Copied to
clipboard

G = C42.158D10order 320 = 26·5

158th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.158D10, C10.322- 1+4, C10.1362+ 1+4, C4⋊D2035C2, C4⋊C4.115D10, C42.C214D5, D10⋊Q838C2, C4.D2032C2, (C4×C20).225C22, (C2×C20).191C23, (C2×C10).244C24, D10.13D437C2, C2.61(D48D10), (C2×D20).173C22, C22.265(C23×D5), D10⋊C4.74C22, C55(C22.56C24), (C2×Dic10).44C22, (C2×Dic5).126C23, C10.D4.55C22, (C22×D5).109C23, C2.33(Q8.10D10), (C5×C42.C2)⋊17C2, (C2×C4×D5).143C22, (C5×C4⋊C4).199C22, (C2×C4).208(C22×D5), SmallGroup(320,1372)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.158D10
C1C5C10C2×C10C22×D5C2×C4×D5D10⋊Q8 — C42.158D10
C5C2×C10 — C42.158D10
C1C22C42.C2

Generators and relations for C42.158D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 950 in 220 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×11], C22, C22 [×12], C5, C2×C4, C2×C4 [×6], C2×C4 [×8], D4 [×6], Q8 [×2], C23 [×4], D5 [×4], C10, C10 [×2], C42, C22⋊C4 [×12], C4⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], Dic5 [×4], C20 [×7], D10 [×12], C2×C10, C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, Dic10 [×2], C4×D5 [×4], D20 [×6], C2×Dic5 [×4], C2×C20, C2×C20 [×6], C22×D5 [×4], C22.56C24, C10.D4 [×4], D10⋊C4 [×12], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×6], C4.D20 [×2], D10.13D4 [×4], C4⋊D20 [×4], D10⋊Q8 [×4], C5×C42.C2, C42.158D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×2], 2- 1+4, C22×D5 [×7], C22.56C24, C23×D5, Q8.10D10, D48D10 [×2], C42.158D10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4G4H4I4J4K5A5B10A···10F20A···20L20M···20T
order122222224···444445510···1020···2020···20
size1111202020204···420202020222···24···48···8

47 irreducible representations

dim1111112224444
type++++++++++-+
imageC1C2C2C2C2C2D5D10D102+ 1+42- 1+4Q8.10D10D48D10
kernelC42.158D10C4.D20D10.13D4C4⋊D20D10⋊Q8C5×C42.C2C42.C2C42C4⋊C4C10C10C2C2
# reps12444122122148

Matrix representation of C42.158D10 in GL8(𝔽41)

119000000
3230000000
001190000
0032300000
000039132026
0000282215
0000261128
0000882230
,
00100000
00010000
400000000
040000000
00004002813
0000040280
000003810
000033801
,
20214010000
203740330000
40121200000
40332140000
00003528341
00001331035
000025253813
00002791919
,
0034340000
00170000
77000000
4034000000
0000101328
000035403237
000000735
000000834

G:=sub<GL(8,GF(41))| [11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,39,28,2,8,0,0,0,0,13,2,6,8,0,0,0,0,20,21,11,22,0,0,0,0,26,5,28,30],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,3,0,0,0,0,0,40,38,38,0,0,0,0,28,28,1,0,0,0,0,0,13,0,0,1],[20,20,40,40,0,0,0,0,21,37,1,33,0,0,0,0,40,40,21,21,0,0,0,0,1,33,20,4,0,0,0,0,0,0,0,0,35,13,25,27,0,0,0,0,28,31,25,9,0,0,0,0,34,0,38,19,0,0,0,0,1,35,13,19],[0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,13,32,7,8,0,0,0,0,28,37,35,34] >;

C42.158D10 in GAP, Magma, Sage, TeX

C_4^2._{158}D_{10}
% in TeX

G:=Group("C4^2.158D10");
// GroupNames label

G:=SmallGroup(320,1372);
// by ID

G=gap.SmallGroup(320,1372);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,675,570,136,12550]);
// 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=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