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G = C4⋊C4.7F5order 320 = 26·5

5th non-split extension by C4⋊C4 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.7F5, C20⋊C813C2, C4⋊Dic5.6C4, C2.6(Q8.F5), D10⋊C8.3C2, C10.10(C8○D4), C2.11(D4.F5), Dic5⋊C810C2, D10⋊C4.10C4, C10.C4214C2, Dic5.28(C4○D4), C22.78(C22×F5), C52(C42.7C22), C10.11(C42⋊C2), (C4×Dic5).244C22, (C2×Dic5).333C23, C2.14(D10.C23), (C4×C5⋊C8)⋊15C2, (C5×C4⋊C4).10C4, (C2×C4).62(C2×F5), (C2×C20).84(C2×C4), (C2×C5⋊C8).30C22, (C2×C4×D5).59C22, C4⋊C47D5.18C2, (C2×C10).44(C22×C4), (C2×Dic5).59(C2×C4), (C22×D5).50(C2×C4), SmallGroup(320,1044)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4⋊C4.7F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C4×C5⋊C8 — C4⋊C4.7F5
C5C2×C10 — C4⋊C4.7F5
C1C22C4⋊C4

Generators and relations for C4⋊C4.7F5
 G = < a,b,c,d | a4=b4=c5=1, d4=a2, bab-1=a-1, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 330 in 96 conjugacy classes, 42 normal (all characteristic)
C1, C2 [×3], C2, C4 [×7], C22, C22 [×3], C5, C8 [×4], C2×C4 [×3], C2×C4 [×5], C23, D5, C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×4], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×3], D10 [×3], C2×C10, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C42⋊C2, C5⋊C8 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.7C22, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C5⋊C8 [×4], C2×C4×D5, C4×C5⋊C8, C20⋊C8, C10.C42, D10⋊C8 [×2], Dic5⋊C8, C4⋊C47D5, C4⋊C4.7F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], F5, C42⋊C2, C8○D4 [×2], C2×F5 [×3], C42.7C22, C22×F5, D10.C23, D4.F5, Q8.F5, C4⋊C4.7F5

