Copied to
clipboard

G = C20⋊(C4○D4)  order 320 = 26·5

2nd semidirect product of C20 and C4○D4 acting via C4○D4/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20⋊Q818C2, C209(C4○D4), C4⋊D425D5, C20⋊D414C2, C207D431C2, C43(D42D5), C22.1(D4×D5), C4⋊C4.175D10, (D4×Dic5)⋊14C2, (C2×Dic5)⋊13D4, D208C419C2, Dic52(C4○D4), Dic5⋊D47C2, Dic54D45C2, (C2×D4).150D10, (C2×C20).34C23, C22⋊C4.45D10, Dic5.17(C2×D4), C10.59(C22×D4), (C2×C10).140C24, (C22×C4).366D10, Dic5.5D416C2, (D4×C10).114C22, (C2×D20).147C22, C4⋊Dic5.203C22, (C22×C10).11C23, C53(C22.26C24), (C22×D5).59C23, C22.161(C23×D5), C23.178(C22×D5), C23.D5.18C22, D10⋊C4.57C22, (C22×C20).235C22, (C4×Dic5).285C22, (C2×Dic5).233C23, C10.D4.12C22, (C2×Dic10).156C22, (C22×Dic5).244C22, C2.32(C2×D4×D5), (C2×C4×Dic5)⋊7C2, (C5×C4⋊D4)⋊5C2, C2.33(D5×C4○D4), (C2×C10).3(C2×D4), (C2×D42D5)⋊8C2, C10.79(C2×C4○D4), (C2×C4×D5).88C22, C2.30(C2×D42D5), (C2×C4).34(C22×D5), (C5×C4⋊C4).136C22, (C2×C5⋊D4).23C22, (C5×C22⋊C4).5C22, SmallGroup(320,1268)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20⋊(C4○D4)
C1C5C10C2×C10C2×Dic5C22×Dic5C2×D42D5 — C20⋊(C4○D4)
C5C2×C10 — C20⋊(C4○D4)
C1C22C4⋊D4

Generators and relations for C20⋊(C4○D4)
 G = < a,b,c,d | a20=b4=d2=1, c2=b2, bab-1=a9, cac-1=a11, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 1102 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×22], D4 [×20], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic5 [×6], Dic5 [×3], C20 [×2], C20 [×3], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C42, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×6], C2×Dic5 [×6], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.26C24, C4×Dic5 [×4], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, D42D5 [×8], C22×Dic5 [×2], C22×Dic5 [×2], C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], Dic54D4 [×2], Dic5.5D4 [×2], C20⋊Q8, D208C4, C2×C4×Dic5, C207D4, D4×Dic5, Dic5⋊D4 [×2], C20⋊D4, C5×C4⋊D4, C2×D42D5 [×2], C20⋊(C4○D4)
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, D4×D5 [×2], D42D5 [×2], C23×D5, C2×D4×D5, C2×D42D5, D5×C4○D4, C20⋊(C4○D4)

Smallest permutation representation of C20⋊(C4○D4)
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)
(1 129 117 69)(2 138 118 78)(3 127 119 67)(4 136 120 76)(5 125 101 65)(6 134 102 74)(7 123 103 63)(8 132 104 72)(9 121 105 61)(10 130 106 70)(11 139 107 79)(12 128 108 68)(13 137 109 77)(14 126 110 66)(15 135 111 75)(16 124 112 64)(17 133 113 73)(18 122 114 62)(19 131 115 71)(20 140 116 80)(21 83 157 56)(22 92 158 45)(23 81 159 54)(24 90 160 43)(25 99 141 52)(26 88 142 41)(27 97 143 50)(28 86 144 59)(29 95 145 48)(30 84 146 57)(31 93 147 46)(32 82 148 55)(33 91 149 44)(34 100 150 53)(35 89 151 42)(36 98 152 51)(37 87 153 60)(38 96 154 49)(39 85 155 58)(40 94 156 47)
(1 69 117 129)(2 80 118 140)(3 71 119 131)(4 62 120 122)(5 73 101 133)(6 64 102 124)(7 75 103 135)(8 66 104 126)(9 77 105 137)(10 68 106 128)(11 79 107 139)(12 70 108 130)(13 61 109 121)(14 72 110 132)(15 63 111 123)(16 74 112 134)(17 65 113 125)(18 76 114 136)(19 67 115 127)(20 78 116 138)(21 83 157 56)(22 94 158 47)(23 85 159 58)(24 96 160 49)(25 87 141 60)(26 98 142 51)(27 89 143 42)(28 100 144 53)(29 91 145 44)(30 82 146 55)(31 93 147 46)(32 84 148 57)(33 95 149 48)(34 86 150 59)(35 97 151 50)(36 88 152 41)(37 99 153 52)(38 90 154 43)(39 81 155 54)(40 92 156 45)
(1 147)(2 148)(3 149)(4 150)(5 151)(6 152)(7 153)(8 154)(9 155)(10 156)(11 157)(12 158)(13 159)(14 160)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 132)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 140)(58 121)(59 122)(60 123)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 100)(77 81)(78 82)(79 83)(80 84)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K···4P4Q4R5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order122222222244444444444···4445510···10101010101010101020···2020202020
size111122442020222244555510···102020222···2444488884···48888

56 irreducible representations

dim11111111111122222222444
type++++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4C4○D4D10D10D10D10D42D5D4×D5D5×C4○D4
kernelC20⋊(C4○D4)Dic54D4Dic5.5D4C20⋊Q8D208C4C2×C4×Dic5C207D4D4×Dic5Dic5⋊D4C20⋊D4C5×C4⋊D4C2×D42D5C2×Dic5C4⋊D4Dic5C20C22⋊C4C4⋊C4C22×C4C2×D4C4C22C2
# reps12211111211242444226444

Matrix representation of C20⋊(C4○D4) in GL6(𝔽41)

010000
40350000
00403900
001100
0000400
0000040
,
3560000
160000
009000
000900
0000320
0000032
,
4000000
0400000
0032000
009900
000090
0000132
,
4000000
0400000
0091800
00323200
000092
0000132

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,35,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,18,32,0,0,0,0,0,0,9,1,0,0,0,0,2,32] >;

C20⋊(C4○D4) in GAP, Magma, Sage, TeX

C_{20}\rtimes (C_4\circ D_4)
% in TeX

G:=Group("C20:(C4oD4)");
// GroupNames label

G:=SmallGroup(320,1268);
// by ID

G=gap.SmallGroup(320,1268);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,570,185,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^4=d^2=1,c^2=b^2,b*a*b^-1=a^9,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

׿
×
𝔽