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## G = C5×C42.78C22order 320 = 26·5

### Direct product of C5 and C42.78C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C42.78C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — D4×C10 — C5×D4⋊C4 — C5×C42.78C22
 Lower central C1 — C2 — C2×C4 — C5×C42.78C22
 Upper central C1 — C2×C10 — C4×C20 — C5×C42.78C22

Generators and relations for C5×C42.78C22
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, be=eb, dcd=c-1, ce=ec, ede-1=b2cd >

Subgroups: 178 in 96 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.78C22, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, D4×C10, Q8×C10, C4×C40, C5×D4⋊C4, C5×Q8⋊C4, C5×C4.4D4, C5×C42.C2, C5×C42.78C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4.4D4, C4○D8, C5×D4, C22×C10, C42.78C22, D4×C10, C5×C4○D4, C5×C4.4D4, C5×C4○D8, C5×C42.78C22

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 4A ··· 4F 4G 4H 4I 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M 10N 10O 10P 20A ··· 20X 20Y ··· 20AJ 40A ··· 40AF order 1 2 2 2 2 4 ··· 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 8 2 ··· 2 8 8 8 1 1 1 1 2 ··· 2 1 ··· 1 8 8 8 8 2 ··· 2 8 ··· 8 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 C4○D4 C4○D8 C5×D4 C5×C4○D4 C5×C4○D8 kernel C5×C42.78C22 C4×C40 C5×D4⋊C4 C5×Q8⋊C4 C5×C4.4D4 C5×C42.C2 C42.78C22 C4×C8 D4⋊C4 Q8⋊C4 C4.4D4 C42.C2 C2×C20 C20 C10 C2×C4 C4 C2 # reps 1 1 2 2 1 1 4 4 8 8 4 4 2 4 8 8 16 32

Matrix representation of C5×C42.78C22 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 37 0 0 0 0 37
,
 32 0 0 0 0 32 0 0 0 0 0 9 0 0 32 0
,
 0 40 0 0 1 0 0 0 0 0 0 40 0 0 1 0
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 26 26 0 0 15 26 0 0 0 0 12 29 0 0 12 12
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[32,0,0,0,0,32,0,0,0,0,0,32,0,0,9,0],[0,1,0,0,40,0,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[26,15,0,0,26,26,0,0,0,0,12,12,0,0,29,12] >;

C5×C42.78C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2._{78}C_2^2
% in TeX

G:=Group("C5xC4^2.78C2^2");
// GroupNames label

G:=SmallGroup(320,989);
// by ID

G=gap.SmallGroup(320,989);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1688,1766,226,7004,172,10085,124]);
// 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,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c*d>;
// generators/relations

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