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G = C10.C42order 160 = 25·5

4th non-split extension by C10 of C42 acting via C42/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.4C42, C10.2M4(2), C5⋊C82C4, C52(C8⋊C4), (C2×C4).2F5, C2.4(C4×F5), (C2×C20).2C4, C2.2(C4.F5), (C4×Dic5).7C2, C22.10(C2×F5), Dic5.10(C2×C4), (C2×Dic5).10C4, C2.1(C22.F5), (C2×Dic5).50C22, (C2×C5⋊C8).2C2, (C2×C10).5(C2×C4), SmallGroup(160,77)

Series: Derived Chief Lower central Upper central

C1C10 — C10.C42
C1C5C10Dic5C2×Dic5C2×C5⋊C8 — C10.C42
C5C10 — C10.C42
C1C22C2×C4

Generators and relations for C10.C42
 G = < a,b,c | a10=c4=1, b4=a5, bab-1=a3, ac=ca, cbc-1=a5b >

2C4
5C4
5C4
10C4
5C8
5C8
5C8
5C8
5C2×C4
5C2×C4
2Dic5
2C20
5C2×C8
5C42
5C2×C8
5C8⋊C4

Character table of C10.C42

 class 12A2B2C4A4B4C4D4E4F4G4H58A8B8C8D8E8F8G8H10A10B10C20A20B20C20D
 size 11112255551010410101010101010104444444
ρ11111111111111111111111111111    trivial
ρ21111-1-11111-1-11-1111-1-1-11111-1-1-1-1    linear of order 2
ρ31111111111111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ41111-1-11111-1-111-1-1-1111-1111-1-1-1-1    linear of order 2
ρ511-1-1-ii1-11-1i-i1i1-11-ii-i-11-1-1i-i-ii    linear of order 4
ρ611-1-1-ii1-11-1i-i1-i-11-1i-ii11-1-1i-i-ii    linear of order 4
ρ7111111-1-1-1-1-1-11-ii-i-iii-ii1111111    linear of order 4
ρ8111111-1-1-1-1-1-11i-iii-i-ii-i1111111    linear of order 4
ρ911-1-1-ii-11-11-ii11ii-i1-1-1-i1-1-1i-i-ii    linear of order 4
ρ1011-1-1i-i-11-11i-i1-1ii-i-111-i1-1-1-iii-i    linear of order 4
ρ1111-1-1i-i1-11-1-ii1-i1-11i-ii-11-1-1-iii-i    linear of order 4
ρ1211-1-1i-i1-11-1-ii1i-11-1-ii-i11-1-1-iii-i    linear of order 4
ρ131111-1-1-1-1-1-1111-i-iiiii-i-i111-1-1-1-1    linear of order 4
ρ141111-1-1-1-1-1-1111ii-i-i-i-iii111-1-1-1-1    linear of order 4
ρ1511-1-1i-i-11-11i-i11-i-ii1-1-1i1-1-1-iii-i    linear of order 4
ρ1611-1-1-ii-11-11-ii1-1-i-ii-111i1-1-1i-i-ii    linear of order 4
ρ172-22-200-2i-2i2i2i00200000000-2-220000    complex lifted from M4(2)
ρ182-2-22002i-2i-2i2i00200000000-22-20000    complex lifted from M4(2)
ρ192-2-2200-2i2i2i-2i00200000000-22-20000    complex lifted from M4(2)
ρ202-22-2002i2i-2i-2i00200000000-2-220000    complex lifted from M4(2)
ρ214444-4-4000000-100000000-1-1-11111    orthogonal lifted from C2×F5
ρ22444444000000-100000000-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ234-44-400000000-10000000011-15-55-5    symplectic lifted from C22.F5, Schur index 2
ρ244-44-400000000-10000000011-1-55-55    symplectic lifted from C22.F5, Schur index 2
ρ2544-4-4-4i4i000000-100000000-111-iii-i    complex lifted from C4×F5
ρ2644-4-44i-4i000000-100000000-111i-i-ii    complex lifted from C4×F5
ρ274-4-4400000000-1000000001-11-5-5--5--5    complex lifted from C4.F5
ρ284-4-4400000000-1000000001-11--5--5-5-5    complex lifted from C4.F5

Smallest permutation representation of C10.C42
Regular action on 160 points
Generators in S160
(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)
(1 106 22 84 6 101 27 89)(2 103 21 87 7 108 26 82)(3 110 30 90 8 105 25 85)(4 107 29 83 9 102 24 88)(5 104 28 86 10 109 23 81)(11 112 34 98 16 117 39 93)(12 119 33 91 17 114 38 96)(13 116 32 94 18 111 37 99)(14 113 31 97 19 118 36 92)(15 120 40 100 20 115 35 95)(41 142 64 128 46 147 69 123)(42 149 63 121 47 144 68 126)(43 146 62 124 48 141 67 129)(44 143 61 127 49 148 66 122)(45 150 70 130 50 145 65 125)(51 152 74 138 56 157 79 133)(52 159 73 131 57 154 78 136)(53 156 72 134 58 151 77 139)(54 153 71 137 59 158 76 132)(55 160 80 140 60 155 75 135)
(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 136 91 126)(82 137 92 127)(83 138 93 128)(84 139 94 129)(85 140 95 130)(86 131 96 121)(87 132 97 122)(88 133 98 123)(89 134 99 124)(90 135 100 125)(101 156 111 146)(102 157 112 147)(103 158 113 148)(104 159 114 149)(105 160 115 150)(106 151 116 141)(107 152 117 142)(108 153 118 143)(109 154 119 144)(110 155 120 145)

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

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

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

C10.C42 is a maximal subgroup of
C10.C4≀C2  (C2×D4).F5  (C2×Q8).F5  C42.5F5  C4×C4.F5  C42.15F5  C42.7F5  Dic5.C42  C5⋊C8⋊D4  D10⋊M4(2)  C23.(C2×F5)  D10.C42  D102M4(2)  C4⋊C4.7F5  Dic5.M4(2)  C20.M4(2)  C4×C22.F5  Dic5.13M4(2)  C20.30M4(2)  C5⋊C87D4  C20.6M4(2)  C30.M4(2)  C30.11C42
C10.C42 is a maximal quotient of
C42.3F5  C20.31M4(2)  C20.23C42  C10.(C4⋊C8)  C30.M4(2)  C30.11C42

Matrix representation of C10.C42 in GL6(𝔽41)

4000000
0400000
0000140
000010
0040010
0004010
,
31370000
17100000
002817307
0017243035
001711624
003401324
,
3040000
31110000
00193038
00022338
00383220
00380319

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[31,17,0,0,0,0,37,10,0,0,0,0,0,0,28,17,17,34,0,0,17,24,11,0,0,0,30,30,6,13,0,0,7,35,24,24],[30,31,0,0,0,0,4,11,0,0,0,0,0,0,19,0,38,38,0,0,3,22,3,0,0,0,0,3,22,3,0,0,38,38,0,19] >;

C10.C42 in GAP, Magma, Sage, TeX

C_{10}.C_4^2
% in TeX

G:=Group("C10.C4^2");
// GroupNames label

G:=SmallGroup(160,77);
// by ID

G=gap.SmallGroup(160,77);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,86,2309,1169]);
// Polycyclic

G:=Group<a,b,c|a^10=c^4=1,b^4=a^5,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^5*b>;
// generators/relations

Export

Subgroup lattice of C10.C42 in TeX
Character table of C10.C42 in TeX

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