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G = D44.2C4order 352 = 25·11

The non-split extension by D44 of C4 acting through Inn(D44)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D44.2C4, C8.18D22, C88.23C22, C44.37C23, Dic22.2C4, (C2×C8)⋊7D11, (C2×C88)⋊10C2, (C8×D11)⋊6C2, C111(C8○D4), C88⋊C27C2, C44.20(C2×C4), D22.1(C2×C4), C4.10(C4×D11), (C2×C4).78D22, C11⋊D4.2C4, C11⋊C8.11C22, C44.C411C2, C22.2(C4×D11), D445C2.6C2, C22.14(C22×C4), (C2×C44).98C22, Dic11.3(C2×C4), C4.37(C22×D11), (C4×D11).15C22, C2.15(C2×C4×D11), (C2×C22).16(C2×C4), SmallGroup(352,96)

Series: Derived Chief Lower central Upper central

C1C22 — D44.2C4
C1C11C22C44C4×D11D445C2 — D44.2C4
C11C22 — D44.2C4
C1C8C2×C8

Generators and relations for D44.2C4
 G = < a,b,c | a44=b2=1, c4=a22, bab=a-1, ac=ca, bc=cb >

Subgroups: 298 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C11, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, D11 [×2], C22, C22, C8○D4, Dic11 [×2], C44 [×2], D22 [×2], C2×C22, C11⋊C8 [×2], C88 [×2], Dic22, C4×D11 [×2], D44, C11⋊D4 [×2], C2×C44, C8×D11 [×2], C88⋊C2 [×2], C44.C4, C2×C88, D445C2, D44.2C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, D11, C8○D4, D22 [×3], C4×D11 [×2], C22×D11, C2×C4×D11, D44.2C4

Smallest permutation representation of D44.2C4
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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 49)(46 48)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(89 121)(90 120)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(122 132)(123 131)(124 130)(125 129)(126 128)(133 161)(134 160)(135 159)(136 158)(137 157)(138 156)(139 155)(140 154)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)
(1 122 86 164 23 100 64 142)(2 123 87 165 24 101 65 143)(3 124 88 166 25 102 66 144)(4 125 45 167 26 103 67 145)(5 126 46 168 27 104 68 146)(6 127 47 169 28 105 69 147)(7 128 48 170 29 106 70 148)(8 129 49 171 30 107 71 149)(9 130 50 172 31 108 72 150)(10 131 51 173 32 109 73 151)(11 132 52 174 33 110 74 152)(12 89 53 175 34 111 75 153)(13 90 54 176 35 112 76 154)(14 91 55 133 36 113 77 155)(15 92 56 134 37 114 78 156)(16 93 57 135 38 115 79 157)(17 94 58 136 39 116 80 158)(18 95 59 137 40 117 81 159)(19 96 60 138 41 118 82 160)(20 97 61 139 42 119 83 161)(21 98 62 140 43 120 84 162)(22 99 63 141 44 121 85 163)

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

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

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

100 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J11A···11E22A···22O44A···44T88A···88AN
order1222244444888888888811···1122···2244···4488···88
size11222221122222111122222222222···22···22···22···2

100 irreducible representations

dim1111111112222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4D11C8○D4D22D22C4×D11C4×D11D44.2C4
kernelD44.2C4C8×D11C88⋊C2C44.C4C2×C88D445C2Dic22D44C11⋊D4C2×C8C11C8C2×C4C4C22C1
# reps12211122454105101040

Matrix representation of D44.2C4 in GL2(𝔽89) generated by

7542
2533
,
33
2786
,
770
077
G:=sub<GL(2,GF(89))| [75,25,42,33],[3,27,3,86],[77,0,0,77] >;

D44.2C4 in GAP, Magma, Sage, TeX

D_{44}._2C_4
% in TeX

G:=Group("D44.2C4");
// GroupNames label

G:=SmallGroup(352,96);
// by ID

G=gap.SmallGroup(352,96);
# by ID

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

G:=Group<a,b,c|a^44=b^2=1,c^4=a^22,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations

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