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G = C42.140D10order 320 = 26·5

140th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.140D10, C10.892- 1+4, C10.722+ 1+4, (C2×Q8).83D10, C4.4D4.9D5, (C2×D4).109D10, (C2×C20).78C23, C22⋊C4.34D10, C20.6Q828C2, Dic5⋊Q823C2, (C4×C20).221C22, (C2×C10).216C24, C4⋊Dic5.50C22, C2.74(D46D10), C23.38(C22×D5), (D4×C10).209C22, C23.D1038C2, (C22×C10).46C23, (Q8×C10).125C22, C22.237(C23×D5), Dic5.14D439C2, C23.D5.53C22, C53(C22.57C24), (C2×Dic5).111C23, (C4×Dic5).140C22, C10.D4.83C22, C23.18D10.6C2, C2.50(D4.10D10), (C2×Dic10).182C22, (C22×Dic5).141C22, (C5×C4.4D4).7C2, (C2×C4).192(C22×D5), (C5×C22⋊C4).63C22, SmallGroup(320,1344)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.140D10
C1C5C10C2×C10C2×Dic5C22×Dic5Dic5.14D4 — C42.140D10
C5C2×C10 — C42.140D10
C1C22C4.4D4

Generators and relations for C42.140D10
 G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 614 in 196 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×8], C20 [×5], C2×C10, C2×C10 [×6], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C422C2 [×4], C4⋊Q8 [×2], Dic10 [×2], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20, C2×C20 [×4], C5×D4, C5×Q8, C22×C10 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×12], C4⋊Dic5 [×4], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C22×Dic5 [×2], D4×C10, Q8×C10, C20.6Q8 [×2], Dic5.14D4 [×4], C23.D10 [×4], C23.18D10 [×2], Dic5⋊Q8 [×2], C5×C4.4D4, C42.140D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D5 [×7], C22.57C24, C23×D5, D46D10, D4.10D10 [×2], C42.140D10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A···4E4F···4M5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order1222224···44···45510···101010101020···2020202020
size1111444···420···20222···288884···48888

47 irreducible representations

dim1111111222224444
type+++++++++++++--
imageC1C2C2C2C2C2C2D5D10D10D10D102+ 1+42- 1+4D46D10D4.10D10
kernelC42.140D10C20.6Q8Dic5.14D4C23.D10C23.18D10Dic5⋊Q8C5×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8C10C10C2C2
# reps1244221228221248

Matrix representation of C42.140D10 in GL8(𝔽41)

1132000000
930000000
3202320000
28937390000
00002133335
000028393339
000036153028
000020212211
,
003410000
404039400000
2323100000
3738700000
0000103838
00000130
0000013400
00002828040
,
407000000
347000000
13635340000
168600000
0000353500
000064000
0000372876
000009340
,
2440000000
317000000
17020170000
38115210000
000032000
000028900
000036351928
00004063122

G:=sub<GL(8,GF(41))| [11,9,32,28,0,0,0,0,32,30,0,9,0,0,0,0,0,0,2,37,0,0,0,0,0,0,32,39,0,0,0,0,0,0,0,0,2,28,36,20,0,0,0,0,13,39,15,21,0,0,0,0,33,33,30,22,0,0,0,0,35,39,28,11],[0,40,23,37,0,0,0,0,0,40,23,38,0,0,0,0,34,39,1,7,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,13,28,0,0,0,0,38,3,40,0,0,0,0,0,38,0,0,40],[40,34,13,16,0,0,0,0,7,7,6,8,0,0,0,0,0,0,35,6,0,0,0,0,0,0,34,0,0,0,0,0,0,0,0,0,35,6,37,0,0,0,0,0,35,40,28,9,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0],[24,3,17,38,0,0,0,0,40,17,0,1,0,0,0,0,0,0,20,15,0,0,0,0,0,0,17,21,0,0,0,0,0,0,0,0,32,28,36,40,0,0,0,0,0,9,35,6,0,0,0,0,0,0,19,31,0,0,0,0,0,0,28,22] >;

C42.140D10 in GAP, Magma, Sage, TeX

C_4^2._{140}D_{10}
% in TeX

G:=Group("C4^2.140D10");
// GroupNames label

G:=SmallGroup(320,1344);
// by ID

G=gap.SmallGroup(320,1344);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,219,184,1571,570,297,136,12550]);
// Polycyclic

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

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