p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊C8⋊13C4, C8⋊8(C4⋊C4), (C4×C8)⋊24C4, C4⋊2(C8⋊C4), (C2×C8).58Q8, C4.45(C4×Q8), (C2×C8).287D4, C4.169(C4×D4), C2.2(C8⋊6D4), (C2×C4).72C42, C2.1(C8⋊4Q8), C22.91(C4×D4), C22.23(C4×Q8), C42.314(C2×C4), (C2×C4).45M4(2), C22.27(C8○D4), C2.C42.9C4, C22.55(C2×C42), C4.74(C42⋊C2), (C22×C8).384C22, C23.260(C22×C4), C22.42(C2×M4(2)), C2.16(C8○2M4(2)), (C22×C4).1613C23, (C2×C42).1049C22, C22.7C42.42C2, C2.9(C4×C4⋊C4), (C2×C4×C8).62C2, (C4×C4⋊C4).7C2, C4.74(C2×C4⋊C4), (C2×C4⋊C4).48C4, (C2×C4⋊C8).53C2, C2.10(C2×C8⋊C4), (C2×C8).137(C2×C4), (C2×C4).333(C2×Q8), (C2×C8⋊C4).26C2, (C2×C4).1505(C2×D4), (C2×C4).923(C4○D4), (C22×C4).110(C2×C4), (C2×C4).603(C22×C4), SmallGroup(128,502)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C8⋊13C4
G = < a,b,c | a4=b8=c4=1, bab-1=cac-1=a-1, cbc-1=b5 >
Subgroups: 188 in 136 conjugacy classes, 92 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C8⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22×C8, C22.7C42, C4×C4⋊C4, C2×C4×C8, C2×C8⋊C4, C2×C4⋊C8, C4⋊C8⋊13C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C8○D4, C4×C4⋊C4, C2×C8⋊C4, C8○2M4(2), C8⋊6D4, C8⋊4Q8, C4⋊C8⋊13C4
►Regular action on 128 points
Generators in S128
(1 116 25 72)(2 65 26 117)(3 118 27 66)(4 67 28 119)(5 120 29 68)(6 69 30 113)(7 114 31 70)(8 71 32 115)(9 24 59 40)(10 33 60 17)(11 18 61 34)(12 35 62 19)(13 20 63 36)(14 37 64 21)(15 22 57 38)(16 39 58 23)(41 75 85 108)(42 109 86 76)(43 77 87 110)(44 111 88 78)(45 79 81 112)(46 105 82 80)(47 73 83 106)(48 107 84 74)(49 104 121 93)(50 94 122 97)(51 98 123 95)(52 96 124 99)(53 100 125 89)(54 90 126 101)(55 102 127 91)(56 92 128 103)
(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 36 93 82)(2 33 94 87)(3 38 95 84)(4 35 96 81)(5 40 89 86)(6 37 90 83)(7 34 91 88)(8 39 92 85)(9 125 76 68)(10 122 77 65)(11 127 78 70)(12 124 79 67)(13 121 80 72)(14 126 73 69)(15 123 74 66)(16 128 75 71)(17 97 43 26)(18 102 44 31)(19 99 45 28)(20 104 46 25)(21 101 47 30)(22 98 48 27)(23 103 41 32)(24 100 42 29)(49 105 116 63)(50 110 117 60)(51 107 118 57)(52 112 119 62)(53 109 120 59)(54 106 113 64)(55 111 114 61)(56 108 115 58)
G:=sub<Sym(128)| (1,116,25,72)(2,65,26,117)(3,118,27,66)(4,67,28,119)(5,120,29,68)(6,69,30,113)(7,114,31,70)(8,71,32,115)(9,24,59,40)(10,33,60,17)(11,18,61,34)(12,35,62,19)(13,20,63,36)(14,37,64,21)(15,22,57,38)(16,39,58,23)(41,75,85,108)(42,109,86,76)(43,77,87,110)(44,111,88,78)(45,79,81,112)(46,105,82,80)(47,73,83,106)(48,107,84,74)(49,104,121,93)(50,94,122,97)(51,98,123,95)(52,96,124,99)(53,100,125,89)(54,90,126,101)(55,102,127,91)(56,92,128,103), (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), (1,36,93,82)(2,33,94,87)(3,38,95,84)(4,35,96,81)(5,40,89,86)(6,37,90,83)(7,34,91,88)(8,39,92,85)(9,125,76,68)(10,122,77,65)(11,127,78,70)(12,124,79,67)(13,121,80,72)(14,126,73,69)(15,123,74,66)(16,128,75,71)(17,97,43,26)(18,102,44,31)(19,99,45,28)(20,104,46,25)(21,101,47,30)(22,98,48,27)(23,103,41,32)(24,100,42,29)(49,105,116,63)(50,110,117,60)(51,107,118,57)(52,112,119,62)(53,109,120,59)(54,106,113,64)(55,111,114,61)(56,108,115,58)>;