Smallest permutation representation of C4⋊C4.7F5
On 160 points
Generators in S160
(1 130 5 134)(2 8 6 4)(3 132 7 136)(9 116 13 120)(10 155 14 159)(11 118 15 114)(12 157 16 153)(17 19 21 23)(18 89 22 93)(20 91 24 95)(25 149 29 145)(26 82 30 86)(27 151 31 147)(28 84 32 88)(33 81 37 85)(34 150 38 146)(35 83 39 87)(36 152 40 148)(41 61 45 57)(42 70 46 66)(43 63 47 59)(44 72 48 68)(49 124 53 128)(50 100 54 104)(51 126 55 122)(52 102 56 98)(58 142 62 138)(60 144 64 140)(65 141 69 137)(67 143 71 139)(73 117 77 113)(74 156 78 160)(75 119 79 115)(76 158 80 154)(90 92 94 96)(97 108 101 112)(99 110 103 106)(105 123 109 127)(107 125 111 121)(129 135 133 131)
(1 18 132 95)(2 19 133 96)(3 20 134 89)(4 21 135 90)(5 22 136 91)(6 23 129 92)(7 24 130 93)(8 17 131 94)(9 101 80 126)(10 102 73 127)(11 103 74 128)(12 104 75 121)(13 97 76 122)(14 98 77 123)(15 99 78 124)(16 100 79 125)(25 139 33 47)(26 140 34 48)(27 141 35 41)(28 142 36 42)(29 143 37 43)(30 144 38 44)(31 137 39 45)(32 138 40 46)(49 114 106 160)(50 115 107 153)(51 116 108 154)(52 117 109 155)(53 118 110 156)(54 119 111 157)(55 120 112 158)(56 113 105 159)(57 151 65 83)(58 152 66 84)(59 145 67 85)(60 146 68 86)(61 147 69 87)(62 148 70 88)(63 149 71 81)(64 150 72 82)
(1 12 57 143 113)(2 144 13 114 58)(3 115 137 59 14)(4 60 116 15 138)(5 16 61 139 117)(6 140 9 118 62)(7 119 141 63 10)(8 64 120 11 142)(17 150 112 103 36)(18 104 151 37 105)(19 38 97 106 152)(20 107 39 145 98)(21 146 108 99 40)(22 100 147 33 109)(23 34 101 110 148)(24 111 35 149 102)(25 52 91 125 87)(26 126 53 88 92)(27 81 127 93 54)(28 94 82 55 128)(29 56 95 121 83)(30 122 49 84 96)(31 85 123 89 50)(32 90 86 51 124)(41 71 73 130 157)(42 131 72 158 74)(43 159 132 75 65)(44 76 160 66 133)(45 67 77 134 153)(46 135 68 154 78)(47 155 136 79 69)(48 80 156 70 129)
(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)| (1,130,5,134)(2,8,6,4)(3,132,7,136)(9,116,13,120)(10,155,14,159)(11,118,15,114)(12,157,16,153)(17,19,21,23)(18,89,22,93)(20,91,24,95)(25,149,29,145)(26,82,30,86)(27,151,31,147)(28,84,32,88)(33,81,37,85)(34,150,38,146)(35,83,39,87)(36,152,40,148)(41,61,45,57)(42,70,46,66)(43,63,47,59)(44,72,48,68)(49,124,53,128)(50,100,54,104)(51,126,55,122)(52,102,56,98)(58,142,62,138)(60,144,64,140)(65,141,69,137)(67,143,71,139)(73,117,77,113)(74,156,78,160)(75,119,79,115)(76,158,80,154)(90,92,94,96)(97,108,101,112)(99,110,103,106)(105,123,109,127)(107,125,111,121)(129,135,133,131), (1,18,132,95)(2,19,133,96)(3,20,134,89)(4,21,135,90)(5,22,136,91)(6,23,129,92)(7,24,130,93)(8,17,131,94)(9,101,80,126)(10,102,73,127)(11,103,74,128)(12,104,75,121)(13,97,76,122)(14,98,77,123)(15,99,78,124)(16,100,79,125)(25,139,33,47)(26,140,34,48)(27,141,35,41)(28,142,36,42)(29,143,37,43)(30,144,38,44)(31,137,39,45)(32,138,40,46)(49,114,106,160)(50,115,107,153)(51,116,108,154)(52,117,109,155)(53,118,110,156)(54,119,111,157)(55,120,112,158)(56,113,105,159)(57,151,65,83)(58,152,66,84)(59,145,67,85)(60,146,68,86)(61,147,69,87)(62,148,70,88)(63,149,71,81)(64,150,72,82), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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( (1,130,5,134)(2,8,6,4)(3,132,7,136)(9,116,13,120)(10,155,14,159)(11,118,15,114)(12,157,16,153)(17,19,21,23)(18,89,22,93)(20,91,24,95)(25,149,29,145)(26,82,30,86)(27,151,31,147)(28,84,32,88)(33,81,37,85)(34,150,38,146)(35,83,39,87)(36,152,40,148)(41,61,45,57)(42,70,46,66)(43,63,47,59)(44,72,48,68)(49,124,53,128)(50,100,54,104)(51,126,55,122)(52,102,56,98)(58,142,62,138)(60,144,64,140)(65,141,69,137)(67,143,71,139)(73,117,77,113)(74,156,78,160)(75,119,79,115)(76,158,80,154)(90,92,94,96)(97,108,101,112)(99,110,103,106)(105,123,109,127)(107,125,111,121)(129,135,133,131), (1,18,132,95)(2,19,133,96)(3,20,134,89)(4,21,135,90)(5,22,136,91)(6,23,129,92)(7,24,130,93)(8,17,131,94)(9,101,80,126)(10,102,73,127)(11,103,74,128)(12,104,75,121)(13,97,76,122)(14,98,77,123)(15,99,78,124)(16,100,79,125)(25,139,33,47)(26,140,34,48)(27,141,35,41)(28,142,36,42)(29,143,37,43)(30,144,38,44)(31,137,39,45)(32,138,40,46)(49,114,106,160)(50,115,107,153)(51,116,108,154)(52,117,109,155)(53,118,110,156)(54,119,111,157)(55,120,112,158)(56,113,105,159)(57,151,65,83)(58,152,66,84)(59,145,67,85)(60,146,68,86)(61,147,69,87)(62,148,70,88)(63,149,71,81)(64,150,72,82), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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([(1,130,5,134),(2,8,6,4),(3,132,7,136),(9,116,13,120),(10,155,14,159),(11,118,15,114),(12,157,16,153),(17,19,21,23),(18,89,22,93),(20,91,24,95),(25,149,29,145),(26,82,30,86),(27,151,31,147),(28,84,32,88),(33,81,37,85),(34,150,38,146),(35,83,39,87),(36,152,40,148),(41,61,45,57),(42,70,46,66),(43,63,47,59),(44,72,48,68),(49,124,53,128),(50,100,54,104),(51,126,55,122),(52,102,56,98),(58,142,62,138),(60,144,64,140),(65,141,69,137),(67,143,71,139),(73,117,77,113),(74,156,78,160),(75,119,79,115),(76,158,80,154),(90,92,94,96),(97,108,101,112),(99,110,103,106),(105,123,109,127),(107,125,111,121),(129,135,133,131)], [(1,18,132,95),(2,19,133,96),(3,20,134,89),(4,21,135,90),(5,22,136,91),(6,23,129,92),(7,24,130,93),(8,17,131,94),(9,101,80,126),(10,102,73,127),(11,103,74,128),(12,104,75,121),(13,97,76,122),(14,98,77,123),(15,99,78,124),(16,100,79,125),(25,139,33,47),(26,140,34,48),(27,141,35,41),(28,142,36,42),(29,143,37,43),(30,144,38,44),(31,137,39,45),(32,138,40,46),(49,114,106,160),(50,115,107,153),(51,116,108,154),(52,117,109,155),(53,118,110,156),(54,119,111,157),(55,120,112,158),(56,113,105,159),(57,151,65,83),(58,152,66,84),(59,145,67,85),(60,146,68,86),(61,147,69,87),(62,148,70,88),(63,149,71,81),(64,150,72,82)], [(1,12,57,143,113),(2,144,13,114,58),(3,115,137,59,14),(4,60,116,15,138),(5,16,61,139,117),(6,140,9,118,62),(7,119,141,63,10),(8,64,120,11,142),(17,150,112,103,36),(18,104,151,37,105),(19,38,97,106,152),(20,107,39,145,98),(21,146,108,99,40),(22,100,147,33,109),(23,34,101,110,148),(24,111,35,149,102),(25,52,91,125,87),(26,126,53,88,92),(27,81,127,93,54),(28,94,82,55,128),(29,56,95,121,83),(30,122,49,84,96),(31,85,123,89,50),(32,90,86,51,124),(41,71,73,130,157),(42,131,72,158,74),(43,159,132,75,65),(44,76,160,66,133),(45,67,77,134,153),(46,135,68,154,78),(47,155,136,79,69),(48,80,156,70,129)], [(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)])

