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G = C42.6F5order 320 = 26·5

3rd non-split extension by C42 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.6F5, Dic5.10M4(2), (C4×D5)⋊7C8, (C4×C20).18C4, C20.22(C2×C8), C4.17(D5⋊C8), D10.13(C2×C8), C10.2(C22×C8), D10⋊C8.6C2, Dic5.14(C2×C8), (C4×Dic5).25C4, (D5×C42).30C2, C10.3(C2×M4(2)), Dic5⋊C814C2, C51(C42.12C4), C2.3(D5⋊M4(2)), C10.1(C42⋊C2), Dic5.23(C4○D4), C22.27(C22×F5), (C2×Dic5).314C23, (C4×Dic5).354C22, C2.1(D10.C23), (C4×C5⋊C8)⋊8C2, C2.4(C2×D5⋊C8), (C2×C4×D5).27C4, (C2×C4).96(C2×F5), (C2×C5⋊C8).16C22, (C2×C20).167(C2×C4), (C2×C4×D5).356C22, (C2×C10).16(C22×C4), (C2×Dic5).164(C2×C4), (C22×D5).116(C2×C4), SmallGroup(320,1016)

Series: Derived Chief Lower central Upper central

C1C10 — C42.6F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C4×C5⋊C8 — C42.6F5
C5C10 — C42.6F5
C1C2×C4C42

Generators and relations for C42.6F5
 G = < a,b,c,d | a4=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 378 in 118 conjugacy classes, 56 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×4], C22×C4 [×3], Dic5 [×6], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.12C4, C4×Dic5 [×3], C4×C20, C2×C5⋊C8 [×4], C2×C4×D5 [×3], C4×C5⋊C8 [×2], D10⋊C8 [×2], Dic5⋊C8 [×2], D5×C42, C42.6F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], F5, C42⋊C2, C22×C8, C2×M4(2), C2×F5 [×3], C42.12C4, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D5⋊M4(2), D10.C23, C42.6F5

Smallest permutation representation of C42.6F5
On 160 points
Generators in S160
(1 18 137 12)(2 19 138 13)(3 20 139 14)(4 21 140 15)(5 22 141 16)(6 23 142 9)(7 24 143 10)(8 17 144 11)(25 54 151 47)(26 55 152 48)(27 56 145 41)(28 49 146 42)(29 50 147 43)(30 51 148 44)(31 52 149 45)(32 53 150 46)(33 69 102 136)(34 70 103 129)(35 71 104 130)(36 72 97 131)(37 65 98 132)(38 66 99 133)(39 67 100 134)(40 68 101 135)(57 117 109 155)(58 118 110 156)(59 119 111 157)(60 120 112 158)(61 113 105 159)(62 114 106 160)(63 115 107 153)(64 116 108 154)(73 127 90 82)(74 128 91 83)(75 121 92 84)(76 122 93 85)(77 123 94 86)(78 124 95 87)(79 125 96 88)(80 126 89 81)
(1 7 5 3)(2 144 6 140)(4 138 8 142)(9 21 13 17)(10 16 14 12)(11 23 15 19)(18 24 22 20)(25 100 29 104)(26 40 30 36)(27 102 31 98)(28 34 32 38)(33 149 37 145)(35 151 39 147)(41 69 45 65)(42 129 46 133)(43 71 47 67)(44 131 48 135)(49 70 53 66)(50 130 54 134)(51 72 55 68)(52 132 56 136)(57 123 61 127)(58 87 62 83)(59 125 63 121)(60 81 64 85)(73 155 77 159)(74 118 78 114)(75 157 79 153)(76 120 80 116)(82 109 86 105)(84 111 88 107)(89 154 93 158)(90 117 94 113)(91 156 95 160)(92 119 96 115)(97 152 101 148)(99 146 103 150)(106 128 110 124)(108 122 112 126)(137 143 141 139)
(1 92 132 50 113)(2 51 93 114 133)(3 115 52 134 94)(4 135 116 95 53)(5 96 136 54 117)(6 55 89 118 129)(7 119 56 130 90)(8 131 120 91 49)(9 26 126 58 103)(10 59 27 104 127)(11 97 60 128 28)(12 121 98 29 61)(13 30 122 62 99)(14 63 31 100 123)(15 101 64 124 32)(16 125 102 25 57)(17 36 112 83 146)(18 84 37 147 105)(19 148 85 106 38)(20 107 149 39 86)(21 40 108 87 150)(22 88 33 151 109)(23 152 81 110 34)(24 111 145 35 82)(41 71 73 143 157)(42 144 72 158 74)(43 159 137 75 65)(44 76 160 66 138)(45 67 77 139 153)(46 140 68 154 78)(47 155 141 79 69)(48 80 156 70 142)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4P4Q4R 5 8A···8P10A10B10C20A···20L
order122222444444444···44458···810101020···20
size11111010111122225···51010410···104444···4

56 irreducible representations

dim1111111112244444
type+++++++
imageC1C2C2C2C2C4C4C4C8M4(2)C4○D4F5C2×F5D5⋊C8D5⋊M4(2)D10.C23
kernelC42.6F5C4×C5⋊C8D10⋊C8Dic5⋊C8D5×C42C4×Dic5C4×C20C2×C4×D5C4×D5Dic5Dic5C42C2×C4C4C2C2
# reps12221224164413444

Matrix representation of C42.6F5 in GL6(𝔽41)

900000
090000
0032000
0003200
0000320
0000032
,
3200000
090000
00223803
00019383
00338190
00303822
,
100000
010000
0000040
0010040
0001040
0000140
,
010000
900000
00442426
002830730
0011341113
0015173737

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,4,28,11,15,0,0,4,30,34,17,0,0,24,7,11,37,0,0,26,30,13,37] >;

C42.6F5 in GAP, Magma, Sage, TeX

C_4^2._6F_5
% in TeX

G:=Group("C4^2.6F5");
// GroupNames label

G:=SmallGroup(320,1016);
// by ID

G=gap.SmallGroup(320,1016);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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