G:=Group( (1,116,25,72)(2,65,26,117)(3,118,27,66)(4,67,28,119)(5,120,29,68)(6,69,30,113)(7,114,31,70)(8,71,32,115)(9,24,59,40)(10,33,60,17)(11,18,61,34)(12,35,62,19)(13,20,63,36)(14,37,64,21)(15,22,57,38)(16,39,58,23)(41,75,85,108)(42,109,86,76)(43,77,87,110)(44,111,88,78)(45,79,81,112)(46,105,82,80)(47,73,83,106)(48,107,84,74)(49,104,121,93)(50,94,122,97)(51,98,123,95)(52,96,124,99)(53,100,125,89)(54,90,126,101)(55,102,127,91)(56,92,128,103), (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), (1,36,93,82)(2,33,94,87)(3,38,95,84)(4,35,96,81)(5,40,89,86)(6,37,90,83)(7,34,91,88)(8,39,92,85)(9,125,76,68)(10,122,77,65)(11,127,78,70)(12,124,79,67)(13,121,80,72)(14,126,73,69)(15,123,74,66)(16,128,75,71)(17,97,43,26)(18,102,44,31)(19,99,45,28)(20,104,46,25)(21,101,47,30)(22,98,48,27)(23,103,41,32)(24,100,42,29)(49,105,116,63)(50,110,117,60)(51,107,118,57)(52,112,119,62)(53,109,120,59)(54,106,113,64)(55,111,114,61)(56,108,115,58) );
G=PermutationGroup([[(1,116,25,72),(2,65,26,117),(3,118,27,66),(4,67,28,119),(5,120,29,68),(6,69,30,113),(7,114,31,70),(8,71,32,115),(9,24,59,40),(10,33,60,17),(11,18,61,34),(12,35,62,19),(13,20,63,36),(14,37,64,21),(15,22,57,38),(16,39,58,23),(41,75,85,108),(42,109,86,76),(43,77,87,110),(44,111,88,78),(45,79,81,112),(46,105,82,80),(47,73,83,106),(48,107,84,74),(49,104,121,93),(50,94,122,97),(51,98,123,95),(52,96,124,99),(53,100,125,89),(54,90,126,101),(55,102,127,91),(56,92,128,103)], [(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)], [(1,36,93,82),(2,33,94,87),(3,38,95,84),(4,35,96,81),(5,40,89,86),(6,37,90,83),(7,34,91,88),(8,39,92,85),(9,125,76,68),(10,122,77,65),(11,127,78,70),(12,124,79,67),(13,121,80,72),(14,126,73,69),(15,123,74,66),(16,128,75,71),(17,97,43,26),(18,102,44,31),(19,99,45,28),(20,104,46,25),(21,101,47,30),(22,98,48,27),(23,103,41,32),(24,100,42,29),(49,105,116,63),(50,110,117,60),(51,107,118,57),(52,112,119,62),(53,109,120,59),(54,106,113,64),(55,111,114,61),(56,108,115,58)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | C4⋊C8⋊13C4 | C22.7C42 | C4×C4⋊C4 | C2×C4×C8 | C2×C8⋊C4 | C2×C4⋊C8 | C2.C42 | C4×C8 | C4⋊C8 | C2×C4⋊C4 | C2×C8 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 8 | 8 | 4 | 2 | 2 | 8 | 4 | 8 |
Matrix representation of C4⋊C8⋊13C4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 15 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,0,0,15,0,0,0,15,0],[13,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0] >;
C4⋊C8⋊13C4 in GAP, Magma, Sage, TeX
C_4\rtimes C_8\rtimes_{13}C_4 % in TeX
G:=Group("C4:C8:13C4"); // GroupNames label
G:=SmallGroup(128,502);
// by ID
G=gap.SmallGroup(128,502);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,142,172]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=c^4=1,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations