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G = C20.38SD16order 320 = 26·5

4th non-split extension by C20 of SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D44Dic10, C20.38SD16, C42.45D10, (C5×D4)⋊4Q8, (C4×D4).3D5, (D4×C20).3C2, C54(D42Q8), C203C818C2, (C2×C20).58D4, C20.26(C2×Q8), C4⋊C4.239D10, C202Q815C2, (C2×D4).186D10, C4.60(C4○D20), C20.46(C4○D4), (C4×C20).79C22, C4.13(D4.D5), C20.Q830C2, C10.51(C2×SD16), C4.10(C2×Dic10), D4⋊Dic5.8C2, C2.7(D4⋊D10), (C2×C20).333C23, C10.62(C22⋊Q8), C10.107(C8⋊C22), (D4×C10).228C22, C4⋊Dic5.137C22, C2.13(C20.48D4), C2.6(C2×D4.D5), (C2×C10).464(C2×D4), (C2×C4).244(C5⋊D4), (C5×C4⋊C4).270C22, (C2×C52C8).90C22, (C2×C4).433(C22×D5), C22.147(C2×C5⋊D4), SmallGroup(320,635)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.38SD16
C1C5C10C20C2×C20C4⋊Dic5C202Q8 — C20.38SD16
C5C10C2×C20 — C20.38SD16
C1C22C42C4×D4

Generators and relations for C20.38SD16
 G = < a,b,c | a20=b8=c2=1, bab-1=a-1, ac=ca, cbc=a10b3 >

Subgroups: 358 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], D4⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×D4, C4⋊Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D42Q8, C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C203C8, C20.Q8 [×2], D4⋊Dic5 [×2], C202Q8, D4×C20, C20.38SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, SD16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×SD16, C8⋊C22, Dic10 [×2], C5⋊D4 [×2], C22×D5, D42Q8, D4.D5 [×2], C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, C2×D4.D5, D4⋊D10, C20.38SD16

Smallest permutation representation of C20.38SD16
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 40 115 78 147 41 132 90)(2 39 116 77 148 60 133 89)(3 38 117 76 149 59 134 88)(4 37 118 75 150 58 135 87)(5 36 119 74 151 57 136 86)(6 35 120 73 152 56 137 85)(7 34 101 72 153 55 138 84)(8 33 102 71 154 54 139 83)(9 32 103 70 155 53 140 82)(10 31 104 69 156 52 121 81)(11 30 105 68 157 51 122 100)(12 29 106 67 158 50 123 99)(13 28 107 66 159 49 124 98)(14 27 108 65 160 48 125 97)(15 26 109 64 141 47 126 96)(16 25 110 63 142 46 127 95)(17 24 111 62 143 45 128 94)(18 23 112 61 144 44 129 93)(19 22 113 80 145 43 130 92)(20 21 114 79 146 42 131 91)
(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 100)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(101 138)(102 139)(103 140)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 129)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order12222244444444455888810···1010···1020···2020···20
size1111442222444404022202020202···24···42···24···4

59 irreducible representations

dim11111122222222222444
type+++++++-++++-+-+
imageC1C2C2C2C2C2D4Q8D5SD16C4○D4D10D10D10C5⋊D4Dic10C4○D20C8⋊C22D4.D5D4⋊D10
kernelC20.38SD16C203C8C20.Q8D4⋊Dic5C202Q8D4×C20C2×C20C5×D4C4×D4C20C20C42C4⋊C4C2×D4C2×C4D4C4C10C4C2
# reps11221122242222888144

Matrix representation of C20.38SD16 in GL4(𝔽41) generated by

20000
43900
0010
0001
,
11100
33000
001932
00411
,
1000
194000
0010
001040
G:=sub<GL(4,GF(41))| [20,4,0,0,0,39,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,1,30,0,0,0,0,19,4,0,0,32,11],[1,19,0,0,0,40,0,0,0,0,1,10,0,0,0,40] >;

C20.38SD16 in GAP, Magma, Sage, TeX

C_{20}._{38}{\rm SD}_{16}
% in TeX

G:=Group("C20.38SD16");
// GroupNames label

G:=SmallGroup(320,635);
// by ID

G=gap.SmallGroup(320,635);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,336,253,120,254,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=a^10*b^3>;
// generators/relations

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