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## G = C11×C4.10D4order 352 = 25·11

### Direct product of C11 and C4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C11×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C44 — C11×M4(2) — C11×C4.10D4
 Lower central C1 — C2 — C22 — C11×C4.10D4
 Upper central C1 — C22 — C2×C44 — C11×C4.10D4

Generators and relations for C11×C4.10D4
G = < a,b,c,d | a11=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Smallest permutation representation of C11×C4.10D4
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 63 73 47)(2 64 74 48)(3 65 75 49)(4 66 76 50)(5 56 77 51)(6 57 67 52)(7 58 68 53)(8 59 69 54)(9 60 70 55)(10 61 71 45)(11 62 72 46)(12 34 24 172)(13 35 25 173)(14 36 26 174)(15 37 27 175)(16 38 28 176)(17 39 29 166)(18 40 30 167)(19 41 31 168)(20 42 32 169)(21 43 33 170)(22 44 23 171)(78 116 106 99)(79 117 107 89)(80 118 108 90)(81 119 109 91)(82 120 110 92)(83 121 100 93)(84 111 101 94)(85 112 102 95)(86 113 103 96)(87 114 104 97)(88 115 105 98)(122 160 150 143)(123 161 151 133)(124 162 152 134)(125 163 153 135)(126 164 154 136)(127 165 144 137)(128 155 145 138)(129 156 146 139)(130 157 147 140)(131 158 148 141)(132 159 149 142)
(1 151 63 161 73 123 47 133)(2 152 64 162 74 124 48 134)(3 153 65 163 75 125 49 135)(4 154 66 164 76 126 50 136)(5 144 56 165 77 127 51 137)(6 145 57 155 67 128 52 138)(7 146 58 156 68 129 53 139)(8 147 59 157 69 130 54 140)(9 148 60 158 70 131 55 141)(10 149 61 159 71 132 45 142)(11 150 62 160 72 122 46 143)(12 94 172 84 24 111 34 101)(13 95 173 85 25 112 35 102)(14 96 174 86 26 113 36 103)(15 97 175 87 27 114 37 104)(16 98 176 88 28 115 38 105)(17 99 166 78 29 116 39 106)(18 89 167 79 30 117 40 107)(19 90 168 80 31 118 41 108)(20 91 169 81 32 119 42 109)(21 92 170 82 33 120 43 110)(22 93 171 83 23 121 44 100)
(1 167 47 30 73 40 63 18)(2 168 48 31 74 41 64 19)(3 169 49 32 75 42 65 20)(4 170 50 33 76 43 66 21)(5 171 51 23 77 44 56 22)(6 172 52 24 67 34 57 12)(7 173 53 25 68 35 58 13)(8 174 54 26 69 36 59 14)(9 175 55 27 70 37 60 15)(10 176 45 28 71 38 61 16)(11 166 46 29 72 39 62 17)(78 160 99 122 106 143 116 150)(79 161 89 123 107 133 117 151)(80 162 90 124 108 134 118 152)(81 163 91 125 109 135 119 153)(82 164 92 126 110 136 120 154)(83 165 93 127 100 137 121 144)(84 155 94 128 101 138 111 145)(85 156 95 129 102 139 112 146)(86 157 96 130 103 140 113 147)(87 158 97 131 104 141 114 148)(88 159 98 132 105 142 115 149)

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

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

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

121 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 8A 8B 8C 8D 11A ··· 11J 22A ··· 22J 22K ··· 22T 44A ··· 44T 44U ··· 44AN 88A ··· 88AN order 1 2 2 4 4 4 4 8 8 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 size 1 1 2 2 2 4 4 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

121 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + - image C1 C2 C2 C4 C11 C22 C22 C44 D4 D4×C11 C4.10D4 C11×C4.10D4 kernel C11×C4.10D4 C11×M4(2) Q8×C22 C2×C44 C4.10D4 M4(2) C2×Q8 C2×C4 C44 C4 C11 C1 # reps 1 2 1 4 10 20 10 40 2 20 1 10

Matrix representation of C11×C4.10D4 in GL4(𝔽89) generated by

 39 0 0 0 0 39 0 0 0 0 39 0 0 0 0 39
,
 0 1 0 0 88 0 0 0 0 0 0 88 0 0 1 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 88 0 0 0
,
 0 0 40 1 0 0 1 49 88 40 0 0 40 1 0 0
G:=sub<GL(4,GF(89))| [39,0,0,0,0,39,0,0,0,0,39,0,0,0,0,39],[0,88,0,0,1,0,0,0,0,0,0,1,0,0,88,0],[0,0,0,88,0,0,1,0,1,0,0,0,0,1,0,0],[0,0,88,40,0,0,40,1,40,1,0,0,1,49,0,0] >;

C11×C4.10D4 in GAP, Magma, Sage, TeX

C_{11}\times C_4._{10}D_4
% in TeX

G:=Group("C11xC4.10D4");
// GroupNames label

G:=SmallGroup(352,50);
// by ID

G=gap.SmallGroup(352,50);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,1063,5283,3970,88]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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