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G = C2×C81C8order 128 = 27

Direct product of C2 and C81C8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C81C8, C42.49Q8, C42.318D4, C42.620C23, (C2×C8)⋊5C8, C87(C2×C8), C4(C81C8), (C4×C8).25C4, C4.82(C2×D8), C4.11(C4⋊C8), (C2×C4).165D8, (C2×C4).68Q16, C4.54(C2×Q16), (C22×C8).33C4, C4.25(C22×C8), C4.23(C2.D8), (C22×C4).73Q8, C4⋊C8.264C22, C23.80(C4⋊C4), C22.21(C4⋊C8), C42.308(C2×C4), (C4×C8).388C22, (C22×C4).810D4, C4.40(C2×M4(2)), (C2×C4).74M4(2), C22.20(C2.D8), C22.11(C8.C4), (C2×C42).1039C22, C2.5(C2×C4⋊C8), (C2×C4×C8).31C2, (C2×C4)(C81C8), (C2×C4⋊C8).17C2, C2.1(C2×C2.D8), (C2×C4).81(C2×C8), (C2×C8).218(C2×C4), C2.2(C2×C8.C4), C22.46(C2×C4⋊C4), (C2×C4).147(C2×Q8), (C2×C4).115(C4⋊C4), (C2×C4).1456(C2×D4), (C2×C4).502(C22×C4), (C22×C4).472(C2×C4), SmallGroup(128,295)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C81C8
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C2×C81C8
C1C2C4 — C2×C81C8
C1C22×C4C2×C42 — C2×C81C8
C1C22C22C42 — C2×C81C8

Generators and relations for C2×C81C8
 G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 140 in 100 conjugacy classes, 76 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C4 [×2], C22, C22 [×6], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×8], C2×C8 [×10], C22×C4 [×3], C4×C8 [×4], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C22×C8 [×2], C81C8 [×4], C2×C4×C8, C2×C4⋊C8 [×2], C2×C81C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2.D8 [×4], C8.C4 [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C2×D8, C2×Q16, C81C8 [×4], C2×C4⋊C8, C2×C2.D8, C2×C8.C4, C2×C81C8

Smallest permutation representation of C2×C81C8
Regular action on 128 points
Generators in S128
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(57 125)(58 126)(59 127)(60 128)(61 121)(62 122)(63 123)(64 124)(65 109)(66 110)(67 111)(68 112)(69 105)(70 106)(71 107)(72 108)(73 119)(74 120)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(89 100)(90 101)(91 102)(92 103)(93 104)(94 97)(95 98)(96 99)
(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)
(1 127 103 70 24 49 82 118)(2 126 104 69 17 56 83 117)(3 125 97 68 18 55 84 116)(4 124 98 67 19 54 85 115)(5 123 99 66 20 53 86 114)(6 122 100 65 21 52 87 113)(7 121 101 72 22 51 88 120)(8 128 102 71 23 50 81 119)(9 32 77 37 64 95 111 48)(10 31 78 36 57 94 112 47)(11 30 79 35 58 93 105 46)(12 29 80 34 59 92 106 45)(13 28 73 33 60 91 107 44)(14 27 74 40 61 90 108 43)(15 26 75 39 62 89 109 42)(16 25 76 38 63 96 110 41)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P8A···8P8Q···8AF
order12···24···44···48···88···8
size11···11···12···22···24···4

56 irreducible representations

dim111111122222222
type+++++-+-+-
imageC1C2C2C2C4C4C8D4Q8D4Q8M4(2)D8Q16C8.C4
kernelC2×C81C8C81C8C2×C4×C8C2×C4⋊C8C4×C8C22×C8C2×C8C42C42C22×C4C22×C4C2×C4C2×C4C2×C4C22
# reps1412441611114448

Matrix representation of C2×C81C8 in GL4(𝔽17) generated by

16000
01600
0010
0001
,
1000
01600
00011
00311
,
2000
01600
00107
0057
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[2,0,0,0,0,16,0,0,0,0,10,5,0,0,7,7] >;

C2×C81C8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_1C_8
% in TeX

G:=Group("C2xC8:1C8");
// GroupNames label

G:=SmallGroup(128,295);
// by ID

G=gap.SmallGroup(128,295);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,288,1123,136,172]);
// Polycyclic

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

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