direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8⋊C8, C23.40C42, C42.740C23, C8⋊9(C2×C8), (C2×C8)⋊8C8, C4○(C8⋊C8), (C4×C8).15C4, C4.13(C4×C8), C42○(C8⋊C8), (C2×C4).85C42, C4.31(C22×C8), (C22×C8).42C4, C22.15(C4×C8), C4.16(C8⋊C4), (C4×C8).358C22, C42.337(C2×C4), C4.55(C2×M4(2)), (C2×C4).87M4(2), C22.20(C2×C42), C22.13(C8⋊C4), (C2×C42).1142C22, C2.1(C2×C4×C8), (C2×C4×C8).6C2, (C2×C4)○(C8⋊C8), C2.1(C2×C8⋊C4), (C2×C4).98(C2×C8), (C2×C8).199(C2×C4), (C2×C4).577(C22×C4), (C22×C4).502(C2×C4), SmallGroup(128,180)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
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 [×6], C4 [×12], C22, C22 [×6], C8 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×16], C23, C42 [×2], C42 [×2], C2×C8 [×20], C2×C8 [×8], C22×C4, C22×C4 [×2], C4×C8 [×12], C2×C42, C22×C8 [×6], C8⋊C8 [×4], C2×C4×C8, C2×C4×C8 [×2], C2×C8⋊C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], M4(2) [×8], C22×C4 [×3], C4×C8 [×4], C8⋊C4 [×8], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C8⋊C8 [×4], C2×C4×C8, C2×C8⋊C4 [×2], C2×C8⋊C8
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 81)(16 82)(17 70)(18 71)(19 72)(20 65)(21 66)(22 67)(23 68)(24 69)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(41 112)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 119)(50 120)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 97)(89 124)(90 125)(91 126)(92 127)(93 128)(94 121)(95 122)(96 123)
(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)
(1 116 79 87 42 57 18 125)(2 113 80 84 43 62 19 122)(3 118 73 81 44 59 20 127)(4 115 74 86 45 64 21 124)(5 120 75 83 46 61 22 121)(6 117 76 88 47 58 23 126)(7 114 77 85 48 63 24 123)(8 119 78 82 41 60 17 128)(9 109 102 67 94 35 50 25)(10 106 103 72 95 40 51 30)(11 111 104 69 96 37 52 27)(12 108 97 66 89 34 53 32)(13 105 98 71 90 39 54 29)(14 110 99 68 91 36 55 26)(15 107 100 65 92 33 56 31)(16 112 101 70 93 38 49 28)
G:=sub<Sym(128)| (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,112)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,119)(50,120)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,97)(89,124)(90,125)(91,126)(92,127)(93,128)(94,121)(95,122)(96,123), (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), (1,116,79,87,42,57,18,125)(2,113,80,84,43,62,19,122)(3,118,73,81,44,59,20,127)(4,115,74,86,45,64,21,124)(5,120,75,83,46,61,22,121)(6,117,76,88,47,58,23,126)(7,114,77,85,48,63,24,123)(8,119,78,82,41,60,17,128)(9,109,102,67,94,35,50,25)(10,106,103,72,95,40,51,30)(11,111,104,69,96,37,52,27)(12,108,97,66,89,34,53,32)(13,105,98,71,90,39,54,29)(14,110,99,68,91,36,55,26)(15,107,100,65,92,33,56,31)(16,112,101,70,93,38,49,28)>;
G:=Group( (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,112)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,119)(50,120)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,97)(89,124)(90,125)(91,126)(92,127)(93,128)(94,121)(95,122)(96,123), (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), (1,116,79,87,42,57,18,125)(2,113,80,84,43,62,19,122)(3,118,73,81,44,59,20,127)(4,115,74,86,45,64,21,124)(5,120,75,83,46,61,22,121)(6,117,76,88,47,58,23,126)(7,114,77,85,48,63,24,123)(8,119,78,82,41,60,17,128)(9,109,102,67,94,35,50,25)(10,106,103,72,95,40,51,30)(11,111,104,69,96,37,52,27)(12,108,97,66,89,34,53,32)(13,105,98,71,90,39,54,29)(14,110,99,68,91,36,55,26)(15,107,100,65,92,33,56,31)(16,112,101,70,93,38,49,28) );
G=PermutationGroup([(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,81),(16,82),(17,70),(18,71),(19,72),(20,65),(21,66),(22,67),(23,68),(24,69),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(41,112),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,119),(50,120),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,97),(89,124),(90,125),(91,126),(92,127),(93,128),(94,121),(95,122),(96,123)], [(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)], [(1,116,79,87,42,57,18,125),(2,113,80,84,43,62,19,122),(3,118,73,81,44,59,20,127),(4,115,74,86,45,64,21,124),(5,120,75,83,46,61,22,121),(6,117,76,88,47,58,23,126),(7,114,77,85,48,63,24,123),(8,119,78,82,41,60,17,128),(9,109,102,67,94,35,50,25),(10,106,103,72,95,40,51,30),(11,111,104,69,96,37,52,27),(12,108,97,66,89,34,53,32),(13,105,98,71,90,39,54,29),(14,110,99,68,91,36,55,26),(15,107,100,65,92,33,56,31),(16,112,101,70,93,38,49,28)])
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