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G = C2.C82order 128 = 27

1st central stem extension by C2 of C82

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2.1C82, C42.66Q8, C42.452D4, C23.39C42, (C2×C8)⋊3C8, C4.21(C4⋊C8), C2.1(C8⋊C8), C22.11(C4×C8), (C22×C8).21C4, C4.27(C22⋊C8), (C2×C4).85M4(2), C22.10(C8⋊C4), (C2×C42).1141C22, C2.1(C22.7C42), C22.14(C2.C42), (C2×C4×C8).1C2, (C2×C4).91(C2×C8), (C2×C4).154(C4⋊C4), (C22×C4).500(C2×C4), (C2×C4).367(C22⋊C4), SmallGroup(128,5)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2.C82
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C2.C82
C1C2 — C2.C82
C1C2×C42 — C2.C82
C1C22C22C2×C42 — C2.C82

Generators and relations for C2.C82
 G = < a,b,c | a2=b8=c8=1, cbc-1=ab=ba, ac=ca >

Subgroups: 136 in 106 conjugacy classes, 76 normal (7 characteristic)
C1, C2, C2 [×6], C4 [×12], C22, C22 [×6], C8 [×12], C2×C4 [×18], C23, C42, C42 [×3], C2×C8 [×12], C2×C8 [×12], C22×C4 [×3], C4×C8 [×6], C2×C42, C22×C8 [×6], C2×C4×C8 [×3], C2.C82
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×12], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×6], M4(2) [×6], C2.C42, C4×C8 [×3], C8⋊C4 [×3], C22⋊C8 [×6], C4⋊C8 [×6], C82, C8⋊C8 [×3], C22.7C42 [×3], C2.C82

Smallest permutation representation of C2.C82
Regular action on 128 points
Generators in S128
(1 95)(2 96)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 103)(18 104)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 81)(32 82)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 120)
(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 66 55 121 105 37 87 103)(2 75 56 13 106 46 88 18)(3 68 49 123 107 39 81 97)(4 77 50 15 108 48 82 20)(5 70 51 125 109 33 83 99)(6 79 52 9 110 42 84 22)(7 72 53 127 111 35 85 101)(8 73 54 11 112 44 86 24)(10 119 43 27 23 93 80 61)(12 113 45 29 17 95 74 63)(14 115 47 31 19 89 76 57)(16 117 41 25 21 91 78 59)(26 100 92 71 60 126 118 34)(28 102 94 65 62 128 120 36)(30 104 96 67 64 122 114 38)(32 98 90 69 58 124 116 40)

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

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

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

80 conjugacy classes

class 1 2A···2G4A···4X8A···8AV
order12···24···48···8
size11···11···12···2

80 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D4Q8M4(2)
kernelC2.C82C2×C4×C8C22×C8C2×C8C42C42C2×C4
# reps1312483112

Matrix representation of C2.C82 in GL4(𝔽17) generated by

1000
0100
00160
00016
,
8000
0800
00101
0017
,
8000
0100
0001
00160
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[8,0,0,0,0,8,0,0,0,0,10,1,0,0,1,7],[8,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0] >;

C2.C82 in GAP, Magma, Sage, TeX

C_2.C_8^2
% in TeX

G:=Group("C2.C8^2");
// GroupNames label

G:=SmallGroup(128,5);
// by ID

G=gap.SmallGroup(128,5);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,136,172]);
// Polycyclic

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

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