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G = C42.150D10order 320 = 26·5

150th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.150D10, C10.282- 1+4, C10.1322+ 1+4, (C4×D20)⋊47C2, C42.C26D5, C4⋊C4.207D10, C422D59C2, D208C436C2, D10⋊Q835C2, D102Q837C2, C4⋊D20.13C2, D10.17(C4○D4), (C2×C20).188C23, (C2×C10).236C24, (C4×C20).196C22, D10.13D434C2, C2.57(D48D10), Dic5.Q833C2, (C2×D20).170C22, C4⋊Dic5.314C22, C22.257(C23×D5), D10⋊C4.10C22, C58(C22.33C24), (C4×Dic5).151C22, (C2×Dic5).268C23, C10.D4.52C22, (C22×D5).102C23, C2.29(Q8.10D10), (C2×Dic10).186C22, (D5×C4⋊C4)⋊36C2, C2.87(D5×C4○D4), C4⋊C4⋊D534C2, (C5×C42.C2)⋊9C2, C10.198(C2×C4○D4), (C2×C4×D5).135C22, (C2×C4).80(C22×D5), (C5×C4⋊C4).191C22, SmallGroup(320,1364)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.150D10
C1C5C10C2×C10C22×D5C2×C4×D5D5×C4⋊C4 — C42.150D10
C5C2×C10 — C42.150D10
C1C22C42.C2

Generators and relations for C42.150D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 830 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×10], C5, C2×C4 [×7], C2×C4 [×11], D4 [×5], Q8, C23 [×3], D5 [×4], C10 [×3], C42, C42, C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×8], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×5], C20 [×7], D10 [×2], D10 [×8], C2×C10, C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2, C42.C2, C422C2 [×2], Dic10, C4×D5 [×6], D20 [×5], C2×Dic5 [×5], C2×C20 [×7], C22×D5 [×3], C22.33C24, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×6], C2×Dic10, C2×C4×D5 [×5], C2×D20 [×3], C4×D20, C422D5, Dic5.Q8, D5×C4⋊C4, D208C4, D10.13D4 [×4], C4⋊D20, D10⋊Q8 [×2], D102Q8, C4⋊C4⋊D5, C5×C42.C2, C42.150D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.33C24, C23×D5, Q8.10D10, D5×C4○D4, D48D10, C42.150D10

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C···4H4I4J4K4L4M4N5A5B10A···10F20A···20L20M···20T
order12222222444···44444445510···1020···2020···20
size111110102020224···4101020202020222···24···48···8

50 irreducible representations

dim111111111111222244444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2D5C4○D4D10D102+ 1+42- 1+4Q8.10D10D5×C4○D4D48D10
kernelC42.150D10C4×D20C422D5Dic5.Q8D5×C4⋊C4D208C4D10.13D4C4⋊D20D10⋊Q8D102Q8C4⋊C4⋊D5C5×C42.C2C42.C2D10C42C4⋊C4C10C10C2C2C2
# reps1111114121112421211444

Matrix representation of C42.150D10 in GL8(𝔽41)

400000000
040000000
003200000
000320000
00001000
00000100
000000400
0000018040
,
10000000
01000000
00900000
0037320000
0000321800
00000900
000002392
0000039032
,
134000000
734000000
0031370000
0035100000
00000010
00000109
000040000
0000018040
,
400000000
341000000
0031370000
0035100000
00000183239
0000093240
000092300
0000139132

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,37,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,18,9,23,39,0,0,0,0,0,0,9,0,0,0,0,0,0,0,2,32],[1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,31,35,0,0,0,0,0,0,37,10,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,18,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,40],[40,34,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,35,0,0,0,0,0,0,37,10,0,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,18,9,23,39,0,0,0,0,32,32,0,1,0,0,0,0,39,40,0,32] >;

C42.150D10 in GAP, Magma, Sage, TeX

C_4^2._{150}D_{10}
% in TeX

G:=Group("C4^2.150D10");
// GroupNames label

G:=SmallGroup(320,1364);
// by ID

G=gap.SmallGroup(320,1364);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,136,12550]);
// 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=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations

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