Copied to
clipboard

## G = C5×C4⋊Q8order 160 = 25·5

### Direct product of C5 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C4⋊Q8
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — Q8×C10 — C5×C4⋊Q8
 Lower central C1 — C22 — C5×C4⋊Q8
 Upper central C1 — C2×C10 — C5×C4⋊Q8

Generators and relations for C5×C4⋊Q8
G = < a,b,c,d | a5=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 84 in 68 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C5, C2×C4, C2×C4 [×6], Q8 [×4], C10, C10 [×2], C42, C4⋊C4 [×4], C2×Q8 [×2], C20 [×6], C20 [×4], C2×C10, C4⋊Q8, C2×C20, C2×C20 [×6], C5×Q8 [×4], C4×C20, C5×C4⋊C4 [×4], Q8×C10 [×2], C5×C4⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], Q8 [×4], C23, C10 [×7], C2×D4, C2×Q8 [×2], C2×C10 [×7], C4⋊Q8, C5×D4 [×2], C5×Q8 [×4], C22×C10, D4×C10, Q8×C10 [×2], C5×C4⋊Q8

Smallest permutation representation of C5×C4⋊Q8
Regular action 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 95 23 90)(2 91 24 86)(3 92 25 87)(4 93 21 88)(5 94 22 89)(6 103 11 96)(7 104 12 97)(8 105 13 98)(9 101 14 99)(10 102 15 100)(16 110 156 115)(17 106 157 111)(18 107 158 112)(19 108 159 113)(20 109 160 114)(26 84 31 77)(27 85 32 78)(28 81 33 79)(29 82 34 80)(30 83 35 76)(36 63 43 56)(37 64 44 57)(38 65 45 58)(39 61 41 59)(40 62 42 60)(46 71 51 66)(47 72 52 67)(48 73 53 68)(49 74 54 69)(50 75 55 70)(116 136 123 143)(117 137 124 144)(118 138 125 145)(119 139 121 141)(120 140 122 142)(126 146 131 151)(127 147 132 152)(128 148 133 153)(129 149 134 154)(130 150 135 155)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 123 16 130)(7 124 17 126)(8 125 18 127)(9 121 19 128)(10 122 20 129)(11 116 156 135)(12 117 157 131)(13 118 158 132)(14 119 159 133)(15 120 160 134)(21 68 28 61)(22 69 29 62)(23 70 30 63)(24 66 26 64)(25 67 27 65)(36 95 55 76)(37 91 51 77)(38 92 52 78)(39 93 53 79)(40 94 54 80)(41 88 48 81)(42 89 49 82)(43 90 50 83)(44 86 46 84)(45 87 47 85)(96 136 115 155)(97 137 111 151)(98 138 112 152)(99 139 113 153)(100 140 114 154)(101 141 108 148)(102 142 109 149)(103 143 110 150)(104 144 106 146)(105 145 107 147)
(1 115 35 96)(2 111 31 97)(3 112 32 98)(4 113 33 99)(5 114 34 100)(6 90 16 83)(7 86 17 84)(8 87 18 85)(9 88 19 81)(10 89 20 82)(11 95 156 76)(12 91 157 77)(13 92 158 78)(14 93 159 79)(15 94 160 80)(21 108 28 101)(22 109 29 102)(23 110 30 103)(24 106 26 104)(25 107 27 105)(36 135 55 116)(37 131 51 117)(38 132 52 118)(39 133 53 119)(40 134 54 120)(41 128 48 121)(42 129 49 122)(43 130 50 123)(44 126 46 124)(45 127 47 125)(56 155 75 136)(57 151 71 137)(58 152 72 138)(59 153 73 139)(60 154 74 140)(61 148 68 141)(62 149 69 142)(63 150 70 143)(64 146 66 144)(65 147 67 145)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 5A 5B 5C 5D 10A ··· 10L 20A ··· 20X 20Y ··· 20AN order 1 2 2 2 4 ··· 4 4 4 4 4 5 5 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4

70 irreducible representations

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

Matrix representation of C5×C4⋊Q8 in GL5(𝔽41)

 37 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 40 2 0 0 0 40 1
,
 40 0 0 0 0 0 6 2 0 0 0 2 35 0 0 0 0 0 33 12 0 0 0 39 8

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,40,0,0,0,2,1],[40,0,0,0,0,0,6,2,0,0,0,2,35,0,0,0,0,0,33,39,0,0,0,12,8] >;

C5×C4⋊Q8 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes Q_8
% in TeX

G:=Group("C5xC4:Q8");
// GroupNames label

G:=SmallGroup(160,189);
// by ID

G=gap.SmallGroup(160,189);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,505,247,1514,374]);
// Polycyclic

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

׿
×
𝔽