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G = C2×C5⋊Q16order 160 = 25·5

Direct product of C2 and C5⋊Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C5⋊Q16, C102Q16, C20.20D4, Q8.7D10, C20.16C23, Dic10.10C22, C53(C2×Q16), (C2×Q8).3D5, (C2×C4).54D10, (C2×C10).43D4, C10.55(C2×D4), C4.9(C5⋊D4), (Q8×C10).3C2, C52C8.9C22, C4.16(C22×D5), (C5×Q8).7C22, (C2×C20).38C22, (C2×Dic10).9C2, C22.24(C5⋊D4), (C2×C52C8).6C2, C2.19(C2×C5⋊D4), SmallGroup(160,164)

Series: Derived Chief Lower central Upper central

C1C20 — C2×C5⋊Q16
C1C5C10C20Dic10C2×Dic10 — C2×C5⋊Q16
C5C10C20 — C2×C5⋊Q16
C1C22C2×C4C2×Q8

Generators and relations for C2×C5⋊Q16
 G = < a,b,c,d | a2=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 152 in 60 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], Q8 [×4], C10, C10 [×2], C2×C8, Q16 [×4], C2×Q8, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×Q16, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×C52C8, C5⋊Q16 [×4], C2×Dic10, Q8×C10, C2×C5⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, Q16 [×2], C2×D4, D10 [×3], C2×Q16, C5⋊D4 [×2], C22×D5, C5⋊Q16 [×2], C2×C5⋊D4, C2×C5⋊Q16

Smallest permutation representation of C2×C5⋊Q16
Regular action on 160 points
Generators in S160
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 151)(10 152)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(41 120)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 100)(50 101)(51 102)(52 103)(53 104)(54 97)(55 98)(56 99)(57 108)(58 109)(59 110)(60 111)(61 112)(62 105)(63 106)(64 107)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 153)(144 154)
(1 149 34 51 63)(2 64 52 35 150)(3 151 36 53 57)(4 58 54 37 152)(5 145 38 55 59)(6 60 56 39 146)(7 147 40 49 61)(8 62 50 33 148)(9 18 104 108 80)(10 73 109 97 19)(11 20 98 110 74)(12 75 111 99 21)(13 22 100 112 76)(14 77 105 101 23)(15 24 102 106 78)(16 79 107 103 17)(25 143 81 94 48)(26 41 95 82 144)(27 137 83 96 42)(28 43 89 84 138)(29 139 85 90 44)(30 45 91 86 140)(31 141 87 92 46)(32 47 93 88 142)(65 153 121 134 119)(66 120 135 122 154)(67 155 123 136 113)(68 114 129 124 156)(69 157 125 130 115)(70 116 131 126 158)(71 159 127 132 117)(72 118 133 128 160)
(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)
(1 115 5 119)(2 114 6 118)(3 113 7 117)(4 120 8 116)(9 27 13 31)(10 26 14 30)(11 25 15 29)(12 32 16 28)(17 138 21 142)(18 137 22 141)(19 144 23 140)(20 143 24 139)(33 158 37 154)(34 157 38 153)(35 156 39 160)(36 155 40 159)(41 77 45 73)(42 76 46 80)(43 75 47 79)(44 74 48 78)(49 127 53 123)(50 126 54 122)(51 125 55 121)(52 124 56 128)(57 136 61 132)(58 135 62 131)(59 134 63 130)(60 133 64 129)(65 149 69 145)(66 148 70 152)(67 147 71 151)(68 146 72 150)(81 102 85 98)(82 101 86 97)(83 100 87 104)(84 99 88 103)(89 111 93 107)(90 110 94 106)(91 109 95 105)(92 108 96 112)

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

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

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

C2×C5⋊Q16 is a maximal subgroup of
D20.7D4  C5⋊Q165C4  Dic54Q16  Dic5⋊Q16  Dic10.11D4  D104Q16  Q8.D20  D10⋊Q16  C52C8.D4  Q8.1D20  C42.59D10  C207Q16  D20.37D4  Dic10.37D4  (C2×C10)⋊Q16  C5⋊(C8.D4)  C42.61D10  C42.214D10  C42.65D10  C42.80D10  C20⋊Q16  C203Q16  (C5×Q8).D4  C40.31D4  C40.43D4  Dic10.16D4  C40.26D4  Dic53Q16  D105Q16  C40.37D4  M4(2).16D10  (C2×C10)⋊8Q16  (C5×D4).32D4  C2×D5×Q16  D20.44D4  D20.35C23
C2×C5⋊Q16 is a maximal quotient of
C4⋊C4.230D10  C20.23Q16  C207Q16  (C2×C10).Q16  Dic10.37D4  (C2×C10)⋊Q16  C20.17D8  C20.Q16  C20⋊Q16  Dic105Q8  C203Q16  C20.11Q16  (C2×C10)⋊8Q16

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20L
order122244444455888810···1020···20
size11112244202022101010102···24···4

34 irreducible representations

dim11111222222224
type++++++++-++-
imageC1C2C2C2C2D4D4D5Q16D10D10C5⋊D4C5⋊D4C5⋊Q16
kernelC2×C5⋊Q16C2×C52C8C5⋊Q16C2×Dic10Q8×C10C20C2×C10C2×Q8C10C2×C4Q8C4C22C2
# reps11411112424444

Matrix representation of C2×C5⋊Q16 in GL5(𝔽41)

400000
040000
004000
00010
00001
,
10000
01000
00100
0003440
00010
,
10000
0291200
0292900
00017
000040
,
10000
032000
00900
00010
00001

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,1,0,0,0,40,0],[1,0,0,0,0,0,29,29,0,0,0,12,29,0,0,0,0,0,1,0,0,0,0,7,40],[1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2×C5⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes Q_{16}
% in TeX

G:=Group("C2xC5:Q16");
// GroupNames label

G:=SmallGroup(160,164);
// by ID

G=gap.SmallGroup(160,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,86,579,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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