Copied to
clipboard

G = C42.157D10order 320 = 26·5

157th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.157D10, C10.312- 1+4, C10.1352+ 1+4, C20⋊Q838C2, C4⋊C4.114D10, C42.C213D5, D102Q839C2, D10⋊Q837C2, C20.6Q830C2, (C2×C20).190C23, (C2×C10).243C24, (C4×C20).224C22, C4.D20.14C2, (C2×D20).38C22, C2.60(D48D10), D10.13D4.4C2, C4⋊Dic5.245C22, C22.264(C23×D5), D10⋊C4.43C22, C54(C22.57C24), (C2×Dic10).43C22, (C2×Dic5).125C23, (C4×Dic5).156C22, C10.D4.86C22, (C22×D5).108C23, C2.61(D4.10D10), C2.32(Q8.10D10), C4⋊C4⋊D538C2, (C5×C42.C2)⋊16C2, (C2×C4×D5).142C22, (C5×C4⋊C4).198C22, (C2×C4).207(C22×D5), SmallGroup(320,1371)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.157D10
C1C5C10C2×C10C22×D5C2×C4×D5D102Q8 — C42.157D10
C5C2×C10 — C42.157D10
C1C22C42.C2

Generators and relations for C42.157D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 710 in 196 conjugacy classes, 91 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic5 [×6], C20 [×7], D10 [×6], C2×C10, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2, C42.C2, C422C2 [×4], C4⋊Q8 [×2], Dic10 [×3], C4×D5 [×2], D20, C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C20.6Q8, C4.D20, C20⋊Q8 [×2], D10.13D4 [×2], D10⋊Q8 [×2], D102Q8 [×2], C4⋊C4⋊D5 [×4], C5×C42.C2, C42.157D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D5 [×7], C22.57C24, C23×D5, Q8.10D10, D48D10, D4.10D10, C42.157D10

Smallest permutation representation of C42.157D10
On 160 points
Generators in S160
(1 159 93 60)(2 150 94 51)(3 141 95 42)(4 152 96 53)(5 143 97 44)(6 154 98 55)(7 145 99 46)(8 156 100 57)(9 147 81 48)(10 158 82 59)(11 149 83 50)(12 160 84 41)(13 151 85 52)(14 142 86 43)(15 153 87 54)(16 144 88 45)(17 155 89 56)(18 146 90 47)(19 157 91 58)(20 148 92 49)(21 72 114 127)(22 63 115 138)(23 74 116 129)(24 65 117 140)(25 76 118 131)(26 67 119 122)(27 78 120 133)(28 69 101 124)(29 80 102 135)(30 71 103 126)(31 62 104 137)(32 73 105 128)(33 64 106 139)(34 75 107 130)(35 66 108 121)(36 77 109 132)(37 68 110 123)(38 79 111 134)(39 70 112 125)(40 61 113 136)
(1 36 83 119)(2 120 84 37)(3 38 85 101)(4 102 86 39)(5 40 87 103)(6 104 88 21)(7 22 89 105)(8 106 90 23)(9 24 91 107)(10 108 92 25)(11 26 93 109)(12 110 94 27)(13 28 95 111)(14 112 96 29)(15 30 97 113)(16 114 98 31)(17 32 99 115)(18 116 100 33)(19 34 81 117)(20 118 82 35)(41 68 150 133)(42 134 151 69)(43 70 152 135)(44 136 153 71)(45 72 154 137)(46 138 155 73)(47 74 156 139)(48 140 157 75)(49 76 158 121)(50 122 159 77)(51 78 160 123)(52 124 141 79)(53 80 142 125)(54 126 143 61)(55 62 144 127)(56 128 145 63)(57 64 146 129)(58 130 147 65)(59 66 148 131)(60 132 149 67)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 103 31 113)(22 112 32 102)(23 101 33 111)(24 110 34 120)(25 119 35 109)(26 108 36 118)(27 117 37 107)(28 106 38 116)(29 115 39 105)(30 104 40 114)(41 147 51 157)(42 156 52 146)(43 145 53 155)(44 154 54 144)(45 143 55 153)(46 152 56 142)(47 141 57 151)(48 150 58 160)(49 159 59 149)(50 148 60 158)(61 72 71 62)(63 70 73 80)(64 79 74 69)(65 68 75 78)(66 77 76 67)(81 94 91 84)(82 83 92 93)(85 90 95 100)(86 99 96 89)(87 88 97 98)(121 132 131 122)(123 130 133 140)(124 139 134 129)(125 128 135 138)(126 137 136 127)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A···4G4H···4M5A5B10A···10F20A···20L20M···20T
order1222224···44···45510···1020···2020···20
size111120204···420···20222···24···48···8

47 irreducible representations

dim11111111122244444
type+++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2D5D10D102+ 1+42- 1+4Q8.10D10D48D10D4.10D10
kernelC42.157D10C20.6Q8C4.D20C20⋊Q8D10.13D4D10⋊Q8D102Q8C4⋊C4⋊D5C5×C42.C2C42.C2C42C4⋊C4C10C10C2C2C2
# reps111222241221212444

Matrix representation of C42.157D10 in GL8(𝔽41)

3928000000
132000000
0039280000
001320000
00009000
00000900
0000298320
0000315032
,
00100000
00010000
400000000
040000000
0000174000
000012400
00009201740
00003932124
,
31316350000
10126110000
63510100000
61131290000
00003312727
00001516142
000036351133
000037333922
,
31316350000
121030350000
35631310000
11612100000
00003312727
0000018214
0000243311
000029262239

G:=sub<GL(8,GF(41))| [39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,9,0,29,31,0,0,0,0,0,9,8,5,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17,1,9,39,0,0,0,0,40,24,20,32,0,0,0,0,0,0,17,1,0,0,0,0,0,0,40,24],[31,10,6,6,0,0,0,0,31,12,35,11,0,0,0,0,6,6,10,31,0,0,0,0,35,11,10,29,0,0,0,0,0,0,0,0,33,15,36,37,0,0,0,0,1,16,35,33,0,0,0,0,27,14,11,39,0,0,0,0,27,2,33,22],[31,12,35,11,0,0,0,0,31,10,6,6,0,0,0,0,6,30,31,12,0,0,0,0,35,35,31,10,0,0,0,0,0,0,0,0,33,0,2,29,0,0,0,0,1,18,4,26,0,0,0,0,27,2,33,22,0,0,0,0,27,14,11,39] >;

C42.157D10 in GAP, Magma, Sage, TeX

C_4^2._{157}D_{10}
% in TeX

G:=Group("C4^2.157D10");
// GroupNames label

G:=SmallGroup(320,1371);
// by ID

G=gap.SmallGroup(320,1371);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,1571,570,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations

׿
×
𝔽