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
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, C4, C22, C22, C8, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C2×C4×C8, C2.C82
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C82, C8⋊C8, C22.7C42, C2.C82
►Regular action on 128 points
Generators in S128
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 97)(48 98)(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)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(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 84 31 121 105 43)(2 75 56 93 32 13 106 102)(3 68 49 86 25 123 107 45)(4 77 50 95 26 15 108 104)(5 70 51 88 27 125 109 47)(6 79 52 89 28 9 110 98)(7 72 53 82 29 127 111 41)(8 73 54 91 30 11 112 100)(10 119 99 33 80 61 90 20)(12 113 101 35 74 63 92 22)(14 115 103 37 76 57 94 24)(16 117 97 39 78 59 96 18)(17 124 116 46 38 69 58 87)(19 126 118 48 40 71 60 81)(21 128 120 42 34 65 62 83)(23 122 114 44 36 67 64 85)
G:=sub<Sym(128)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,25)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(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)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(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), (1,66,55,84,31,121,105,43)(2,75,56,93,32,13,106,102)(3,68,49,86,25,123,107,45)(4,77,50,95,26,15,108,104)(5,70,51,88,27,125,109,47)(6,79,52,89,28,9,110,98)(7,72,53,82,29,127,111,41)(8,73,54,91,30,11,112,100)(10,119,99,33,80,61,90,20)(12,113,101,35,74,63,92,22)(14,115,103,37,76,57,94,24)(16,117,97,39,78,59,96,18)(17,124,116,46,38,69,58,87)(19,126,118,48,40,71,60,81)(21,128,120,42,34,65,62,83)(23,122,114,44,36,67,64,85)>;
G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,25)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(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)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(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), (1,66,55,84,31,121,105,43)(2,75,56,93,32,13,106,102)(3,68,49,86,25,123,107,45)(4,77,50,95,26,15,108,104)(5,70,51,88,27,125,109,47)(6,79,52,89,28,9,110,98)(7,72,53,82,29,127,111,41)(8,73,54,91,30,11,112,100)(10,119,99,33,80,61,90,20)(12,113,101,35,74,63,92,22)(14,115,103,37,76,57,94,24)(16,117,97,39,78,59,96,18)(17,124,116,46,38,69,58,87)(19,126,118,48,40,71,60,81)(21,128,120,42,34,65,62,83)(23,122,114,44,36,67,64,85) );
G=PermutationGroup([[(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,126),(10,127),(11,128),(12,121),(13,122),(14,123),(15,124),(16,125),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,25),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,97),(48,98),(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),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(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)], [(1,66,55,84,31,121,105,43),(2,75,56,93,32,13,106,102),(3,68,49,86,25,123,107,45),(4,77,50,95,26,15,108,104),(5,70,51,88,27,125,109,47),(6,79,52,89,28,9,110,98),(7,72,53,82,29,127,111,41),(8,73,54,91,30,11,112,100),(10,119,99,33,80,61,90,20),(12,113,101,35,74,63,92,22),(14,115,103,37,76,57,94,24),(16,117,97,39,78,59,96,18),(17,124,116,46,38,69,58,87),(19,126,118,48,40,71,60,81),(21,128,120,42,34,65,62,83),(23,122,114,44,36,67,64,85)]])
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 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D4 | Q8 | M4(2) |
| kernel | C2.C82 | C2×C4×C8 | C22×C8 | C2×C8 | C42 | C42 | C2×C4 |
| # reps | 1 | 3 | 12 | 48 | 3 | 1 | 12 |
Matrix representation of C2.C82 ►in GL4(𝔽17) generated by
| 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 | 1 |
| 0 | 0 | 16 | 0 |
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