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G = C42.30D10order 320 = 26·5

30th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.30D10, C4⋊C815D5, C4.56(Q8×D5), C203C815C2, C4.209(D4×D5), (C4×D5).19Q8, (C4×D5).105D4, C20.368(C2×D4), (C2×C8).184D10, C20.114(C2×Q8), D10.13(C4⋊C4), C10.37(C8○D4), (C4×C20).65C22, C20.8Q824C2, C42⋊D5.1C2, D10⋊C4.24C4, Dic5.14(C4⋊C4), (C2×C20).836C23, (C2×C40).215C22, C10.D4.24C4, C53(C42.6C22), C2.14(D20.2C4), C2.15(D20.3C4), (C4×Dic5).205C22, (C5×C4⋊C8)⋊20C2, C2.10(D5×C4⋊C4), (D5×C2×C8).18C2, C10.32(C2×C4⋊C4), (C2×C4).36(C4×D5), C22.114(C2×C4×D5), (C2×C20).217(C2×C4), (C2×C8⋊D5).13C2, (C2×C4×D5).351C22, (C2×Dic5).99(C2×C4), (C22×D5).76(C2×C4), (C2×C4).778(C22×D5), (C2×C10).192(C22×C4), (C2×C52C8).316C22, SmallGroup(320,466)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.30D10
C1C5C10C20C2×C20C2×C4×D5C42⋊D5 — C42.30D10
C5C2×C10 — C42.30D10
C1C2×C4C4⋊C8

Generators and relations for C42.30D10
 G = < a,b,c,d | a4=b4=1, c10=d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 350 in 114 conjugacy classes, 55 normal (47 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×7], C23, D5 [×2], C10 [×3], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4⋊C8, C4⋊C8 [×3], C42⋊C2, C22×C8, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.6C22, C8×D5 [×2], C8⋊D5 [×2], C2×C52C8 [×2], C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C203C8, C20.8Q8 [×2], C5×C4⋊C8, C42⋊D5, D5×C2×C8, C2×C8⋊D5, C42.30D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8○D4 [×2], C4×D5 [×2], C22×D5, C42.6C22, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D20.3C4, D20.2C4, C42.30D10

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

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F20A···20H20I···20P40A···40P
order12222244444444445588888888888810···1020···2020···2040···40
size1111101011114410102020222222441010101020202···22···24···44···4

68 irreducible representations

dim11111111122222222444
type++++++++-++++-
imageC1C2C2C2C2C2C2C4C4D4Q8D5D10D10C8○D4C4×D5D20.3C4D4×D5Q8×D5D20.2C4
kernelC42.30D10C203C8C20.8Q8C5×C4⋊C8C42⋊D5D5×C2×C8C2×C8⋊D5C10.D4D10⋊C4C4×D5C4×D5C4⋊C8C42C2×C8C10C2×C4C2C4C4C2
# reps112111144222248816224

Matrix representation of C42.30D10 in GL6(𝔽41)

4000000
0400000
003800
00403800
00003823
0000373
,
100000
010000
0032000
0003200
000090
000009
,
34350000
700000
0014000
0001400
0000380
000013
,
710000
34340000
0014000
00102700
0000380
0000038

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,40,0,0,0,0,8,38,0,0,0,0,0,0,38,37,0,0,0,0,23,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,38,1,0,0,0,0,0,3],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,14,10,0,0,0,0,0,27,0,0,0,0,0,0,38,0,0,0,0,0,0,38] >;

C42.30D10 in GAP, Magma, Sage, TeX

C_4^2._{30}D_{10}
% in TeX

G:=Group("C4^2.30D10");
// GroupNames label

G:=SmallGroup(320,466);
// by ID

G=gap.SmallGroup(320,466);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,219,58,136,12550]);
// Polycyclic

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

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