Copied to
clipboard

G = C42.148D10order 320 = 26·5

148th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.148D10, C10.952- 1+4, C10.1302+ 1+4, (C4×D5)⋊1Q8, C20⋊Q835C2, C4.39(Q8×D5), D10.4(C2×Q8), C42.C24D5, C20.50(C2×Q8), C4⋊C4.111D10, C202Q832C2, Dic5.6(C2×Q8), (C2×C20).87C23, D10⋊Q8.2C2, C4.Dic1033C2, C42⋊D5.6C2, C10.42(C22×Q8), (C2×C10).233C24, (C4×C20).193C22, D102Q8.12C2, C2.55(D48D10), Dic5.Q832C2, C4⋊Dic5.240C22, C22.254(C23×D5), D10⋊C4.39C22, C54(C23.41C23), (C4×Dic5).148C22, (C2×Dic5).121C23, (C2×Dic10).42C22, (C22×D5).230C23, C2.57(D4.10D10), C10.D4.122C22, C2.25(C2×Q8×D5), (D5×C4⋊C4).11C2, (C5×C42.C2)⋊6C2, C4⋊C47D5.12C2, (C2×C4×D5).133C22, (C5×C4⋊C4).188C22, (C2×C4).203(C22×D5), SmallGroup(320,1361)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.148D10
C1C5C10C2×C10C22×D5C2×C4×D5D5×C4⋊C4 — C42.148D10
C5C2×C10 — C42.148D10
C1C22C42.C2

Generators and relations for C42.148D10
 G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >

Subgroups: 686 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C22×C4 [×3], C2×Q8 [×4], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2, C42.C2 [×3], C4⋊Q8 [×4], Dic10 [×4], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C22×D5, C23.41C23, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10 [×4], C2×C4×D5, C2×C4×D5 [×2], C202Q8, C42⋊D5, C20⋊Q8, C20⋊Q8 [×2], Dic5.Q8 [×2], C4.Dic10, D5×C4⋊C4, C4⋊C47D5, D10⋊Q8 [×2], D102Q8 [×2], C5×C42.C2, C42.148D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4, 2- 1+4, C22×D5 [×7], C23.41C23, Q8×D5 [×2], C23×D5, C2×Q8×D5, D48D10, D4.10D10, C42.148D10

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4H4I4J4K···4P5A5B10A···10F20A···20L20M···20T
order122222444···4444···45510···1020···2020···20
size11111010224···4101020···20222···24···48···8

50 irreducible representations

dim11111111111222244444
type+++++++++++-++++--+-
imageC1C2C2C2C2C2C2C2C2C2C2Q8D5D10D102+ 1+42- 1+4Q8×D5D48D10D4.10D10
kernelC42.148D10C202Q8C42⋊D5C20⋊Q8Dic5.Q8C4.Dic10D5×C4⋊C4C4⋊C47D5D10⋊Q8D102Q8C5×C42.C2C4×D5C42.C2C42C4⋊C4C10C10C4C2C2
# reps111321112214221211444

Matrix representation of C42.148D10 in GL6(𝔽41)

100000
010000
00197937
00034329
00267220
00266157
,
010000
4000000
0022800
00133900
000333928
003334132
,
6390000
39350000
003471713
0010817
00211011
003419136
,
3520000
260000
003411317
00134178
00211011
003439613

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,0,26,26,0,0,7,34,7,6,0,0,9,32,22,15,0,0,37,9,0,7],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,2,13,0,33,0,0,28,39,33,34,0,0,0,0,39,13,0,0,0,0,28,2],[6,39,0,0,0,0,39,35,0,0,0,0,0,0,34,1,21,34,0,0,7,0,10,19,0,0,17,8,1,13,0,0,13,17,1,6],[35,2,0,0,0,0,2,6,0,0,0,0,0,0,34,1,21,34,0,0,1,34,10,39,0,0,13,17,1,6,0,0,17,8,1,13] >;

C42.148D10 in GAP, Magma, Sage, TeX

C_4^2._{148}D_{10}
% in TeX

G:=Group("C4^2.148D10");
// GroupNames label

G:=SmallGroup(320,1361);
// by ID

G=gap.SmallGroup(320,1361);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,192,12550]);
// Polycyclic

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

׿
×
𝔽