153 19 167)(6 152 20 166)(7 151 21 165)(8 150 22 164)(9 149 23 163)(10 148 24 162)(11 147 25 161)(12 146 26 160)(13 145 27 159)(14 144 28 158)(29 84 43 70)(30 83 44 69)(31 82 45 68)(32 81 46 67)(33 80 47 66)(34 79 48 65)(35 78 49 64)(36 77 50 63)(37 76 51 62)(38 75 52 61)(39 74 53 60)(40 73 54 59)(41 72 55 58)(42 71 56 57)(85 139 99 125)(86 138 100 124)(87 137 101 123)(88 136 102 122)(89 135 103 121)(90 134 104 120)(91 133 105 119)(92 132 106 118)(93 131 107 117)(94 130 108 116)(95 129 109 115)(96 128 110 114)(97 127 111 113)(98 126 112 140)(169 202 183 216)(170 201 184 215)(171 200 185 214)(172 199 186 213)(173 198 187 212)(174 197 188 211)(175 224 189 210)(176 223 190 209)(177 222 191 208)(178 221 192 207)(179 220 193 206)(180 219 194 205)(181 218 195 204)(182 217 196 203)

G:=sub<Sym(224)| (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,39,117,186)(2,38,118,185)(3,37,119,184)(4,36,120,183)(5,35,121,182)(6,34,122,181)(7,33,123,180)(8,32,124,179)(9,31,125,178)(10,30,126,177)(11,29,127,176)(12,56,128,175)(13,55,129,174)(14,54,130,173)(15,53,131,172)(16,52,132,171)(17,51,133,170)(18,50,134,169)(19,49,135,196)(20,48,136,195)(21,47,137,194)(22,46,138,193)(23,45,139,192)(24,44,140,191)(25,43,113,190)(26,42,114,189)(27,41,115,188)(28,40,116,187)(57,167,224,103)(58,166,197,102)(59,165,198,101)(60,164,199,100)(61,163,200,99)(62,162,201,98)(63,161,202,97)(64,160,203,96)(65,159,204,95)(66,158,205,94)(67,157,206,93)(68,156,207,92)(69,155,208,91)(70,154,209,90)(71,153,210,89)(72,152,211,88)(73,151,212,87)(74,150,213,86)(75,149,214,85)(76,148,215,112)(77,147,216,111)(78,146,217,110)(79,145,218,109)(80,144,219,108)(81,143,220,107)(82,142,221,106)(83,141,222,105)(84,168,223,104), (1,157,15,143)(2,156,16,142)(3,155,17,141)(4,154,18,168)(5,153,19,167)(6,152,20,166)(7,151,21,165)(8,150,22,164)(9,149,23,163)(10,148,24,162)(11,147,25,161)(12,146,26,160)(13,145,27,159)(14,144,28,158)(29,84,43,70)(30,83,44,69)(31,82,45,68)(32,81,46,67)(33,80,47,66)(34,79,48,65)(35,78,49,64)(36,77,50,63)(37,76,51,62)(38,75,52,61)(39,74,53,60)(40,73,54,59)(41,72,55,58)(42,71,56,57)(85,139,99,125)(86,138,100,124)(87,137,101,123)(88,136,102,122)(89,135,103,121)(90,134,104,120)(91,133,105,119)(92,132,106,118)(93,131,107,117)(94,130,108,116)(95,129,109,115)(96,128,110,114)(97,127,111,113)(98,126,112,140)(169,202,183,216)(170,201,184,215)(171,200,185,214)(172,199,186,213)(173,198,187,212)(174,197,188,211)(175,224,189,210)(176,223,190,209)(177,222,191,208)(178,221,192,207)(179,220,193,206)(180,219,194,205)(181,218,195,204)(182,217,196,203)>;

