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G = D22.C8order 352 = 25·11

The non-split extension by D22 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D22.C8, C1764C2, C163D11, C8.20D22, Dic11.C8, C111M5(2), C88.20C22, C11⋊C164C2, C11⋊C8.2C4, C22.2(C2×C8), C2.3(C8×D11), C44.22(C2×C4), (C4×D11).2C4, (C8×D11).2C2, C4.17(C4×D11), SmallGroup(352,4)

Series: Derived Chief Lower central Upper central

C1C22 — D22.C8
C1C11C22C44C88C8×D11 — D22.C8
C11C22 — D22.C8
C1C8C16

Generators and relations for D22.C8
 G = < a,b,c | a22=b2=1, c8=a11, bab=a-1, ac=ca, cbc-1=a11b >

22C2
11C4
11C22
2D11
11C2×C4
11C8
11C2×C8
11C16
11M5(2)

Smallest permutation representation of D22.C8
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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 12)(23 24)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(63 66)(64 65)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(87 88)(89 91)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(111 115)(112 114)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(133 141)(134 140)(135 139)(136 138)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 176)(157 175)(158 174)(159 173)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)
(1 172 77 130 24 143 65 96 12 161 88 119 35 154 54 107)(2 173 78 131 25 144 66 97 13 162 67 120 36 133 55 108)(3 174 79 132 26 145 45 98 14 163 68 121 37 134 56 109)(4 175 80 111 27 146 46 99 15 164 69 122 38 135 57 110)(5 176 81 112 28 147 47 100 16 165 70 123 39 136 58 89)(6 155 82 113 29 148 48 101 17 166 71 124 40 137 59 90)(7 156 83 114 30 149 49 102 18 167 72 125 41 138 60 91)(8 157 84 115 31 150 50 103 19 168 73 126 42 139 61 92)(9 158 85 116 32 151 51 104 20 169 74 127 43 140 62 93)(10 159 86 117 33 152 52 105 21 170 75 128 44 141 63 94)(11 160 87 118 34 153 53 106 22 171 76 129 23 142 64 95)

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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(23,24)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(63,66)(64,65)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(87,88)(89,91)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(111,115)(112,114)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(133,141)(134,140)(135,139)(136,138)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167), (1,172,77,130,24,143,65,96,12,161,88,119,35,154,54,107)(2,173,78,131,25,144,66,97,13,162,67,120,36,133,55,108)(3,174,79,132,26,145,45,98,14,163,68,121,37,134,56,109)(4,175,80,111,27,146,46,99,15,164,69,122,38,135,57,110)(5,176,81,112,28,147,47,100,16,165,70,123,39,136,58,89)(6,155,82,113,29,148,48,101,17,166,71,124,40,137,59,90)(7,156,83,114,30,149,49,102,18,167,72,125,41,138,60,91)(8,157,84,115,31,150,50,103,19,168,73,126,42,139,61,92)(9,158,85,116,32,151,51,104,20,169,74,127,43,140,62,93)(10,159,86,117,33,152,52,105,21,170,75,128,44,141,63,94)(11,160,87,118,34,153,53,106,22,171,76,129,23,142,64,95)>;

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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(23,24)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(63,66)(64,65)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(87,88)(89,91)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(111,115)(112,114)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(133,141)(134,140)(135,139)(136,138)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167), (1,172,77,130,24,143,65,96,12,161,88,119,35,154,54,107)(2,173,78,131,25,144,66,97,13,162,67,120,36,133,55,108)(3,174,79,132,26,145,45,98,14,163,68,121,37,134,56,109)(4,175,80,111,27,146,46,99,15,164,69,122,38,135,57,110)(5,176,81,112,28,147,47,100,16,165,70,123,39,136,58,89)(6,155,82,113,29,148,48,101,17,166,71,124,40,137,59,90)(7,156,83,114,30,149,49,102,18,167,72,125,41,138,60,91)(8,157,84,115,31,150,50,103,19,168,73,126,42,139,61,92)(9,158,85,116,32,151,51,104,20,169,74,127,43,140,62,93)(10,159,86,117,33,152,52,105,21,170,75,128,44,141,63,94)(11,160,87,118,34,153,53,106,22,171,76,129,23,142,64,95) );

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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,12),(23,24),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(63,66),(64,65),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(87,88),(89,91),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102),(111,115),(112,114),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125),(133,141),(134,140),(135,139),(136,138),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,176),(157,175),(158,174),(159,173),(160,172),(161,171),(162,170),(163,169),(164,168),(165,167)], [(1,172,77,130,24,143,65,96,12,161,88,119,35,154,54,107),(2,173,78,131,25,144,66,97,13,162,67,120,36,133,55,108),(3,174,79,132,26,145,45,98,14,163,68,121,37,134,56,109),(4,175,80,111,27,146,46,99,15,164,69,122,38,135,57,110),(5,176,81,112,28,147,47,100,16,165,70,123,39,136,58,89),(6,155,82,113,29,148,48,101,17,166,71,124,40,137,59,90),(7,156,83,114,30,149,49,102,18,167,72,125,41,138,60,91),(8,157,84,115,31,150,50,103,19,168,73,126,42,139,61,92),(9,158,85,116,32,151,51,104,20,169,74,127,43,140,62,93),(10,159,86,117,33,152,52,105,21,170,75,128,44,141,63,94),(11,160,87,118,34,153,53,106,22,171,76,129,23,142,64,95)])

100 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F11A···11E16A16B16C16D16E16F16G16H22A···22E44A···44J88A···88T176A···176AN
order12244488888811···11161616161616161622···2244···4488···88176···176
size11221122111122222···22222222222222···22···22···22···2

100 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C4C4C8C8D11M5(2)D22C4×D11C8×D11D22.C8
kernelD22.C8C11⋊C16C176C8×D11C11⋊C8C4×D11Dic11D22C16C11C8C4C2C1
# reps11112244545102040

Matrix representation of D22.C8 in GL2(𝔽353) generated by

204130
223130
,
130204
130223
,
10234
319251
G:=sub<GL(2,GF(353))| [204,223,130,130],[130,130,204,223],[102,319,34,251] >;

D22.C8 in GAP, Magma, Sage, TeX

D_{22}.C_8
% in TeX

G:=Group("D22.C8");
// GroupNames label

G:=SmallGroup(352,4);
// by ID

G=gap.SmallGroup(352,4);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,31,50,69,11525]);
// Polycyclic

G:=Group<a,b,c|a^22=b^2=1,c^8=a^11,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^11*b>;
// generators/relations

Export

Subgroup lattice of D22.C8 in TeX

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