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## G = C4×C4○D20order 320 = 26·5

### Direct product of C4 and C4○D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×C4○D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×C4○D20 — C4×C4○D20
 Lower central C5 — C10 — C4×C4○D20
 Upper central C1 — C42 — C2×C42

Generators and relations for C4×C4○D20
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >

Subgroups: 894 in 310 conjugacy classes, 159 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×10], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×8], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C2×C42 [×2], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C22×D5 [×2], C22×C10, C4×C4○D4, C4×Dic5 [×6], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×2], C4×C20 [×2], C4×C20 [×2], C2×Dic10, C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C22×C20 [×2], C4×Dic10 [×2], D5×C42 [×2], C42⋊D5 [×2], C4×D20 [×2], C23.21D10, C4×C5⋊D4 [×4], C2×C4×C20, C2×C4○D20, C4×C4○D20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C4×C4○D4, C2×C4×D5 [×6], C4○D20 [×4], C23×D5, D5×C22×C4, C2×C4○D20 [×2], C4×C4○D20

Smallest permutation representation of C4×C4○D20
On 160 points
Generators in S160
(1 119 122 74)(2 120 123 75)(3 101 124 76)(4 102 125 77)(5 103 126 78)(6 104 127 79)(7 105 128 80)(8 106 129 61)(9 107 130 62)(10 108 131 63)(11 109 132 64)(12 110 133 65)(13 111 134 66)(14 112 135 67)(15 113 136 68)(16 114 137 69)(17 115 138 70)(18 116 139 71)(19 117 140 72)(20 118 121 73)(21 156 60 92)(22 157 41 93)(23 158 42 94)(24 159 43 95)(25 160 44 96)(26 141 45 97)(27 142 46 98)(28 143 47 99)(29 144 48 100)(30 145 49 81)(31 146 50 82)(32 147 51 83)(33 148 52 84)(34 149 53 85)(35 150 54 86)(36 151 55 87)(37 152 56 88)(38 153 57 89)(39 154 58 90)(40 155 59 91)
(1 96 11 86)(2 97 12 87)(3 98 13 88)(4 99 14 89)(5 100 15 90)(6 81 16 91)(7 82 17 92)(8 83 18 93)(9 84 19 94)(10 85 20 95)(21 105 31 115)(22 106 32 116)(23 107 33 117)(24 108 34 118)(25 109 35 119)(26 110 36 120)(27 111 37 101)(28 112 38 102)(29 113 39 103)(30 114 40 104)(41 61 51 71)(42 62 52 72)(43 63 53 73)(44 64 54 74)(45 65 55 75)(46 66 56 76)(47 67 57 77)(48 68 58 78)(49 69 59 79)(50 70 60 80)(121 159 131 149)(122 160 132 150)(123 141 133 151)(124 142 134 152)(125 143 135 153)(126 144 136 154)(127 145 137 155)(128 146 138 156)(129 147 139 157)(130 148 140 158)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 111)(102 110)(103 109)(104 108)(105 107)(112 120)(113 119)(114 118)(115 117)(121 137)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(138 140)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4L 4M ··· 4R 4S ··· 4AD 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 D10 C4×D5 C4○D20 kernel C4×C4○D20 C4×Dic10 D5×C42 C42⋊D5 C4×D20 C23.21D10 C4×C5⋊D4 C2×C4×C20 C2×C4○D20 C4○D20 C2×C42 C20 C42 C22×C4 C2×C4 C4 # reps 1 2 2 2 2 1 4 1 1 16 2 8 8 6 16 32

Matrix representation of C4×C4○D20 in GL3(𝔽41) generated by

 9 0 0 0 40 0 0 0 40
,
 1 0 0 0 32 0 0 0 32
,
 40 0 0 0 30 39 0 16 14
,
 40 0 0 0 1 0 0 8 40
G:=sub<GL(3,GF(41))| [9,0,0,0,40,0,0,0,40],[1,0,0,0,32,0,0,0,32],[40,0,0,0,30,16,0,39,14],[40,0,0,0,1,8,0,0,40] >;

C4×C4○D20 in GAP, Magma, Sage, TeX

C_4\times C_4\circ D_{20}
% in TeX

G:=Group("C4xC4oD20");
// GroupNames label

G:=SmallGroup(320,1146);
// by ID

G=gap.SmallGroup(320,1146);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations

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