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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,39,117,186)(2,38,118,185)(3,37,119,184)(4,36,120,183)(5,35,121,182)(6,34,122,181)(7,33,123,180)(8,32,124,179)(9,31,125,178)(10,30,126,177)(11,29,127,176)(12,56,128,175)(13,55,129,174)(14,54,130,173)(15,53,131,172)(16,52,132,171)(17,51,133,170)(18,50,134,169)(19,49,135,196)(20,48,136,195)(21,47,137,194)(22,46,138,193)(23,45,139,192)(24,44,140,191)(25,43,113,190)(26,42,114,189)(27,41,115,188)(28,40,116,187)(57,167,224,103)(58,166,197,102)(59,165,198,101)(60,164,199,100)(61,163,200,99)(62,162,201,98)(63,161,202,97)(64,160,203,96)(65,159,204,95)(66,158,205,94)(67,157,206,93)(68,156,207,92)(69,155,208,91)(70,154,209,90)(71,153,210,89)(72,152,211,88)(73,151,212,87)(74,150,213,86)(75,149,214,85)(76,148,215,112)(77,147,216,111)(78,146,217,110)(79,145,218,109)(80,144,219,108)(81,143,220,107)(82,142,221,106)(83,141,222,105)(84,168,223,104), (1,157,15,143)(2,156,16,142)(3,155,17,141)(4,154,18,168)(5,153,19,167)(6,152,20,166)(7,151,21,165)(8,150,22,164)(9,149,23,163)(10,148,24,162)(11,147,25,161)(12,146,26,160)(13,145,27,159)(14,144,28,158)(29,84,43,70)(30,83,44,69)(31,82,45,68)(32,81,46,67)(33,80,47,66)(34,79,48,65)(35,78,49,64)(36,77,50,63)(37,76,51,62)(38,75,52,61)(39,74,53,60)(40,73,54,59)(41,72,55,58)(42,71,56,57)(85,139,99,125)(86,138,100,124)(87,137,101,123)(88,136,102,122)(89,135,103,121)(90,134,104,120)(91,133,105,119)(92,132,106,118)(93,131,107,117)(94,130,108,116)(95,129,109,115)(96,128,110,114)(97,127,111,113)(98,126,112,140)(169,202,183,216)(170,201,184,215)(171,200,185,214)(172,199,186,213)(173,198,187,212)(174,197,188,211)(175,224,189,210)(176,223,190,209)(177,222,191,208)(178,221,192,207)(179,220,193,206)(180,219,194,205)(181,218,195,204)(182,217,196,203) );

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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,39,117,186),(2,38,118,185),(3,37,119,184),(4,36,120,183),(5,35,121,182),(6,34,122,181),(7,33,123,180),(8,32,124,179),(9,31,125,178),(10,30,126,177),(11,29,127,176),(12,56,128,175),(13,55,129,174),(14,54,130,173),(15,53,131,172),(16,52,132,171),(17,51,133,170),(18,50,134,169),(19,49,135,196),(20,48,136,195),(21,47,137,194),(22,46,138,193),(23,45,139,192),(24,44,140,191),(25,43,113,190),(26,42,114,189),(27,41,115,188),(28,40,116,187),(57,167,224,103),(58,166,197,102),(59,165,198,101),(60,164,199,100),(61,163,200,99),(62,162,201,98),(63,161,202,97),(64,160,203,96),(65,159,204,95),(66,158,205,94),(67,157,206,93),(68,156,207,92),(69,155,208,91),(70,154,209,90),(71,153,210,89),(72,152,211,88),(73,151,212,87),(74,150,213,86),(75,149,214,85),(76,148,215,112),(77,147,216,111),(78,146,217,110),(79,145,218,109),(80,144,219,108),(81,143,220,107),(82,142,221,106),(83,141,222,105),(84,168,223,104)], [(1,157,15,143),(2,156,16,142),(3,155,17,141),(4,154,18,168),(5,153,19,167),(6,152,20,166),(7,151,21,165),(8,150,22,164),(9,149,23,163),(10,148,24,162),(11,147,25,161),(12,146,26,160),(13,145,27,159),(14,144,28,158),(29,84,43,70),(30,83,44,69),(31,82,45,68),(32,81,46,67),(33,80,47,66),(34,79,48,65),(35,78,49,64),(36,77,50,63),(37,76,51,62),(38,75,52,61),(39,74,53,60),(40,73,54,59),(41,72,55,58),(42,71,56,57),(85,139,99,125),(86,138,100,124),(87,137,101,123),(88,136,102,122),(89,135,103,121),(90,134,104,120),(91,133,105,119),(92,132,106,118),(93,131,107,117),(94,130,108,116),(95,129,109,115),(96,128,110,114),(97,127,111,113),(98,126,112,140),(169,202,183,216),(170,201,184,215),(171,200,185,214),(172,199,186,213),(173,198,187,212),(174,197,188,211),(175,224,189,210),(176,223,190,209),(177,222,191,208),(178,221,192,207),(179,220,193,206),(180,219,194,205),(181,218,195,204),(182,217,196,203)])

