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G = C55(C8×D4)  order 320 = 26·5

The semidirect product of C5 and C8×D4 acting via C8×D4/C22⋊C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C55(C8×D4), C5⋊D42C8, D104(C2×C8), C52C829D4, C221(C8×D5), C22⋊C816D5, Dic52(C2×C8), C10.56(C4×D4), C4.194(D4×D5), (C8×Dic5)⋊15C2, C20.353(C2×D4), (C2×C8).194D10, D101C816C2, C23.23(C4×D5), C10.48(C8○D4), C10.31(C22×C8), C20.8Q818C2, C23.D5.11C4, D10⋊C4.19C4, C20.297(C4○D4), (C2×C40).170C22, (C2×C20).820C23, C10.D4.19C4, (C22×C4).303D10, C4.123(D42D5), C2.3(Dic54D4), C2.2(D20.2C4), (C22×C20).337C22, (C4×Dic5).302C22, C2.9(D5×C2×C8), (D5×C2×C8)⋊13C2, (C2×C10)⋊6(C2×C8), (C2×C4).63(C4×D5), (C5×C22⋊C8)⋊14C2, C22.44(C2×C4×D5), (C4×C5⋊D4).13C2, (C2×C5⋊D4).12C4, (C2×C20).326(C2×C4), (C22×C52C8)⋊16C2, (C2×C4×D5).342C22, (C2×Dic5).94(C2×C4), (C22×D5).72(C2×C4), (C2×C4).762(C22×D5), (C22×C10).106(C2×C4), (C2×C10).176(C22×C4), (C2×C52C8).308C22, SmallGroup(320,352)

Series: Derived Chief Lower central Upper central

C1C10 — C55(C8×D4)
C1C5C10C20C2×C20C2×C4×D5C4×C5⋊D4 — C55(C8×D4)
C5C10 — C55(C8×D4)
C1C2×C4C22⋊C8

Generators and relations for C55(C8×D4)
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 398 in 134 conjugacy classes, 61 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×6], C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8 [×2], C52C8 [×2], C52C8, C40 [×2], C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C8×D4, C8×D5 [×2], C2×C52C8 [×2], C2×C52C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D101C8, C5×C22⋊C8, D5×C2×C8, C22×C52C8, C4×C5⋊D4, C55(C8×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C22×C8, C8○D4, C4×D5 [×2], C22×D5, C8×D4, C8×D5 [×2], C2×C4×D5, D4×D5, D42D5, Dic54D4, D5×C2×C8, D20.2C4, C55(C8×D4)

Smallest permutation representation of C55(C8×D4)
On 160 points
Generators in S160
(1 145 114 34 99)(2 100 35 115 146)(3 147 116 36 101)(4 102 37 117 148)(5 149 118 38 103)(6 104 39 119 150)(7 151 120 40 97)(8 98 33 113 152)(9 92 124 63 41)(10 42 64 125 93)(11 94 126 57 43)(12 44 58 127 95)(13 96 128 59 45)(14 46 60 121 89)(15 90 122 61 47)(16 48 62 123 91)(17 28 56 139 136)(18 129 140 49 29)(19 30 50 141 130)(20 131 142 51 31)(21 32 52 143 132)(22 133 144 53 25)(23 26 54 137 134)(24 135 138 55 27)(65 157 106 88 80)(66 73 81 107 158)(67 159 108 82 74)(68 75 83 109 160)(69 153 110 84 76)(70 77 85 111 154)(71 155 112 86 78)(72 79 87 105 156)
(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)
(1 111 48 25)(2 112 41 26)(3 105 42 27)(4 106 43 28)(5 107 44 29)(6 108 45 30)(7 109 46 31)(8 110 47 32)(9 23 100 155)(10 24 101 156)(11 17 102 157)(12 18 103 158)(13 19 104 159)(14 20 97 160)(15 21 98 153)(16 22 99 154)(33 69 90 132)(34 70 91 133)(35 71 92 134)(36 72 93 135)(37 65 94 136)(38 66 95 129)(39 67 96 130)(40 68 89 131)(49 149 81 58)(50 150 82 59)(51 151 83 60)(52 152 84 61)(53 145 85 62)(54 146 86 63)(55 147 87 64)(56 148 88 57)(73 127 140 118)(74 128 141 119)(75 121 142 120)(76 122 143 113)(77 123 144 114)(78 124 137 115)(79 125 138 116)(80 126 139 117)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 81)(17 152)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(33 139)(34 140)(35 141)(36 142)(37 143)(38 144)(39 137)(40 138)(41 108)(42 109)(43 110)(44 111)(45 112)(46 105)(47 106)(48 107)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 121)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 89)(80 90)(113 136)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L5A5B8A···8H8I···8P8Q8R8S8T10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order122222224444444···4558···88···8888810···101010101020···202020202040···40
size111122101011112210···10222···25···5101010102···244442···244444···4

80 irreducible representations

dim1111111111111222222222444
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4C8D4D5C4○D4D10D10C8○D4C4×D5C4×D5C8×D5D4×D5D42D5D20.2C4
kernelC55(C8×D4)C8×Dic5C20.8Q8D101C8C5×C22⋊C8D5×C2×C8C22×C52C8C4×C5⋊D4C10.D4D10⋊C4C23.D5C2×C5⋊D4C5⋊D4C52C8C22⋊C8C20C2×C8C22×C4C10C2×C4C23C22C4C4C2
# reps111111112222162224244416224

Matrix representation of C55(C8×D4) in GL4(𝔽41) generated by

34100
40000
0010
0001
,
382100
0300
00270
00027
,
13400
04000
0019
001840
,
40000
04000
004032
0001
G:=sub<GL(4,GF(41))| [34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[38,0,0,0,21,3,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,34,40,0,0,0,0,1,18,0,0,9,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,32,1] >;

C55(C8×D4) in GAP, Magma, Sage, TeX

C_5\rtimes_5(C_8\times D_4)
% in TeX

G:=Group("C5:5(C8xD4)");
// GroupNames label

G:=SmallGroup(320,352);
// by ID

G=gap.SmallGroup(320,352);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