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G = C5×C4⋊SD16order 320 = 26·5

Direct product of C5 and C4⋊SD16

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

Aliases: C5×C4⋊SD16, C2015SD16, C4⋊C88C10, Q81(C5×D4), (C5×Q8)⋊12D4, (C4×Q8)⋊3C10, C43(C5×SD16), (Q8×C20)⋊23C2, C4.32(D4×C10), C20.393(C2×D4), C41D4.3C10, (C2×C20).322D4, D4⋊C410C10, C2.7(C10×SD16), (C10×SD16)⋊28C2, (C2×SD16)⋊11C10, C42.15(C2×C10), C10.87(C2×SD16), C22.84(D4×C10), C20.342(C4○D4), (C4×C20).257C22, (C2×C20).919C23, (C2×C40).299C22, C10.143(C4⋊D4), C10.135(C8⋊C22), (D4×C10).186C22, (Q8×C10).262C22, (C5×C4⋊C8)⋊27C2, C4.41(C5×C4○D4), C4⋊C4.52(C2×C10), (C2×C8).36(C2×C10), (C2×C4).128(C5×D4), C2.12(C5×C4⋊D4), C2.10(C5×C8⋊C22), (C5×D4⋊C4)⋊34C2, (C2×D4).10(C2×C10), (C5×C41D4).10C2, (C2×C10).640(C2×D4), (C2×Q8).47(C2×C10), (C5×C4⋊C4).373C22, (C2×C4).94(C22×C10), SmallGroup(320,961)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C4⋊SD16
C1C2C4C2×C4C2×C20D4×C10C5×C41D4 — C5×C4⋊SD16
C1C2C2×C4 — C5×C4⋊SD16
C1C2×C10C4×C20 — C5×C4⋊SD16

Generators and relations for C5×C4⋊SD16
 G = < a,b,c,d | a5=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c3 >

Subgroups: 266 in 128 conjugacy classes, 58 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], Q8, C23 [×2], C10 [×3], C10 [×2], C42, C42, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8, C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C40 [×2], C2×C20 [×3], C2×C20 [×2], C5×D4 [×8], C5×Q8 [×2], C5×Q8, C22×C10 [×2], C4⋊SD16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40 [×2], C5×SD16 [×4], D4×C10 [×2], D4×C10 [×2], Q8×C10, C5×D4⋊C4 [×2], C5×C4⋊C8, Q8×C20, C5×C41D4, C10×SD16 [×2], C5×C4⋊SD16
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×4], C23, C10 [×7], SD16 [×2], C2×D4 [×2], C4○D4, C2×C10 [×7], C4⋊D4, C2×SD16, C8⋊C22, C5×D4 [×4], C22×C10, C4⋊SD16, C5×SD16 [×2], D4×C10 [×2], C5×C4○D4, C5×C4⋊D4, C10×SD16, C5×C8⋊C22, C5×C4⋊SD16

Smallest permutation representation of C5×C4⋊SD16
On 160 points
Generators in S160
(1 9 61 20 53)(2 10 62 21 54)(3 11 63 22 55)(4 12 64 23 56)(5 13 57 24 49)(6 14 58 17 50)(7 15 59 18 51)(8 16 60 19 52)(25 85 74 33 66)(26 86 75 34 67)(27 87 76 35 68)(28 88 77 36 69)(29 81 78 37 70)(30 82 79 38 71)(31 83 80 39 72)(32 84 73 40 65)(41 157 120 149 112)(42 158 113 150 105)(43 159 114 151 106)(44 160 115 152 107)(45 153 116 145 108)(46 154 117 146 109)(47 155 118 147 110)(48 156 119 148 111)(89 126 142 97 134)(90 127 143 98 135)(91 128 144 99 136)(92 121 137 100 129)(93 122 138 101 130)(94 123 139 102 131)(95 124 140 103 132)(96 125 141 104 133)
(1 122 30 106)(2 107 31 123)(3 124 32 108)(4 109 25 125)(5 126 26 110)(6 111 27 127)(7 128 28 112)(8 105 29 121)(9 138 82 43)(10 44 83 139)(11 140 84 45)(12 46 85 141)(13 142 86 47)(14 48 87 143)(15 144 88 41)(16 42 81 137)(17 119 35 135)(18 136 36 120)(19 113 37 129)(20 130 38 114)(21 115 39 131)(22 132 40 116)(23 117 33 133)(24 134 34 118)(49 89 67 147)(50 148 68 90)(51 91 69 149)(52 150 70 92)(53 93 71 151)(54 152 72 94)(55 95 65 145)(56 146 66 96)(57 97 75 155)(58 156 76 98)(59 99 77 157)(60 158 78 100)(61 101 79 159)(62 160 80 102)(63 103 73 153)(64 154 74 104)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 140)(42 143)(43 138)(44 141)(45 144)(46 139)(47 142)(48 137)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)(73 77)(74 80)(76 78)(81 87)(83 85)(84 88)(89 147)(90 150)(91 145)(92 148)(93 151)(94 146)(95 149)(96 152)(97 155)(98 158)(99 153)(100 156)(101 159)(102 154)(103 157)(104 160)(105 127)(106 122)(107 125)(108 128)(109 123)(110 126)(111 121)(112 124)(113 135)(114 130)(115 133)(116 136)(117 131)(118 134)(119 129)(120 132)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4I5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20P20Q···20AJ40A···40P
order12222244444···45555888810···1010···1020···2020···2040···40
size11118822224···4111144441···18···82···24···44···4

95 irreducible representations

dim1111111111112222222244
type+++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4SD16C4○D4C5×D4C5×D4C5×SD16C5×C4○D4C8⋊C22C5×C8⋊C22
kernelC5×C4⋊SD16C5×D4⋊C4C5×C4⋊C8Q8×C20C5×C41D4C10×SD16C4⋊SD16D4⋊C4C4⋊C8C4×Q8C41D4C2×SD16C2×C20C5×Q8C20C20C2×C4Q8C4C4C10C2
# reps12111248444822428816814

Matrix representation of C5×C4⋊SD16 in GL4(𝔽41) generated by

10000
01000
00370
00037
,
91800
03200
0010
0001
,
403900
1100
002626
001526
,
1000
404000
0010
00040
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,37,0,0,0,0,37],[9,0,0,0,18,32,0,0,0,0,1,0,0,0,0,1],[40,1,0,0,39,1,0,0,0,0,26,15,0,0,26,26],[1,40,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

C5×C4⋊SD16 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C5xC4:SD16");
// GroupNames label

G:=SmallGroup(320,961);
// by ID

G=gap.SmallGroup(320,961);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,856,10085,2539,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations

׿
×
𝔽