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122224444444444458···8888810101020···20
size11112022445555101020410···10202020204448···8

38 irreducible representations

dim11111111112244488
type+++++++++-+
imageC1C2C2C2C2C2C2C4C4C4C4○D4C8○D4F5C2×F5D10.C23D4.F5Q8.F5
kernelC4⋊C4.7F5C4×C5⋊C8C20⋊C8C10.C42D10⋊C8Dic5⋊C8C4⋊C47D5C4⋊Dic5D10⋊C4C5×C4⋊C4Dic5C10C4⋊C4C2×C4C2C2C2
# reps11112112424813411

Matrix representation of C4⋊C4.7F5 in GL6(𝔽41)

900000
0320000
00193038
00022338
00383220
00380319
,
010000
100000
0032000
0003200
0000320
0000032
,
100000
010000
0000040
0010040
0001040
0000140
,
2700000
0270000
00662422
003028728
0013341311
0019173535

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,19,0,38,38,0,0,3,22,3,0,0,0,0,3,22,3,0,0,38,38,0,19],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[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],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,6,30,13,19,0,0,6,28,34,17,0,0,24,7,13,35,0,0,22,28,11,35] >;

C4⋊C4.7F5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._7F_5
% in TeX

G:=Group("C4:C4.7F5");
// GroupNames label

G:=SmallGroup(320,1044);
// by ID

G=gap.SmallGroup(320,1044);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,184,136,6278,1595]);
// Polycyclic

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

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