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G = C42.177D10order 320 = 26·5

177th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.177D10, C10.392- 1+4, C4⋊Q815D5, C4⋊C4.221D10, (Q8×Dic5)⋊24C2, (C4×D20).28C2, D102Q845C2, (C4×Dic10)⋊54C2, (C2×Q8).149D10, C20.139(C4○D4), C4.19(D42D5), (C4×C20).217C22, (C2×C20).109C23, (C2×C10).276C24, C4.41(Q82D5), C20.23D4.10C2, (C2×D20).282C22, C4⋊Dic5.386C22, (Q8×C10).143C22, C22.297(C23×D5), C58(C22.50C24), (C4×Dic5).173C22, (C2×Dic5).146C23, (C22×D5).121C23, D10⋊C4.155C22, C2.40(Q8.10D10), (C2×Dic10).312C22, C10.D4.168C22, (C5×C4⋊Q8)⋊18C2, C4⋊C47D543C2, C4⋊C4⋊D546C2, C10.123(C2×C4○D4), C2.66(C2×D42D5), C2.31(C2×Q82D5), (C2×C4×D5).158C22, (C5×C4⋊C4).219C22, (C2×C4).601(C22×D5), SmallGroup(320,1404)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.177D10
C1C5C10C2×C10C22×D5C2×C4×D5D102Q8 — C42.177D10
C5C2×C10 — C42.177D10
C1C22C4⋊Q8

Generators and relations for C42.177D10
 G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c9 >

Subgroups: 678 in 212 conjugacy classes, 99 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×4], C20 [×5], D10 [×6], C2×C10, C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5 [×2], C22.50C24, C4×Dic5 [×6], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, Q8×C10 [×2], C4×Dic10, C4×D20, C4⋊C47D5 [×2], D102Q8 [×2], C4⋊C4⋊D5 [×4], Q8×Dic5 [×2], C20.23D4 [×2], C5×C4⋊Q8, C42.177D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.50C24, D42D5 [×2], Q82D5 [×2], C23×D5, C2×D42D5, C2×Q82D5, Q8.10D10, C42.177D10

Smallest permutation representation of C42.177D10
On 160 points
Generators in S160
(1 134 89 119)(2 120 90 135)(3 136 91 101)(4 102 92 137)(5 138 93 103)(6 104 94 139)(7 140 95 105)(8 106 96 121)(9 122 97 107)(10 108 98 123)(11 124 99 109)(12 110 100 125)(13 126 81 111)(14 112 82 127)(15 128 83 113)(16 114 84 129)(17 130 85 115)(18 116 86 131)(19 132 87 117)(20 118 88 133)(21 160 62 46)(22 47 63 141)(23 142 64 48)(24 49 65 143)(25 144 66 50)(26 51 67 145)(27 146 68 52)(28 53 69 147)(29 148 70 54)(30 55 71 149)(31 150 72 56)(32 57 73 151)(33 152 74 58)(34 59 75 153)(35 154 76 60)(36 41 77 155)(37 156 78 42)(38 43 79 157)(39 158 80 44)(40 45 61 159)
(1 58 99 142)(2 143 100 59)(3 60 81 144)(4 145 82 41)(5 42 83 146)(6 147 84 43)(7 44 85 148)(8 149 86 45)(9 46 87 150)(10 151 88 47)(11 48 89 152)(12 153 90 49)(13 50 91 154)(14 155 92 51)(15 52 93 156)(16 157 94 53)(17 54 95 158)(18 159 96 55)(19 56 97 160)(20 141 98 57)(21 117 72 122)(22 123 73 118)(23 119 74 124)(24 125 75 120)(25 101 76 126)(26 127 77 102)(27 103 78 128)(28 129 79 104)(29 105 80 130)(30 131 61 106)(31 107 62 132)(32 133 63 108)(33 109 64 134)(34 135 65 110)(35 111 66 136)(36 137 67 112)(37 113 68 138)(38 139 69 114)(39 115 70 140)(40 121 71 116)
(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 89 98)(2 97 90 9)(3 8 91 96)(4 95 92 7)(5 6 93 94)(11 20 99 88)(12 87 100 19)(13 18 81 86)(14 85 82 17)(15 16 83 84)(21 24 62 65)(22 64 63 23)(25 40 66 61)(26 80 67 39)(27 38 68 79)(28 78 69 37)(29 36 70 77)(30 76 71 35)(31 34 72 75)(32 74 73 33)(41 148 155 54)(42 53 156 147)(43 146 157 52)(44 51 158 145)(45 144 159 50)(46 49 160 143)(47 142 141 48)(55 154 149 60)(56 59 150 153)(57 152 151 58)(101 106 136 121)(102 140 137 105)(103 104 138 139)(107 120 122 135)(108 134 123 119)(109 118 124 133)(110 132 125 117)(111 116 126 131)(112 130 127 115)(113 114 128 129)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4I4J···4Q4R4S5A5B10A···10F20A···20L20M···20T
order12222244444···44···4445510···1020···2020···20
size1111202022224···410···102020222···24···48···8

53 irreducible representations

dim111111111222224444
type+++++++++++++--+
imageC1C2C2C2C2C2C2C2C2D5C4○D4D10D10D102- 1+4D42D5Q82D5Q8.10D10
kernelC42.177D10C4×Dic10C4×D20C4⋊C47D5D102Q8C4⋊C4⋊D5Q8×Dic5C20.23D4C5×C4⋊Q8C4⋊Q8C20C42C4⋊C4C2×Q8C10C4C4C2
# reps111224221282841444

Matrix representation of C42.177D10 in GL6(𝔽41)

3200000
690000
0040000
0004000
0000400
0000040
,
900000
35320000
0021900
0012000
0000400
0000040
,
26370000
15150000
00254000
00111600
0000407
0000347
,
1540000
5260000
0016100
00322500
0000400
0000341

G:=sub<GL(6,GF(41))| [32,6,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,35,0,0,0,0,0,32,0,0,0,0,0,0,21,1,0,0,0,0,9,20,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[26,15,0,0,0,0,37,15,0,0,0,0,0,0,25,11,0,0,0,0,40,16,0,0,0,0,0,0,40,34,0,0,0,0,7,7],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,16,32,0,0,0,0,1,25,0,0,0,0,0,0,40,34,0,0,0,0,0,1] >;

C42.177D10 in GAP, Magma, Sage, TeX

C_4^2._{177}D_{10}
% in TeX

G:=Group("C4^2.177D10");
// GroupNames label

G:=SmallGroup(320,1404);
// by ID

G=gap.SmallGroup(320,1404);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,1571,297,136,12550]);
// Polycyclic

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

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