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G = C20.Q8order 160 = 25·5

2nd non-split extension by C20 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.2Q8, C10.4SD16, C4.2Dic10, C52C82C4, C4⋊C4.2D5, C52(C4.Q8), C4.12(C4×D5), C20.23(C2×C4), (C2×C10).29D4, (C2×C4).34D10, C2.1(Q8⋊D5), C4⋊Dic5.9C2, C10.10(C4⋊C4), C2.1(D4.D5), (C2×C20).9C22, C2.4(C10.D4), C22.13(C5⋊D4), (C5×C4⋊C4).2C2, (C2×C52C8).2C2, SmallGroup(160,15)

Series: Derived Chief Lower central Upper central

C1C20 — C20.Q8
C1C5C10C2×C10C2×C20C2×C52C8 — C20.Q8
C5C10C20 — C20.Q8
C1C22C2×C4C4⋊C4

Generators and relations for C20.Q8
 G = < a,b,c | a20=b4=1, c2=a5b2, bab-1=a11, cac-1=a9, cbc-1=a15b-1 >

4C4
20C4
2C2×C4
5C8
5C8
10C2×C4
4C20
4Dic5
5C2×C8
5C4⋊C4
2C2×Dic5
2C2×C20
5C4.Q8

Smallest permutation representation of C20.Q8
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)
(1 142 74 138)(2 153 75 129)(3 144 76 140)(4 155 77 131)(5 146 78 122)(6 157 79 133)(7 148 80 124)(8 159 61 135)(9 150 62 126)(10 141 63 137)(11 152 64 128)(12 143 65 139)(13 154 66 130)(14 145 67 121)(15 156 68 132)(16 147 69 123)(17 158 70 134)(18 149 71 125)(19 160 72 136)(20 151 73 127)(21 91 103 45)(22 82 104 56)(23 93 105 47)(24 84 106 58)(25 95 107 49)(26 86 108 60)(27 97 109 51)(28 88 110 42)(29 99 111 53)(30 90 112 44)(31 81 113 55)(32 92 114 46)(33 83 115 57)(34 94 116 48)(35 85 117 59)(36 96 118 50)(37 87 119 41)(38 98 120 52)(39 89 101 43)(40 100 102 54)
(1 50 79 81 11 60 69 91)(2 59 80 90 12 49 70 100)(3 48 61 99 13 58 71 89)(4 57 62 88 14 47 72 98)(5 46 63 97 15 56 73 87)(6 55 64 86 16 45 74 96)(7 44 65 95 17 54 75 85)(8 53 66 84 18 43 76 94)(9 42 67 93 19 52 77 83)(10 51 68 82 20 41 78 92)(21 157 108 138 31 147 118 128)(22 146 109 127 32 156 119 137)(23 155 110 136 33 145 120 126)(24 144 111 125 34 154 101 135)(25 153 112 134 35 143 102 124)(26 142 113 123 36 152 103 133)(27 151 114 132 37 141 104 122)(28 160 115 121 38 150 105 131)(29 149 116 130 39 159 106 140)(30 158 117 139 40 148 107 129)

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

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

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

C20.Q8 is a maximal subgroup of
Dic56SD16  D4⋊Dic10  D4.2Dic10  D10.16SD16  D10⋊SD16  C5⋊(C82D4)  C405C4⋊C2  D4⋊D56C4  Dic57SD16  C5⋊Q165C4  Q8⋊Dic10  Q8.Dic10  D10.11SD16  D102SD16  (C2×C8).D10  C52C8.D4  Dic10⋊Q8  C405Q8  C403Q8  D5×C4.Q8  C8⋊(C4×D5)  D10.12SD16  D10.17SD16  D20⋊Q8  C404Q8  Dic10.2Q8  C8.6Dic10  C8.27(C4×D5)  C4020(C2×C4)  C2.D8⋊D5  C2.D87D5  D20.2Q8  C20.47(C4⋊C4)  C4⋊C4.228D10  C4⋊C4.231D10  C20.64(C4⋊C4)  C4⋊C4.233D10  C20.76(C4⋊C4)  C4⋊C4.236D10  C20.38SD16  D4.3Dic10  C42.48D10  C4×D4.D5  C20.48SD16  Q8.3Dic10  C4×Q8⋊D5  C42.59D10  C4⋊D4.D5  (C2×D4).D10  C4⋊D4⋊D5  C52C823D4  C22⋊Q8.D5  C10.(C4○D8)  C52C824D4  C5⋊(C8.D4)  Dic10.4Q8  C42.215D10  C42.68D10  D20.4Q8  C20.SD16  C42.76D10  D205Q8  Dic106Q8  C60.7Q8  C30.SD16  C60.2Q8
C20.Q8 is a maximal quotient of
C20.39SD16  C8.Dic10  C40.6Q8  C20.31C42  C60.7Q8  C30.SD16  C60.2Q8

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20L
order122244444455888810···1020···20
size11112244202022101010102···24···4

34 irreducible representations

dim111112222222244
type++++-+++--+
imageC1C2C2C2C4Q8D4D5SD16D10Dic10C4×D5C5⋊D4D4.D5Q8⋊D5
kernelC20.Q8C2×C52C8C4⋊Dic5C5×C4⋊C4C52C8C20C2×C10C4⋊C4C10C2×C4C4C4C22C2C2
# reps111141124244422

Matrix representation of C20.Q8 in GL4(𝔽41) generated by

7100
40000
004020
0041
,
32000
03200
0090
00532
,
212600
22000
00028
002211
G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,40,4,0,0,20,1],[32,0,0,0,0,32,0,0,0,0,9,5,0,0,0,32],[21,2,0,0,26,20,0,0,0,0,0,22,0,0,28,11] >;

C20.Q8 in GAP, Magma, Sage, TeX

C_{20}.Q_8
% in TeX

G:=Group("C20.Q8");
// GroupNames label

G:=SmallGroup(160,15);
// by ID

G=gap.SmallGroup(160,15);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,313,31,297,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^2=a^5*b^2,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=a^15*b^-1>;
// generators/relations

Export

Subgroup lattice of C20.Q8 in TeX

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