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G = C2×C8⋊C8order 128 = 27

Direct product of C2 and C8⋊C8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C8⋊C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C2×C8⋊C8
 Lower central C1 — C2 — C2×C8⋊C8
 Upper central C1 — C2×C42 — C2×C8⋊C8
 Jennings C1 — C22 — C22 — C42 — C2×C8⋊C8

Generators and relations for C2×C8⋊C8
G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 140 in 124 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C42, C22×C8, C8⋊C8, C2×C4×C8, C2×C4×C8, C2×C8⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C42, C22×C8, C2×M4(2), C8⋊C8, C2×C4×C8, C2×C8⋊C4, C2×C8⋊C8

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

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 8A ··· 8AV order 1 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 2 type + + + image C1 C2 C2 C4 C4 C8 M4(2) kernel C2×C8⋊C8 C8⋊C8 C2×C4×C8 C4×C8 C22×C8 C2×C8 C2×C4 # reps 1 4 3 12 12 32 16

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

 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 9 0 0 0 0 8
,
 2 0 0 0 0 4 0 0 0 0 0 16 0 0 13 0
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,9,0,0,0,0,8],[2,0,0,0,0,4,0,0,0,0,0,13,0,0,16,0] >;

C2×C8⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes C_8
% in TeX

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

G:=SmallGroup(128,180);
// by ID

G=gap.SmallGroup(128,180);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,477,120,136,172]);
// 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^5>;
// generators/relations

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