Copied to
clipboard

G = (C2×C20).33D4order 320 = 26·5

7th non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).33D4, (C2×C4).22D20, (C22×D5).2Q8, C22.44(Q8×D5), C2.4(C204D4), C10.2(C41D4), (C22×C4).73D10, C22.84(C2×D20), C51(C23.4Q8), C2.9(D102Q8), C2.C4214D5, C10.28(C22⋊Q8), (C23×D5).9C22, (C22×C20).47C22, C23.365(C22×D5), C22.91(D42D5), (C22×C10).302C23, C2.9(C22.D20), C10.13(C22.D4), (C22×Dic5).24C22, (C2×C4⋊Dic5)⋊4C2, (C2×C10).98(C2×D4), (C2×C10).71(C2×Q8), (C2×D10⋊C4).19C2, (C2×C10).136(C4○D4), (C5×C2.C42)⋊12C2, SmallGroup(320,304)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).33D4
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — (C2×C20).33D4
C5C22×C10 — (C2×C20).33D4
C1C23C2.C42

Generators and relations for (C2×C20).33D4
 G = < a,b,c,d | a2=b4=c20=1, d2=a, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=ac-1 >

Subgroups: 790 in 186 conjugacy classes, 65 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×9], C22, C22 [×6], C22 [×10], C5, C2×C4 [×6], C2×C4 [×15], C23, C23 [×8], D5 [×2], C10, C10 [×6], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C24, Dic5 [×3], C20 [×6], D10 [×10], C2×C10, C2×C10 [×6], C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×Dic5 [×9], C2×C20 [×6], C2×C20 [×6], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.4Q8, C4⋊Dic5 [×6], D10⋊C4 [×6], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C5×C2.C42, C2×C4⋊Dic5 [×3], C2×D10⋊C4 [×3], (C2×C20).33D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, D20 [×6], C22×D5, C23.4Q8, C2×D20 [×3], D42D5 [×3], Q8×D5, C204D4, C22.D20 [×3], D102Q8 [×3], (C2×C20).33D4

Smallest permutation representation of (C2×C20).33D4
On 160 points
Generators in S160
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 136)(72 137)(73 138)(74 139)(75 140)(76 121)(77 122)(78 123)(79 124)(80 125)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)
(1 113 52 124)(2 82 53 80)(3 115 54 126)(4 84 55 62)(5 117 56 128)(6 86 57 64)(7 119 58 130)(8 88 59 66)(9 101 60 132)(10 90 41 68)(11 103 42 134)(12 92 43 70)(13 105 44 136)(14 94 45 72)(15 107 46 138)(16 96 47 74)(17 109 48 140)(18 98 49 76)(19 111 50 122)(20 100 51 78)(21 85 145 63)(22 118 146 129)(23 87 147 65)(24 120 148 131)(25 89 149 67)(26 102 150 133)(27 91 151 69)(28 104 152 135)(29 93 153 71)(30 106 154 137)(31 95 155 73)(32 108 156 139)(33 97 157 75)(34 110 158 121)(35 99 159 77)(36 112 160 123)(37 81 141 79)(38 114 142 125)(39 83 143 61)(40 116 144 127)
(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 36 37 20)(2 19 38 35)(3 34 39 18)(4 17 40 33)(5 32 21 16)(6 15 22 31)(7 30 23 14)(8 13 24 29)(9 28 25 12)(10 11 26 27)(41 42 150 151)(43 60 152 149)(44 148 153 59)(45 58 154 147)(46 146 155 57)(47 56 156 145)(48 144 157 55)(49 54 158 143)(50 142 159 53)(51 52 160 141)(61 98 126 110)(62 109 127 97)(63 96 128 108)(64 107 129 95)(65 94 130 106)(66 105 131 93)(67 92 132 104)(68 103 133 91)(69 90 134 102)(70 101 135 89)(71 88 136 120)(72 119 137 87)(73 86 138 118)(74 117 139 85)(75 84 140 116)(76 115 121 83)(77 82 122 114)(78 113 123 81)(79 100 124 112)(80 111 125 99)

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A···4F4G···4L5A5B10A···10N20A···20X
order12···2224···44···45510···1020···20
size11···120204···420···20222···24···4

62 irreducible representations

dim111122222244
type+++++-+++--
imageC1C2C2C2D4Q8D5C4○D4D10D20D42D5Q8×D5
kernel(C2×C20).33D4C5×C2.C42C2×C4⋊Dic5C2×D10⋊C4C2×C20C22×D5C2.C42C2×C10C22×C4C2×C4C22C22
# reps1133622662462

Matrix representation of (C2×C20).33D4 in GL6(𝔽41)

100000
010000
001000
000100
0000400
0000040
,
30130000
19110000
002900
0043900
0000940
00003932
,
1400000
8340000
0023000
00271600
000019
0000040
,
4000000
3310000
00393200
0014200
000019
00001840

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[30,19,0,0,0,0,13,11,0,0,0,0,0,0,2,4,0,0,0,0,9,39,0,0,0,0,0,0,9,39,0,0,0,0,40,32],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,2,27,0,0,0,0,30,16,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,39,14,0,0,0,0,32,2,0,0,0,0,0,0,1,18,0,0,0,0,9,40] >;

(C2×C20).33D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{33}D_4
% in TeX

G:=Group("(C2xC20).33D4");
// GroupNames label

G:=SmallGroup(320,304);
// by ID

G=gap.SmallGroup(320,304);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,226,12550]);
// Polycyclic

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

׿
×
𝔽