Copied to
clipboard

G = C40.28D4order 320 = 26·5

28th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.28D4, (C2×Q16)⋊6D5, C4.29(D4×D5), (C10×Q16)⋊6C2, (C8×Dic5)⋊7C2, C52C8.34D4, (C2×D40).11C2, (C2×C8).244D10, C20.189(C2×D4), C55(C8.12D4), C8.19(C5⋊D4), (C2×Q8).67D10, C10.82(C4○D8), C20.23D46C2, (C2×C40).96C22, C22.282(D4×D5), C2.26(C20⋊D4), C10.35(C41D4), (C2×C20).465C23, (C2×Dic5).163D4, (Q8×C10).94C22, C2.19(Q8.D10), (C2×D20).131C22, (C4×Dic5).276C22, (C2×Q8⋊D5)⋊21C2, C4.16(C2×C5⋊D4), (C2×C10).376(C2×D4), (C2×C4).553(C22×D5), (C2×C52C8).287C22, SmallGroup(320,818)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.28D4
C1C5C10C2×C10C2×C20C2×D20C2×D40 — C40.28D4
C5C10C2×C20 — C40.28D4
C1C22C2×C4C2×Q16

Generators and relations for C40.28D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a-1, cbc=a20b-1 >

Subgroups: 622 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C52C8 [×2], C40 [×2], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5 [×2], C8.12D4, D40 [×2], C2×C52C8, C4×Dic5, D10⋊C4 [×4], Q8⋊D5 [×4], C2×C40, C5×Q16 [×2], C2×D20 [×2], Q8×C10 [×2], C8×Dic5, C2×D40, C2×Q8⋊D5 [×2], C20.23D4 [×2], C10×Q16, C40.28D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C4○D8 [×2], C5⋊D4 [×2], C22×D5, C8.12D4, D4×D5 [×2], C2×C5⋊D4, Q8.D10 [×2], C20⋊D4, C40.28D4

Smallest permutation representation of C40.28D4
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)
(1 135 100 57)(2 144 101 66)(3 153 102 75)(4 122 103 44)(5 131 104 53)(6 140 105 62)(7 149 106 71)(8 158 107 80)(9 127 108 49)(10 136 109 58)(11 145 110 67)(12 154 111 76)(13 123 112 45)(14 132 113 54)(15 141 114 63)(16 150 115 72)(17 159 116 41)(18 128 117 50)(19 137 118 59)(20 146 119 68)(21 155 120 77)(22 124 81 46)(23 133 82 55)(24 142 83 64)(25 151 84 73)(26 160 85 42)(27 129 86 51)(28 138 87 60)(29 147 88 69)(30 156 89 78)(31 125 90 47)(32 134 91 56)(33 143 92 65)(34 152 93 74)(35 121 94 43)(36 130 95 52)(37 139 96 61)(38 148 97 70)(39 157 98 79)(40 126 99 48)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 123)(50 122)(51 121)(52 160)(53 159)(54 158)(55 157)(56 156)(57 155)(58 154)(59 153)(60 152)(61 151)(62 150)(63 149)(64 148)(65 147)(66 146)(67 145)(68 144)(69 143)(70 142)(71 141)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 132)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444558888888810···102020202020···2040···40
size11114040228810101010222222101010102···244448···84···4

50 irreducible representations

dim11111122222222444
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D10D10C4○D8C5⋊D4D4×D5D4×D5Q8.D10
kernelC40.28D4C8×Dic5C2×D40C2×Q8⋊D5C20.23D4C10×Q16C52C8C40C2×Dic5C2×Q16C2×C8C2×Q8C10C8C4C22C2
# reps11122122222488228

Matrix representation of C40.28D4 in GL4(𝔽41) generated by

34700
34100
00030
001517
,
3300
243800
003239
00409
,
34700
40700
00400
0091
G:=sub<GL(4,GF(41))| [34,34,0,0,7,1,0,0,0,0,0,15,0,0,30,17],[3,24,0,0,3,38,0,0,0,0,32,40,0,0,39,9],[34,40,0,0,7,7,0,0,0,0,40,9,0,0,0,1] >;

C40.28D4 in GAP, Magma, Sage, TeX

C_{40}._{28}D_4
% in TeX

G:=Group("C40.28D4");
// GroupNames label

G:=SmallGroup(320,818);
// by ID

G=gap.SmallGroup(320,818);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,555,184,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^20*b^-1>;
// generators/relations

׿
×
𝔽