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## G = C10.54(C4×D4)  order 320 = 26·5

### 6th non-split extension by C10 of C4×D4 acting via C4×D4/C22⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.54(C4×D4)
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — C10.54(C4×D4)
 Lower central C5 — C2×C10 — C10.54(C4×D4)
 Upper central C1 — C23 — C2.C42

Generators and relations for C10.54(C4×D4)
G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=a5c-1 >

Subgroups: 766 in 190 conjugacy classes, 67 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.C22, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C2×D10⋊C4, C10.54(C4×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C422C2, C4×D5, C22×D5, C24.C22, C2×C4×D5, C4○D20, D4×D5, D42D5, Q82D5, C42⋊D5, Dic54D4, D10⋊D4, Dic5.5D4, D208C4, D10.13D4, C4⋊C4⋊D5, C10.54(C4×D4)

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 2 2 2 2 4 4 4 4 10 ··· 10 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 C4×D5 C4○D20 D4×D5 D4⋊2D5 Q8⋊2D5 kernel C10.54(C4×D4) C10.10C42 C5×C2.C42 C2×C4×Dic5 C2×C10.D4 C2×D10⋊C4 D10⋊C4 C2×Dic5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 1 1 1 1 3 8 4 2 8 6 8 16 4 2 2

Matrix representation of C10.54(C4×D4) in GL6(𝔽41)

 0 35 0 0 0 0 7 34 0 0 0 0 0 0 34 34 0 0 0 0 7 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 22 32 0 0 0 0 22 19 0 0 0 0 0 0 11 9 0 0 0 0 14 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 34 1 0 0 0 0 34 7 0 0 0 0 0 0 9 0 0 0 0 0 19 32 0 0 0 0 0 0 18 32 0 0 0 0 27 23
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 18 32 0 0 0 0 4 23

G:=sub<GL(6,GF(41))| [0,7,0,0,0,0,35,34,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,22,0,0,0,0,32,19,0,0,0,0,0,0,11,14,0,0,0,0,9,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,9,19,0,0,0,0,0,32,0,0,0,0,0,0,18,27,0,0,0,0,32,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,18,4,0,0,0,0,32,23] >;

C10.54(C4×D4) in GAP, Magma, Sage, TeX

C_{10}._{54}(C_4\times D_4)
% in TeX

G:=Group("C10.54(C4xD4)");
// GroupNames label

G:=SmallGroup(320,296);
// by ID

G=gap.SmallGroup(320,296);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,422,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

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