Copied to
clipboard

## G = C5×C42.28C22order 320 = 26·5

### Direct product of C5 and C42.28C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C42.28C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — D4×C10 — C5×D4⋊C4 — C5×C42.28C22
 Lower central C1 — C2 — C2×C4 — C5×C42.28C22
 Upper central C1 — C2×C10 — C4×C20 — C5×C42.28C22

Generators and relations for C5×C42.28C22
G = < a,b,c,d,e | a5=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 194 in 100 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.28C22, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, D4×C10, Q8×C10, Q8×C10, C5×C8⋊C4, C5×D4⋊C4, C5×Q8⋊C4, C5×C4.4D4, C5×C4⋊Q8, C5×C42.28C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4.4D4, C8⋊C22, C8.C22, C5×D4, C22×C10, C42.28C22, D4×C10, C5×C4○D4, C5×C4.4D4, C5×C8⋊C22, C5×C8.C22, C5×C42.28C22

Smallest permutation representation of C5×C42.28C22
On 160 points
Generators in S160
(1 111 31 103 23)(2 112 32 104 24)(3 105 25 97 17)(4 106 26 98 18)(5 107 27 99 19)(6 108 28 100 20)(7 109 29 101 21)(8 110 30 102 22)(9 90 42 114 34)(10 91 43 115 35)(11 92 44 116 36)(12 93 45 117 37)(13 94 46 118 38)(14 95 47 119 39)(15 96 48 120 40)(16 89 41 113 33)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 85 153 73 145)(66 86 154 74 146)(67 87 155 75 147)(68 88 156 76 148)(69 81 157 77 149)(70 82 158 78 150)(71 83 159 79 151)(72 84 160 80 152)
(1 72 121 38)(2 69 122 35)(3 66 123 40)(4 71 124 37)(5 68 125 34)(6 65 126 39)(7 70 127 36)(8 67 128 33)(9 107 88 141)(10 112 81 138)(11 109 82 143)(12 106 83 140)(13 111 84 137)(14 108 85 142)(15 105 86 139)(16 110 87 144)(17 146 51 120)(18 151 52 117)(19 148 53 114)(20 145 54 119)(21 150 55 116)(22 147 56 113)(23 152 49 118)(24 149 50 115)(25 154 59 96)(26 159 60 93)(27 156 61 90)(28 153 62 95)(29 158 63 92)(30 155 64 89)(31 160 57 94)(32 157 58 91)(41 102 75 136)(42 99 76 133)(43 104 77 130)(44 101 78 135)(45 98 79 132)(46 103 80 129)(47 100 73 134)(48 97 74 131)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(2 124)(3 7)(4 122)(6 128)(8 126)(9 84)(10 16)(11 82)(12 14)(13 88)(15 86)(17 21)(18 50)(20 56)(22 54)(24 52)(25 29)(26 58)(28 64)(30 62)(32 60)(33 35)(34 72)(36 70)(37 39)(38 68)(40 66)(41 43)(42 80)(44 78)(45 47)(46 76)(48 74)(51 55)(59 63)(65 71)(67 69)(73 79)(75 77)(81 87)(83 85)(89 91)(90 160)(92 158)(93 95)(94 156)(96 154)(97 101)(98 130)(100 136)(102 134)(104 132)(105 109)(106 138)(108 144)(110 142)(112 140)(113 115)(114 152)(116 150)(117 119)(118 148)(120 146)(123 127)(131 135)(139 143)(145 151)(147 149)(153 159)(155 157)
(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)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M 10N 10O 10P 20A ··· 20H 20I ··· 20P 20Q ··· 20AB 40A ··· 40P order 1 2 2 2 2 4 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 8 2 2 4 4 8 8 8 1 1 1 1 4 4 4 4 1 ··· 1 8 8 8 8 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 C4○D4 C5×D4 C5×C4○D4 C8⋊C22 C8.C22 C5×C8⋊C22 C5×C8.C22 kernel C5×C42.28C22 C5×C8⋊C4 C5×D4⋊C4 C5×Q8⋊C4 C5×C4.4D4 C5×C4⋊Q8 C42.28C22 C8⋊C4 D4⋊C4 Q8⋊C4 C4.4D4 C4⋊Q8 C2×C20 C20 C2×C4 C4 C10 C10 C2 C2 # reps 1 1 2 2 1 1 4 4 8 8 4 4 2 4 8 16 1 1 4 4

Matrix representation of C5×C42.28C22 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 0 0 0 0 0 0 37 0 0 0 0 0 0 37 0 0 0 0 0 0 37
,
 0 9 0 0 0 0 9 0 0 0 0 0 0 0 16 26 13 1 0 0 0 21 37 4 0 0 8 8 35 31 0 0 8 0 15 10
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 40 2 25 0 0 2 1 14 18 0 0 0 0 0 1 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 1 39 2 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 7 3 26 30 0 0 33 33 6 10 0 0 40 20 38 36 0 0 0 21 37 4

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,16,0,8,8,0,0,26,21,8,0,0,0,13,37,35,15,0,0,1,4,31,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,2,14,0,40,0,0,25,18,1,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1,40,0,0,0,0,39,0,0,1,0,0,2,0,1,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,7,33,40,0,0,0,3,33,20,21,0,0,26,6,38,37,0,0,30,10,36,4] >;

C5×C42.28C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2._{28}C_2^2
% in TeX

G:=Group("C5xC4^2.28C2^2");
// GroupNames label

G:=SmallGroup(320,990);
// by ID

G=gap.SmallGroup(320,990);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,568,1766,1731,226,7004,172,10085,124]);
// Polycyclic

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

׿
×
𝔽