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## G = C10×C4.A4order 480 = 25·3·5

### Direct product of C10 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C10×C4.A4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C10×SL2(𝔽3) — C10×C4.A4
 Lower central Q8 — C10×C4.A4
 Upper central C1 — C2×C20

Generators and relations for C10×C4.A4
G = < a,b,c,d,e | a10=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

Subgroups: 262 in 98 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C10, C10, C10, C12, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, SL2(𝔽3), C2×C12, C30, C2×C4○D4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, C2×SL2(𝔽3), C4.A4, C60, C2×C30, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C2×C4.A4, C5×SL2(𝔽3), C2×C60, C10×C4○D4, C10×SL2(𝔽3), C5×C4.A4, C10×C4.A4
Quotients: C1, C2, C3, C22, C5, C6, C10, A4, C2×C6, C15, C2×C10, C2×A4, C30, C4.A4, C22×A4, C5×A4, C2×C30, C2×C4.A4, C10×A4, C5×C4.A4, A4×C2×C10, C10×C4.A4

Smallest permutation representation of C10×C4.A4
On 160 points
Generators in S160
(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)
(1 27 124 76)(2 28 125 77)(3 29 126 78)(4 30 127 79)(5 21 128 80)(6 22 129 71)(7 23 130 72)(8 24 121 73)(9 25 122 74)(10 26 123 75)(11 97 49 59)(12 98 50 60)(13 99 41 51)(14 100 42 52)(15 91 43 53)(16 92 44 54)(17 93 45 55)(18 94 46 56)(19 95 47 57)(20 96 48 58)(31 131 160 117)(32 132 151 118)(33 133 152 119)(34 134 153 120)(35 135 154 111)(36 136 155 112)(37 137 156 113)(38 138 157 114)(39 139 158 115)(40 140 159 116)(61 147 109 88)(62 148 110 89)(63 149 101 90)(64 150 102 81)(65 141 103 82)(66 142 104 83)(67 143 105 84)(68 144 106 85)(69 145 107 86)(70 146 108 87)
(1 67 124 105)(2 68 125 106)(3 69 126 107)(4 70 127 108)(5 61 128 109)(6 62 129 110)(7 63 130 101)(8 64 121 102)(9 65 122 103)(10 66 123 104)(11 140 49 116)(12 131 50 117)(13 132 41 118)(14 133 42 119)(15 134 43 120)(16 135 44 111)(17 136 45 112)(18 137 46 113)(19 138 47 114)(20 139 48 115)(21 147 80 88)(22 148 71 89)(23 149 72 90)(24 150 73 81)(25 141 74 82)(26 142 75 83)(27 143 76 84)(28 144 77 85)(29 145 78 86)(30 146 79 87)(31 98 160 60)(32 99 151 51)(33 100 152 52)(34 91 153 53)(35 92 154 54)(36 93 155 55)(37 94 156 56)(38 95 157 57)(39 96 158 58)(40 97 159 59)
(1 139 124 115)(2 140 125 116)(3 131 126 117)(4 132 127 118)(5 133 128 119)(6 134 129 120)(7 135 130 111)(8 136 121 112)(9 137 122 113)(10 138 123 114)(11 106 49 68)(12 107 50 69)(13 108 41 70)(14 109 42 61)(15 110 43 62)(16 101 44 63)(17 102 45 64)(18 103 46 65)(19 104 47 66)(20 105 48 67)(21 152 80 33)(22 153 71 34)(23 154 72 35)(24 155 73 36)(25 156 74 37)(26 157 75 38)(27 158 76 39)(28 159 77 40)(29 160 78 31)(30 151 79 32)(51 146 99 87)(52 147 100 88)(53 148 91 89)(54 149 92 90)(55 150 93 81)(56 141 94 82)(57 142 95 83)(58 143 96 84)(59 144 97 85)(60 145 98 86)
(11 106 116)(12 107 117)(13 108 118)(14 109 119)(15 110 120)(16 101 111)(17 102 112)(18 103 113)(19 104 114)(20 105 115)(31 98 86)(32 99 87)(33 100 88)(34 91 89)(35 92 90)(36 93 81)(37 94 82)(38 95 83)(39 96 84)(40 97 85)(41 70 132)(42 61 133)(43 62 134)(44 63 135)(45 64 136)(46 65 137)(47 66 138)(48 67 139)(49 68 140)(50 69 131)(51 146 151)(52 147 152)(53 148 153)(54 149 154)(55 150 155)(56 141 156)(57 142 157)(58 143 158)(59 144 159)(60 145 160)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A ··· 6F 10A ··· 10L 10M ··· 10T 12A ··· 12H 15A ··· 15H 20A ··· 20P 20Q ··· 20X 30A ··· 30X 60A ··· 60AF order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 4 4 1 1 1 1 6 6 1 1 1 1 4 ··· 4 1 ··· 1 6 ··· 6 4 ··· 4 4 ··· 4 1 ··· 1 6 ··· 6 4 ··· 4 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 C4.A4 C5×C4.A4 A4 C2×A4 C2×A4 C5×A4 C10×A4 C10×A4 kernel C10×C4.A4 C10×SL2(𝔽3) C5×C4.A4 C10×C4○D4 C2×C4.A4 Q8×C10 C5×C4○D4 C2×SL2(𝔽3) C4.A4 C2×C4○D4 C2×Q8 C4○D4 C10 C2 C2×C20 C20 C2×C10 C2×C4 C4 C22 # reps 1 1 2 2 4 2 4 4 8 8 8 16 12 48 1 2 1 4 8 4

Matrix representation of C10×C4.A4 in GL3(𝔽61) generated by

 60 0 0 0 3 0 0 0 3
,
 1 0 0 0 11 0 0 0 11
,
 1 0 0 0 1 59 0 1 60
,
 1 0 0 0 34 28 0 48 27
,
 47 0 0 0 1 0 0 1 13
G:=sub<GL(3,GF(61))| [60,0,0,0,3,0,0,0,3],[1,0,0,0,11,0,0,0,11],[1,0,0,0,1,1,0,59,60],[1,0,0,0,34,48,0,28,27],[47,0,0,0,1,1,0,0,13] >;

C10×C4.A4 in GAP, Magma, Sage, TeX

C_{10}\times C_4.A_4
% in TeX

G:=Group("C10xC4.A4");
// GroupNames label

G:=SmallGroup(480,1130);
// by ID

G=gap.SmallGroup(480,1130);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1688,1068,172,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=e^3=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations

׿
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