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## G = C2×C8⋊2C8order 128 = 27

### Direct product of C2 and C8⋊2C8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C8⋊2C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C2×C8⋊2C8
 Lower central C1 — C2 — C4 — C2×C8⋊2C8
 Upper central C1 — C22×C4 — C2×C42 — C2×C8⋊2C8
 Jennings C1 — C22 — C22 — C42 — C2×C8⋊2C8

Generators and relations for C2×C82C8
G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 140 in 100 conjugacy classes, 76 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], 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], C82C8 [×4], C2×C4×C8, C2×C4⋊C8 [×2], C2×C82C8
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], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C4.Q8 [×4], C8.C4 [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C2×SD16 [×2], C82C8 [×4], C2×C4⋊C8, C2×C4.Q8, C2×C8.C4, C2×C82C8

Smallest permutation representation of C2×C82C8
Regular action on 128 points
Generators in S128
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 81)(49 77)(50 78)(51 79)(52 80)(53 73)(54 74)(55 75)(56 76)(57 69)(58 70)(59 71)(60 72)(61 65)(62 66)(63 67)(64 68)(89 98)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 97)(105 128)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 127)
(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 65 96 127 39 16 86 75)(2 68 89 122 40 11 87 78)(3 71 90 125 33 14 88 73)(4 66 91 128 34 9 81 76)(5 69 92 123 35 12 82 79)(6 72 93 126 36 15 83 74)(7 67 94 121 37 10 84 77)(8 70 95 124 38 13 85 80)(17 57 101 108 25 120 41 51)(18 60 102 111 26 115 42 54)(19 63 103 106 27 118 43 49)(20 58 104 109 28 113 44 52)(21 61 97 112 29 116 45 55)(22 64 98 107 30 119 46 50)(23 59 99 110 31 114 47 53)(24 62 100 105 32 117 48 56)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 8A ··· 8P 8Q ··· 8AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 D4 Q8 D4 Q8 M4(2) SD16 C8.C4 kernel C2×C8⋊2C8 C8⋊2C8 C2×C4×C8 C2×C4⋊C8 C4×C8 C22×C8 C2×C8 C42 C42 C22×C4 C22×C4 C2×C4 C2×C4 C22 # reps 1 4 1 2 4 4 16 1 1 1 1 4 8 8

Matrix representation of C2×C82C8 in GL4(𝔽17) generated by

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

C2×C82C8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_2C_8
% in TeX

G:=Group("C2xC8:2C8");
// GroupNames label

G:=SmallGroup(128,294);
// by ID

G=gap.SmallGroup(128,294);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1123,136,172]);
// 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^3>;
// generators/relations

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