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G = C42.156D10order 320 = 26·5

156th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.156D10, C10.1342+ 1+4, C204D415C2, C4⋊D2034C2, C4⋊C4.113D10, C42.C212D5, D208C438C2, C42⋊D522C2, C20.132(C4○D4), (C2×C20).189C23, (C4×C20).201C22, (C2×C10).242C24, C4.21(Q82D5), D10.13D436C2, C2.59(D48D10), (C2×D20).172C22, C22.263(C23×D5), C56(C22.34C24), (C4×Dic5).155C22, (C2×Dic5).272C23, (C22×D5).107C23, D10⋊C4.113C22, C10.D4.125C22, C10.119(C2×C4○D4), (C5×C42.C2)⋊15C2, C2.26(C2×Q82D5), (C2×C4×D5).141C22, (C5×C4⋊C4).197C22, (C2×C4).594(C22×D5), SmallGroup(320,1370)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.156D10
C1C5C10C2×C10C22×D5C2×C4×D5D10.13D4 — C42.156D10
C5C2×C10 — C42.156D10
C1C22C42.C2

Generators and relations for C42.156D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 1070 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4, C2×C4 [×6], C2×C4 [×9], D4 [×12], C23 [×5], D5 [×5], C10, C10 [×2], C42, C42, C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×5], C2×D4 [×10], Dic5 [×3], C20 [×2], C20 [×6], D10 [×15], C2×C10, C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C41D4, C4×D5 [×6], D20 [×12], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×6], C22×D5, C22×D5 [×4], C22.34C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×6], C2×C4×D5, C2×C4×D5 [×4], C2×D20 [×10], C42⋊D5, C204D4, D208C4 [×2], D10.13D4 [×4], C4⋊D20 [×6], C5×C42.C2, C42.156D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.34C24, Q82D5 [×2], C23×D5, C2×Q82D5, D48D10 [×2], C42.156D10

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D···2H4A4B4C···4H4I4J4K4L4M5A5B10A···10F20A···20L20M···20T
order12222···2444···4444445510···1020···2020···20
size111120···20224···41010101020222···24···48···8

50 irreducible representations

dim11111112222444
type+++++++++++++
imageC1C2C2C2C2C2C2D5C4○D4D10D102+ 1+4Q82D5D48D10
kernelC42.156D10C42⋊D5C204D4D208C4D10.13D4C4⋊D20C5×C42.C2C42.C2C20C42C4⋊C4C10C4C2
# reps111246124212248

Matrix representation of C42.156D10 in GL8(𝔽41)

004000000
000400000
10000000
01000000
000011132121
000019304038
000061228
000039401339
,
00100000
00010000
400000000
040000000
0000111300
0000193000
0000003913
000000282
,
401880000
403333340000
881400000
3334180000
000032100
0000374000
00001341820
000022262121
,
33334010000
78810000
14033330000
3340780000
000022916
000019194034
00001412320
00003939139

G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,19,6,39,0,0,0,0,13,30,1,40,0,0,0,0,21,40,2,13,0,0,0,0,21,38,28,39],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,13,2],[40,40,8,33,0,0,0,0,1,33,8,34,0,0,0,0,8,33,1,1,0,0,0,0,8,34,40,8,0,0,0,0,0,0,0,0,3,37,13,22,0,0,0,0,21,40,4,26,0,0,0,0,0,0,18,21,0,0,0,0,0,0,20,21],[33,7,1,33,0,0,0,0,33,8,40,40,0,0,0,0,40,8,33,7,0,0,0,0,1,1,33,8,0,0,0,0,0,0,0,0,22,19,14,39,0,0,0,0,9,19,12,39,0,0,0,0,1,40,32,13,0,0,0,0,6,34,0,9] >;

C42.156D10 in GAP, Magma, Sage, TeX

C_4^2._{156}D_{10}
% in TeX

G:=Group("C4^2.156D10");
// GroupNames label

G:=SmallGroup(320,1370);
// by ID

G=gap.SmallGroup(320,1370);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,184,675,570,80,12550]);
// 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*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations

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