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## G = C10×C4.4D4order 320 = 26·5

### Direct product of C10 and C4.4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C10×C4.4D4
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C4.4D4 — C10×C4.4D4
 Lower central C1 — C22 — C10×C4.4D4
 Upper central C1 — C22×C10 — C10×C4.4D4

Generators and relations for C10×C4.4D4
G = < a,b,c,d | a10=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 530 in 330 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C2×C4.4D4, C4×C20, C5×C22⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, Q8×C10, C23×C10, C2×C4×C20, C10×C22⋊C4, C5×C4.4D4, D4×C2×C10, Q8×C2×C10, C10×C4.4D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C24, C2×C10, C4.4D4, C22×D4, C2×C4○D4, C5×D4, C22×C10, C2×C4.4D4, D4×C10, C5×C4○D4, C23×C10, C5×C4.4D4, D4×C2×C10, C10×C4○D4, C10×C4.4D4

Smallest permutation representation of C10×C4.4D4
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 26 51 35)(2 27 52 36)(3 28 53 37)(4 29 54 38)(5 30 55 39)(6 21 56 40)(7 22 57 31)(8 23 58 32)(9 24 59 33)(10 25 60 34)(11 62 159 43)(12 63 160 44)(13 64 151 45)(14 65 152 46)(15 66 153 47)(16 67 154 48)(17 68 155 49)(18 69 156 50)(19 70 157 41)(20 61 158 42)(71 125 104 138)(72 126 105 139)(73 127 106 140)(74 128 107 131)(75 129 108 132)(76 130 109 133)(77 121 110 134)(78 122 101 135)(79 123 102 136)(80 124 103 137)(81 117 94 150)(82 118 95 141)(83 119 96 142)(84 120 97 143)(85 111 98 144)(86 112 99 145)(87 113 100 146)(88 114 91 147)(89 115 92 148)(90 116 93 149)
(1 136 48 142)(2 137 49 143)(3 138 50 144)(4 139 41 145)(5 140 42 146)(6 131 43 147)(7 132 44 148)(8 133 45 149)(9 134 46 150)(10 135 47 141)(11 88 21 74)(12 89 22 75)(13 90 23 76)(14 81 24 77)(15 82 25 78)(16 83 26 79)(17 84 27 80)(18 85 28 71)(19 86 29 72)(20 87 30 73)(31 108 160 92)(32 109 151 93)(33 110 152 94)(34 101 153 95)(35 102 154 96)(36 103 155 97)(37 104 156 98)(38 105 157 99)(39 106 158 100)(40 107 159 91)(51 123 67 119)(52 124 68 120)(53 125 69 111)(54 126 70 112)(55 127 61 113)(56 128 62 114)(57 129 63 115)(58 130 64 116)(59 121 65 117)(60 122 66 118)
(1 83 51 96)(2 84 52 97)(3 85 53 98)(4 86 54 99)(5 87 55 100)(6 88 56 91)(7 89 57 92)(8 90 58 93)(9 81 59 94)(10 82 60 95)(11 131 159 128)(12 132 160 129)(13 133 151 130)(14 134 152 121)(15 135 153 122)(16 136 154 123)(17 137 155 124)(18 138 156 125)(19 139 157 126)(20 140 158 127)(21 147 40 114)(22 148 31 115)(23 149 32 116)(24 150 33 117)(25 141 34 118)(26 142 35 119)(27 143 36 120)(28 144 37 111)(29 145 38 112)(30 146 39 113)(41 72 70 105)(42 73 61 106)(43 74 62 107)(44 75 63 108)(45 76 64 109)(46 77 65 110)(47 78 66 101)(48 79 67 102)(49 80 68 103)(50 71 69 104)

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

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

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

140 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M 4N 4O 4P 5A 5B 5C 5D 10A ··· 10AB 10AC ··· 10AR 20A ··· 20AV 20AW ··· 20BL order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 4 4 4 1 1 1 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 C4○D4 C5×D4 C5×C4○D4 kernel C10×C4.4D4 C2×C4×C20 C10×C22⋊C4 C5×C4.4D4 D4×C2×C10 Q8×C2×C10 C2×C4.4D4 C2×C42 C2×C22⋊C4 C4.4D4 C22×D4 C22×Q8 C2×C20 C2×C10 C2×C4 C22 # reps 1 1 4 8 1 1 4 4 16 32 4 4 4 8 16 32

Matrix representation of C10×C4.4D4 in GL5(𝔽41)

 40 0 0 0 0 0 25 0 0 0 0 0 25 0 0 0 0 0 23 0 0 0 0 0 23
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 23 0 0 0 32 40
,
 1 0 0 0 0 0 40 39 0 0 0 1 1 0 0 0 0 0 32 39 0 0 0 40 9
,
 1 0 0 0 0 0 1 2 0 0 0 0 40 0 0 0 0 0 9 0 0 0 0 1 32

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,23,0,0,0,0,0,23],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,32,0,0,0,23,40],[1,0,0,0,0,0,40,1,0,0,0,39,1,0,0,0,0,0,32,40,0,0,0,39,9],[1,0,0,0,0,0,1,0,0,0,0,2,40,0,0,0,0,0,9,1,0,0,0,0,32] >;

C10×C4.4D4 in GAP, Magma, Sage, TeX

C_{10}\times C_4._4D_4
% in TeX

G:=Group("C10xC4.4D4");
// GroupNames label

G:=SmallGroup(320,1528);
// by ID

G=gap.SmallGroup(320,1528);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,436]);
// Polycyclic

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

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