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## G = (C2×D20)⋊22C4order 320 = 26·5

### 7th semidirect product of C2×D20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×D20)⋊22C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×D20 — (C2×D20)⋊22C4
 Lower central C5 — C2×C10 — (C2×D20)⋊22C4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×D20)⋊22C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, dcd-1=ac=ca, ad=da, cbc=b-1, dbd-1=b11 >

Subgroups: 1182 in 258 conjugacy classes, 83 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.3C22, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, C2×C4×Dic5, C2×D10⋊C4, C10×C4⋊C4, C22×D20, (C2×D20)⋊22C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C41D4, C4×D5, D20, C5⋊D4, C22×D5, C24.3C22, D10⋊C4, C2×C4×D5, C2×D20, D4×D5, Q82D5, C2×C5⋊D4, D208C4, C4⋊D20, C2×D10⋊C4, C20⋊D4, C20.23D4, (C2×D20)⋊22C4

Smallest permutation representation of (C2×D20)⋊22C4
On 160 points
Generators in S160
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 81)(20 82)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 121)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 131)
(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 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 122)(102 121)(103 140)(104 139)(105 138)(106 137)(107 136)(108 135)(109 134)(110 133)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(141 149)(142 148)(143 147)(144 146)(150 160)(151 159)(152 158)(153 157)(154 156)
(1 109 158 41)(2 120 159 52)(3 111 160 43)(4 102 141 54)(5 113 142 45)(6 104 143 56)(7 115 144 47)(8 106 145 58)(9 117 146 49)(10 108 147 60)(11 119 148 51)(12 110 149 42)(13 101 150 53)(14 112 151 44)(15 103 152 55)(16 114 153 46)(17 105 154 57)(18 116 155 48)(19 107 156 59)(20 118 157 50)(21 85 122 71)(22 96 123 62)(23 87 124 73)(24 98 125 64)(25 89 126 75)(26 100 127 66)(27 91 128 77)(28 82 129 68)(29 93 130 79)(30 84 131 70)(31 95 132 61)(32 86 133 72)(33 97 134 63)(34 88 135 74)(35 99 136 65)(36 90 137 76)(37 81 138 67)(38 92 139 78)(39 83 140 69)(40 94 121 80)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 20 20 2 2 2 2 4 4 4 4 10 ··· 10 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 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D4 D5 C4○D4 D10 C4×D5 D20 C5⋊D4 D4×D5 Q8⋊2D5 kernel (C2×D20)⋊22C4 C2×C4×Dic5 C2×D10⋊C4 C10×C4⋊C4 C22×D20 C2×D20 C2×Dic5 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 4 1 1 8 4 4 2 4 6 8 8 8 4 4

Matrix representation of (C2×D20)⋊22C4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 35 1 0 0 0 0 40 0 0 0 0 0 0 0 7 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 35 1 0 0 0 0 6 6 0 0 0 0 0 0 34 40 0 0 0 0 7 7 0 0 0 0 0 0 40 0 0 0 0 0 0 1
,
 2 13 0 0 0 0 28 39 0 0 0 0 0 0 24 1 0 0 0 0 40 17 0 0 0 0 0 0 9 0 0 0 0 0 0 32

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,7,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[35,6,0,0,0,0,1,6,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[2,28,0,0,0,0,13,39,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;

(C2×D20)⋊22C4 in GAP, Magma, Sage, TeX

(C_2\times D_{20})\rtimes_{22}C_4
% in TeX

G:=Group("(C2xD20):22C4");
// GroupNames label

G:=SmallGroup(320,615);
// by ID

G=gap.SmallGroup(320,615);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,422,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11>;
// generators/relations

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