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G = C2×C82Q8order 128 = 27

Direct product of C2 and C82Q8

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

Aliases: C2×C82Q8, C42.363D4, C42.717C23, C85(C2×Q8), (C2×C8)⋊12Q8, C4.20(C2×D8), (C2×C4).93D8, C4.14(C4⋊Q8), C4.14(C2×Q16), (C2×C4).44Q16, C4⋊C4.98C23, C2.12(C22×D8), C22.74(C2×D8), C4.10(C22×Q8), (C2×C8).563C23, (C2×C4).357C24, (C4×C8).409C22, (C22×C4).616D4, C23.884(C2×D4), C4⋊Q8.283C22, C22.46(C4⋊Q8), C2.12(C22×Q16), C22.51(C2×Q16), C2.D8.178C22, (C22×C8).538C22, C22.617(C22×D4), (C22×C4).1566C23, (C2×C42).1132C22, (C2×C4×C8).39C2, C2.27(C2×C4⋊Q8), (C2×C4⋊Q8).50C2, (C2×C4).861(C2×D4), (C2×C4).246(C2×Q8), (C2×C2.D8).30C2, (C2×C4⋊C4).630C22, SmallGroup(128,1891)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C82Q8
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C2×C82Q8
C1C2C2×C4 — C2×C82Q8
C1C23C2×C42 — C2×C82Q8
C1C2C2C2×C4 — C2×C82Q8

Generators and relations for C2×C82Q8
 G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 372 in 212 conjugacy classes, 132 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×16], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×16], C4×C8 [×4], C2.D8 [×16], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C2×C4×C8, C2×C2.D8 [×4], C82Q8 [×8], C2×C4⋊Q8 [×2], C2×C82Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D8 [×4], Q16 [×4], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C2×D8 [×6], C2×Q16 [×6], C22×D4, C22×Q8 [×2], C82Q8 [×4], C2×C4⋊Q8, C22×D8, C22×Q16, C2×C82Q8

Smallest permutation representation of C2×C82Q8
Regular action on 128 points
Generators in S128
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 97)(32 98)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 121)
(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 59 43 34)(2 60 44 35)(3 61 45 36)(4 62 46 37)(5 63 47 38)(6 64 48 39)(7 57 41 40)(8 58 42 33)(9 80 18 53)(10 73 19 54)(11 74 20 55)(12 75 21 56)(13 76 22 49)(14 77 23 50)(15 78 24 51)(16 79 17 52)(25 71 113 94)(26 72 114 95)(27 65 115 96)(28 66 116 89)(29 67 117 90)(30 68 118 91)(31 69 119 92)(32 70 120 93)(81 103 108 126)(82 104 109 127)(83 97 110 128)(84 98 111 121)(85 99 112 122)(86 100 105 123)(87 101 106 124)(88 102 107 125)
(1 93 43 70)(2 92 44 69)(3 91 45 68)(4 90 46 67)(5 89 47 66)(6 96 48 65)(7 95 41 72)(8 94 42 71)(9 108 18 81)(10 107 19 88)(11 106 20 87)(12 105 21 86)(13 112 22 85)(14 111 23 84)(15 110 24 83)(16 109 17 82)(25 58 113 33)(26 57 114 40)(27 64 115 39)(28 63 116 38)(29 62 117 37)(30 61 118 36)(31 60 119 35)(32 59 120 34)(49 122 76 99)(50 121 77 98)(51 128 78 97)(52 127 79 104)(53 126 80 103)(54 125 73 102)(55 124 74 101)(56 123 75 100)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4T8A···8P
order12···24···44···48···8
size11···12···28···82···2

44 irreducible representations

dim1111122222
type++++++-++-
imageC1C2C2C2C2D4Q8D4D8Q16
kernelC2×C82Q8C2×C4×C8C2×C2.D8C82Q8C2×C4⋊Q8C42C2×C8C22×C4C2×C4C2×C4
# reps1148228288

Matrix representation of C2×C82Q8 in GL5(𝔽17)

160000
016000
001600
00010
00001
,
10000
0161500
01100
000314
00033
,
10000
0161500
01100
00001
000160
,
10000
09300
01800
00055
000512

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,9,1,0,0,0,3,8,0,0,0,0,0,5,5,0,0,0,5,12] >;

C2×C82Q8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_2Q_8
% in TeX

G:=Group("C2xC8:2Q8");
// GroupNames label

G:=SmallGroup(128,1891);
// by ID

G=gap.SmallGroup(128,1891);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,520,4037,124]);
// Polycyclic

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

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