C28.44D4 is a maximal subgroup of
C28.14Q16  C4×C56⋊C2  C42.264D14  C4×Dic28  C42.14D14  C42.16D14  C42.20D14  Dic28⋊C4  C23.34D28  C23.35D28  C23.10D28  D28.32D4  D2814D4  Dic1414D4  C22⋊Dic28  D4.D7⋊C4  Dic76SD16  C28⋊Q8⋊C2  (C8×Dic7)⋊C2  D4⋊(C4×D7)  D42D7⋊C4  D14⋊SD16  C7⋊C81D4  C7⋊Q16⋊C4  Dic74Q16  Dic7.1Q16  C56⋊C4.C2  D7×Q8⋊C4  (Q8×D7)⋊C4  D14⋊Q16  C7⋊C8.D4  Dic14.3Q8  C28⋊SD16  D28.19D4  C42.36D14  C4⋊Dic28  C28.7Q16  Dic144Q8  Dic14⋊Q8  Dic14.Q8  D14.2SD16  C28.(C4○D4)  Dic142Q8  Dic14.2Q8  D14.2Q16  C2.D87D7  C23.23D28  C5630D4  C56.82D4  C23.46D28  C23.49D28  C562D4  C56.4D4  (C2×D8).D7  Dic14⋊D4  Dic73SD16  (C7×Q8).D4  Dic147D4  Dic14.16D4  Dic73Q16  D145Q16
C28.44D4 is a maximal quotient of
C4.8Dic28  C23.30D28  C4.Dic28  C28.47D8  C28.9C42

62 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F7A7B7C8A8B8C8D14A···14I28A···28L56A···56X
order1222444444777888814···1428···2856···56
size1111222828282822222222···22···22···2

62 irreducible representations

dim1111122222222222
type+++++++-++-
imageC1C2C2C2C4D4D4D7SD16Q16D14C4×D7C7⋊D4D28C56⋊C2Dic28
kernelC28.44D4C4⋊Dic7C2×C56C2×Dic14Dic14C28C2×C14C2×C8C14C14C2×C4C4C4C22C2C2
# reps111141132236661212

Matrix representation of C28.44D4 in GL3(𝔽113) generated by

11200
0913
010019
,
1500
03574
01478
,
100
09976
010014
G:=sub<GL(3,GF(113))| [112,0,0,0,9,100,0,13,19],[15,0,0,0,35,14,0,74,78],[1,0,0,0,99,100,0,76,14] >;

C28.44D4 in GAP, Magma, Sage, TeX

C_{28}._{44}D_4
% in TeX

G:=Group("C28.44D4");
// GroupNames label

G:=SmallGroup(224,22);
// by ID

G=gap.SmallGroup(224,22);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,73,79,362,86,6917]);
// Polycyclic

G:=Group<a,b,c|a^28=b^4=1,c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^21*b^-1>;
// generators/relations

Export

Subgroup lattice of C28.44D4 in TeX

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