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G = C42.151D10order 320 = 26·5

151st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.151D10, C10.292- (1+4), C42.C27D5, C4⋊C4.112D10, D10⋊Q836C2, C42⋊D537C2, (C2×C20).89C23, Dic53Q836C2, D10.38(C4○D4), D208C4.12C2, (C4×C20).240C22, (C2×C10).237C24, Dic5.46(C4○D4), Dic5.Q834C2, D10.13D4.2C2, (C2×D20).171C22, C4⋊Dic5.242C22, C22.258(C23×D5), C59(C22.46C24), (C2×Dic5).269C23, (C4×Dic5).235C22, C10.D4.53C22, (C22×D5).232C23, D10⋊C4.137C22, C2.30(Q8.10D10), (C2×Dic10).187C22, (D5×C4⋊C4)⋊37C2, C2.88(D5×C4○D4), C4⋊C4⋊D535C2, C4⋊C47D536C2, C10.199(C2×C4○D4), (C5×C42.C2)⋊10C2, (C2×C4×D5).136C22, (C5×C4⋊C4).192C22, (C2×C4).204(C22×D5), SmallGroup(320,1365)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.151D10
C1C5C10C2×C10C22×D5C2×C4×D5C42⋊D5 — C42.151D10
C5C2×C10 — C42.151D10

Subgroups: 710 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×7], C5, C2×C4 [×7], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×7], D10 [×2], D10 [×5], C2×C10, C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C42.C2 [×2], C422C2 [×2], Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×6], C2×C20 [×7], C22×D5 [×2], C22.46C24, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×6], C2×Dic10, C2×C4×D5 [×4], C2×D20, C42⋊D5 [×2], Dic53Q8, Dic5.Q8 [×2], D5×C4⋊C4, C4⋊C47D5, D208C4, D10.13D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], C5×C42.C2, C42.151D10

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D5 [×7], C22.46C24, C23×D5, Q8.10D10, D5×C4○D4 [×2], C42.151D10

Generators and relations
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c9 >

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

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

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

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

Matrix representation G ⊆ GL6(𝔽41)

100000
010000
009000
000900
0000032
000090
,
100000
010000
0040000
000100
000090
000009
,
1340000
7340000
0004000
001000
000001
000010
,
4000000
3410000
0032000
0003200
0000040
0000400

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,7,0,0,0,0,34,34,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

53 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4I4J···4O4P4Q4R5A5B10A···10F20A···20L20M···20T
order122222244444···44···44445510···1020···2020···20
size111110102022224···410···10202020222···24···48···8

53 irreducible representations

dim1111111111122222444
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D102- (1+4)Q8.10D10D5×C4○D4
kernelC42.151D10C42⋊D5Dic53Q8Dic5.Q8D5×C4⋊C4C4⋊C47D5D208C4D10.13D4D10⋊Q8C4⋊C4⋊D5C5×C42.C2C42.C2Dic5D10C42C4⋊C4C10C2C2
# reps12121112221244212148

In GAP, Magma, Sage, TeX

C_4^2._{151}D_{10}
% in TeX

G:=Group("C4^2.151D10");
// GroupNames label

G:=SmallGroup(320,1365);
// by ID

G=gap.SmallGroup(320,1365);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,346,297,136,12550]);
// Polycyclic

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

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