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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 [×7], C2 [×2], C4 [×10], C22 [×7], C22 [×10], C5, C2×C4 [×2], C2×C4 [×20], C23, C23 [×8], D5 [×2], C10 [×7], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×3], C24, Dic5 [×6], C20 [×4], D10 [×10], C2×C10 [×7], C2.C42, C2.C42, C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×Dic5 [×6], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.C22, C4×Dic5 [×2], C10.D4 [×2], D10⋊C4 [×4], D10⋊C4 [×4], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C2×D10⋊C4 [×3], C10.54(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C4×D5 [×2], C22×D5, C24.C22, C2×C4×D5, C4○D20 [×2], D4×D5 [×2], 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 60 35 75)(2 59 36 74)(3 58 37 73)(4 57 38 72)(5 56 39 71)(6 55 40 80)(7 54 31 79)(8 53 32 78)(9 52 33 77)(10 51 34 76)(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)(91 139 116 154)(92 138 117 153)(93 137 118 152)(94 136 119 151)(95 135 120 160)(96 134 111 159)(97 133 112 158)(98 132 113 157)(99 131 114 156)(100 140 115 155)
(1 130 25 115)(2 129 26 114)(3 128 27 113)(4 127 28 112)(5 126 29 111)(6 125 30 120)(7 124 21 119)(8 123 22 118)(9 122 23 117)(10 121 24 116)(11 65 160 60)(12 64 151 59)(13 63 152 58)(14 62 153 57)(15 61 154 56)(16 70 155 55)(17 69 156 54)(18 68 157 53)(19 67 158 52)(20 66 159 51)(31 109 46 94)(32 108 47 93)(33 107 48 92)(34 106 49 91)(35 105 50 100)(36 104 41 99)(37 103 42 98)(38 102 43 97)(39 101 44 96)(40 110 45 95)(71 144 86 139)(72 143 87 138)(73 142 88 137)(74 141 89 136)(75 150 90 135)(76 149 81 134)(77 148 82 133)(78 147 83 132)(79 146 84 131)(80 145 85 140)
(1 95 6 100)(2 96 7 91)(3 97 8 92)(4 98 9 93)(5 99 10 94)(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), (1,60,35,75)(2,59,36,74)(3,58,37,73)(4,57,38,72)(5,56,39,71)(6,55,40,80)(7,54,31,79)(8,53,32,78)(9,52,33,77)(10,51,34,76)(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)(91,139,116,154)(92,138,117,153)(93,137,118,152)(94,136,119,151)(95,135,120,160)(96,134,111,159)(97,133,112,158)(98,132,113,157)(99,131,114,156)(100,140,115,155), (1,130,25,115)(2,129,26,114)(3,128,27,113)(4,127,28,112)(5,126,29,111)(6,125,30,120)(7,124,21,119)(8,123,22,118)(9,122,23,117)(10,121,24,116)(11,65,160,60)(12,64,151,59)(13,63,152,58)(14,62,153,57)(15,61,154,56)(16,70,155,55)(17,69,156,54)(18,68,157,53)(19,67,158,52)(20,66,159,51)(31,109,46,94)(32,108,47,93)(33,107,48,92)(34,106,49,91)(35,105,50,100)(36,104,41,99)(37,103,42,98)(38,102,43,97)(39,101,44,96)(40,110,45,95)(71,144,86,139)(72,143,87,138)(73,142,88,137)(74,141,89,136)(75,150,90,135)(76,149,81,134)(77,148,82,133)(78,147,83,132)(79,146,84,131)(80,145,85,140), (1,95,6,100)(2,96,7,91)(3,97,8,92)(4,98,9,93)(5,99,10,94)(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), (1,60,35,75)(2,59,36,74)(3,58,37,73)(4,57,38,72)(5,56,39,71)(6,55,40,80)(7,54,31,79)(8,53,32,78)(9,52,33,77)(10,51,34,76)(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)(91,139,116,154)(92,138,117,153)(93,137,118,152)(94,136,119,151)(95,135,120,160)(96,134,111,159)(97,133,112,158)(98,132,113,157)(99,131,114,156)(100,140,115,155), (1,130,25,115)(2,129,26,114)(3,128,27,113)(4,127,28,112)(5,126,29,111)(6,125,30,120)(7,124,21,119)(8,123,22,118)(9,122,23,117)(10,121,24,116)(11,65,160,60)(12,64,151,59)(13,63,152,58)(14,62,153,57)(15,61,154,56)(16,70,155,55)(17,69,156,54)(18,68,157,53)(19,67,158,52)(20,66,159,51)(31,109,46,94)(32,108,47,93)(33,107,48,92)(34,106,49,91)(35,105,50,100)(36,104,41,99)(37,103,42,98)(38,102,43,97)(39,101,44,96)(40,110,45,95)(71,144,86,139)(72,143,87,138)(73,142,88,137)(74,141,89,136)(75,150,90,135)(76,149,81,134)(77,148,82,133)(78,147,83,132)(79,146,84,131)(80,145,85,140), (1,95,6,100)(2,96,7,91)(3,97,8,92)(4,98,9,93)(5,99,10,94)(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)], [(1,60,35,75),(2,59,36,74),(3,58,37,73),(4,57,38,72),(5,56,39,71),(6,55,40,80),(7,54,31,79),(8,53,32,78),(9,52,33,77),(10,51,34,76),(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),(91,139,116,154),(92,138,117,153),(93,137,118,152),(94,136,119,151),(95,135,120,160),(96,134,111,159),(97,133,112,158),(98,132,113,157),(99,131,114,156),(100,140,115,155)], [(1,130,25,115),(2,129,26,114),(3,128,27,113),(4,127,28,112),(5,126,29,111),(6,125,30,120),(7,124,21,119),(8,123,22,118),(9,122,23,117),(10,121,24,116),(11,65,160,60),(12,64,151,59),(13,63,152,58),(14,62,153,57),(15,61,154,56),(16,70,155,55),(17,69,156,54),(18,68,157,53),(19,67,158,52),(20,66,159,51),(31,109,46,94),(32,108,47,93),(33,107,48,92),(34,106,49,91),(35,105,50,100),(36,104,41,99),(37,103,42,98),(38,102,43,97),(39,101,44,96),(40,110,45,95),(71,144,86,139),(72,143,87,138),(73,142,88,137),(74,141,89,136),(75,150,90,135),(76,149,81,134),(77,148,82,133),(78,147,83,132),(79,146,84,131),(80,145,85,140)], [(1,95,6,100),(2,96,7,91),(3,97,8,92),(4,98,9,93),(5,99,10,94),(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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