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G = C2×Q8⋊C8order 128 = 27

Direct product of C2 and Q8⋊C8

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

Aliases: C2×Q8⋊C8, C42.314D4, C42.596C23, C4(Q8⋊C8), Q83(C2×C8), (C2×Q8)⋊4C8, C4.2(C22×C8), (C4×Q8).11C4, (C2×C4).67Q16, C4.50(C2×Q16), C22.35C4≀C2, C4.90(C2×SD16), C4.2(C2×M4(2)), C4⋊C8.243C22, C4.10(C22⋊C8), C42.255(C2×C4), (C4×C8).361C22, (C2×C4).126SD16, (C22×C4).653D4, (C2×C4).43M4(2), (C22×Q8).18C4, C4.32(Q8⋊C4), (C4×Q8).247C22, C22.41(C22⋊C8), C23.218(C22⋊C4), (C2×C42).1033C22, C22.30(Q8⋊C4), (C2×C4×C8).9C2, C2.2(C2×C4≀C2), (C2×C4⋊C8).9C2, (C2×C4×Q8).3C2, (C2×C4)(Q8⋊C8), (C2×C4⋊C4).37C4, (C2×C4).51(C2×C8), C4⋊C4.174(C2×C4), C2.11(C2×C22⋊C8), C2.1(C2×Q8⋊C4), (C2×C4).1136(C2×D4), (C2×Q8).171(C2×C4), (C2×C4).301(C22×C4), (C22×C4).394(C2×C4), C22.95(C2×C22⋊C4), (C2×C4).236(C22⋊C4), SmallGroup(128,207)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×Q8⋊C8
C1C2C22C2×C4C42C2×C42C2×C4×Q8 — C2×Q8⋊C8
C1C2C4 — C2×Q8⋊C8
C1C22×C4C2×C42 — C2×Q8⋊C8
C1C22C22C42 — C2×Q8⋊C8

Generators and relations for C2×Q8⋊C8
 G = < a,b,c,d | a2=b4=d8=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 220 in 140 conjugacy classes, 76 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×12], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×10], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×3], C4×C8 [×2], C4×C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C22×C8 [×2], C22×Q8, Q8⋊C8 [×4], C2×C4×C8, C2×C4⋊C8, C2×C4×Q8, C2×Q8⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], Q8⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C22×C8, C2×M4(2), C2×SD16, C2×Q16, Q8⋊C8 [×4], C2×C22⋊C8, C2×Q8⋊C4, C2×C4≀C2, C2×Q8⋊C8

Smallest permutation representation of C2×Q8⋊C8
Regular action on 128 points
Generators in S128
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 125)(10 126)(11 127)(12 128)(13 121)(14 122)(15 123)(16 124)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 93)(34 94)(35 95)(36 96)(37 89)(38 90)(39 91)(40 92)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)(57 77)(58 78)(59 79)(60 80)(61 73)(62 74)(63 75)(64 76)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)
(1 39 111 79)(2 80 112 40)(3 33 105 73)(4 74 106 34)(5 35 107 75)(6 76 108 36)(7 37 109 77)(8 78 110 38)(9 49 113 41)(10 42 114 50)(11 51 115 43)(12 44 116 52)(13 53 117 45)(14 46 118 54)(15 55 119 47)(16 48 120 56)(17 69 84 126)(18 127 85 70)(19 71 86 128)(20 121 87 72)(21 65 88 122)(22 123 81 66)(23 67 82 124)(24 125 83 68)(25 61 101 93)(26 94 102 62)(27 63 103 95)(28 96 104 64)(29 57 97 89)(30 90 98 58)(31 59 99 91)(32 92 100 60)
(1 115 111 11)(2 44 112 52)(3 117 105 13)(4 46 106 54)(5 119 107 15)(6 48 108 56)(7 113 109 9)(8 42 110 50)(10 38 114 78)(12 40 116 80)(14 34 118 74)(16 36 120 76)(17 30 84 98)(18 91 85 59)(19 32 86 100)(20 93 87 61)(21 26 88 102)(22 95 81 63)(23 28 82 104)(24 89 83 57)(25 121 101 72)(27 123 103 66)(29 125 97 68)(31 127 99 70)(33 53 73 45)(35 55 75 47)(37 49 77 41)(39 51 79 43)(58 126 90 69)(60 128 92 71)(62 122 94 65)(64 124 96 67)
(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)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4X8A···8P8Q···8X
order12···24···44···44···48···88···8
size11···11···12···24···42···24···4

56 irreducible representations

dim111111111222222
type+++++++-
imageC1C2C2C2C2C4C4C4C8D4D4M4(2)SD16Q16C4≀C2
kernelC2×Q8⋊C8Q8⋊C8C2×C4×C8C2×C4⋊C8C2×C4×Q8C2×C4⋊C4C4×Q8C22×Q8C2×Q8C42C22×C4C2×C4C2×C4C2×C4C22
# reps1411124216224448

Matrix representation of C2×Q8⋊C8 in GL4(𝔽17) generated by

16000
01600
0010
0001
,
1000
0100
0001
00160
,
16000
0100
00125
0055
,
15000
01600
0002
0020
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,12,5,0,0,5,5],[15,0,0,0,0,16,0,0,0,0,0,2,0,0,2,0] >;

C2×Q8⋊C8 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes C_8
% in TeX

G:=Group("C2xQ8:C8");
// GroupNames label

G:=SmallGroup(128,207);
// by ID

G=gap.SmallGroup(128,207);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,570,136,172]);
// Polycyclic

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

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