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## G = C7⋊C3×C22order 462 = 2·3·7·11

### Direct product of C22 and C7⋊C3

Aliases: C7⋊C3×C22, C154⋊C3, C14⋊C33, C774C6, C72C66, SmallGroup(462,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C7⋊C3×C22
 Chief series C1 — C7 — C77 — C11×C7⋊C3 — C7⋊C3×C22
 Lower central C7 — C7⋊C3×C22
 Upper central C1 — C22

Generators and relations for C7⋊C3×C22
G = < a,b,c | a22=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C7⋊C3×C22
On 154 points
Generators in S154
(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)
(1 113 100 84 34 143 61)(2 114 101 85 35 144 62)(3 115 102 86 36 145 63)(4 116 103 87 37 146 64)(5 117 104 88 38 147 65)(6 118 105 67 39 148 66)(7 119 106 68 40 149 45)(8 120 107 69 41 150 46)(9 121 108 70 42 151 47)(10 122 109 71 43 152 48)(11 123 110 72 44 153 49)(12 124 89 73 23 154 50)(13 125 90 74 24 133 51)(14 126 91 75 25 134 52)(15 127 92 76 26 135 53)(16 128 93 77 27 136 54)(17 129 94 78 28 137 55)(18 130 95 79 29 138 56)(19 131 96 80 30 139 57)(20 132 97 81 31 140 58)(21 111 98 82 32 141 59)(22 112 99 83 33 142 60)
(23 124 89)(24 125 90)(25 126 91)(26 127 92)(27 128 93)(28 129 94)(29 130 95)(30 131 96)(31 132 97)(32 111 98)(33 112 99)(34 113 100)(35 114 101)(36 115 102)(37 116 103)(38 117 104)(39 118 105)(40 119 106)(41 120 107)(42 121 108)(43 122 109)(44 123 110)(45 149 68)(46 150 69)(47 151 70)(48 152 71)(49 153 72)(50 154 73)(51 133 74)(52 134 75)(53 135 76)(54 136 77)(55 137 78)(56 138 79)(57 139 80)(58 140 81)(59 141 82)(60 142 83)(61 143 84)(62 144 85)(63 145 86)(64 146 87)(65 147 88)(66 148 67)

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

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

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

110 conjugacy classes

 class 1 2 3A 3B 6A 6B 7A 7B 11A ··· 11J 14A 14B 22A ··· 22J 33A ··· 33T 66A ··· 66T 77A ··· 77T 154A ··· 154T order 1 2 3 3 6 6 7 7 11 ··· 11 14 14 22 ··· 22 33 ··· 33 66 ··· 66 77 ··· 77 154 ··· 154 size 1 1 7 7 7 7 3 3 1 ··· 1 3 3 1 ··· 1 7 ··· 7 7 ··· 7 3 ··· 3 3 ··· 3

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C6 C11 C22 C33 C66 C7⋊C3 C2×C7⋊C3 C11×C7⋊C3 C7⋊C3×C22 kernel C7⋊C3×C22 C11×C7⋊C3 C154 C77 C2×C7⋊C3 C7⋊C3 C14 C7 C22 C11 C2 C1 # reps 1 1 2 2 10 10 20 20 2 2 20 20

Matrix representation of C7⋊C3×C22 in GL4(𝔽463) generated by

 107 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 382 383 1 0 1 0 0 0 0 1 0
,
 21 0 0 0 0 1 0 0 0 80 462 462 0 0 1 0
G:=sub<GL(4,GF(463))| [107,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,382,1,0,0,383,0,1,0,1,0,0],[21,0,0,0,0,1,80,0,0,0,462,1,0,0,462,0] >;

C7⋊C3×C22 in GAP, Magma, Sage, TeX

C_7\rtimes C_3\times C_{22}
% in TeX

G:=Group("C7:C3xC22");
// GroupNames label

G:=SmallGroup(462,4);
// by ID

G=gap.SmallGroup(462,4);
# by ID

G:=PCGroup([4,-2,-3,-11,-7,1063]);
// Polycyclic

G:=Group<a,b,c|a^22=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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