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## G = C10.90(C4×D4)  order 320 = 26·5

### 16th non-split extension by C10 of C4×D4 acting via C4×D4/C4⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.90(C4×D4)
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — C10.90(C4×D4)
 Lower central C5 — C2×C10 — C10.90(C4×D4)
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C10.90(C4×D4)
G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=dad-1=a-1, ac=ca, cbc-1=a5b, bd=db, dcd-1=a5c-1 >

Subgroups: 750 in 190 conjugacy classes, 69 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×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.C22, C4×Dic5, D10⋊C4, D10⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C23×D5, C10.10C42, C2×C4×Dic5, C2×D10⋊C4, C10×C4⋊C4, C10.90(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, C5⋊D4, C22×D5, C24.C22, C2×C4×D5, C4○D20, D4×D5, D42D5, Q82D5, C2×C5⋊D4, C4⋊C47D5, D208C4, D10.13D4, C4⋊C4⋊D5, C4×C5⋊D4, Dic5⋊D4, C20.23D4, C10.90(C4×D4)

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

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

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

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

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 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C4 D4 D4 D5 C4○D4 D10 C4×D5 C5⋊D4 C4○D20 D4×D5 D4⋊2D5 Q8⋊2D5 kernel C10.90(C4×D4) C10.10C42 C2×C4×Dic5 C2×D10⋊C4 C10×C4⋊C4 D10⋊C4 C2×Dic5 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 2 1 3 1 8 2 2 2 8 6 8 8 8 2 2 4

Matrix representation of C10.90(C4×D4) in GL6(𝔽41)

 7 7 0 0 0 0 35 0 0 0 0 0 0 0 1 34 0 0 0 0 7 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 38 37 0 0 0 0 23 3 0 0 0 0 0 0 14 14 0 0 0 0 30 27 0 0 0 0 0 0 9 8 0 0 0 0 0 32
,
 24 7 0 0 0 0 35 17 0 0 0 0 0 0 24 40 0 0 0 0 1 17 0 0 0 0 0 0 32 0 0 0 0 0 10 9
,
 3 4 0 0 0 0 18 38 0 0 0 0 0 0 3 3 0 0 0 0 24 38 0 0 0 0 0 0 9 0 0 0 0 0 0 9

G:=sub<GL(6,GF(41))| [7,35,0,0,0,0,7,0,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[38,23,0,0,0,0,37,3,0,0,0,0,0,0,14,30,0,0,0,0,14,27,0,0,0,0,0,0,9,0,0,0,0,0,8,32],[24,35,0,0,0,0,7,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,32,10,0,0,0,0,0,9],[3,18,0,0,0,0,4,38,0,0,0,0,0,0,3,24,0,0,0,0,3,38,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C10.90(C4×D4) in GAP, Magma, Sage, TeX

C_{10}._{90}(C_4\times D_4)
% in TeX

G:=Group("C10.90(C4xD4)");
// GroupNames label

G:=SmallGroup(320,617);
// by ID

G=gap.SmallGroup(320,617);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,758,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=a^5,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*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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