p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊5(C4⋊C4), C2.D8⋊7C4, C4.7(C4×Q8), C2.11(C4×D8), (C2×C4).58D8, (C2×C8).44Q8, (C2×C8).348D4, C2.11(C4×Q16), (C2×C4).30Q16, C2.4(C8⋊7D4), C2.2(C8⋊2Q8), C22.44(C2×D8), (C22×C4).556D4, C22.175(C4×D4), C23.789(C2×D4), C22.32(C4⋊Q8), C4.75(C22⋊Q8), C4.5(C42.C2), C2.3(C8.5Q8), C22.37(C2×Q16), C2.4(C8.18D4), C22.72(C4○D8), C22.4Q16.15C2, (C22×C8).487C22, C22.135(C4⋊D4), (C2×C42).1068C22, (C22×C4).1391C23, C23.65C23.11C2, C2.9(C23.65C23), (C2×C4×C8).35C2, C4.40(C2×C4⋊C4), C4⋊C4.88(C2×C4), (C2×C2.D8).8C2, (C2×C8).171(C2×C4), (C2×C4).203(C2×Q8), (C2×C4).1348(C2×D4), (C2×C4⋊C4).75C22, (C2×C4).586(C4○D4), (C2×C4).409(C22×C4), SmallGroup(128,674)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊5(C4⋊C4)
G = < a,b,c | a8=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 228 in 124 conjugacy classes, 68 normal (36 characteristic)
C1, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C2.D8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C23.65C23, C2×C4×C8, C2×C2.D8, C8⋊5(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×D8, C2×Q16, C4○D8, C23.65C23, C4×D8, C4×Q16, C8⋊7D4, C8.18D4, C8.5Q8, C8⋊2Q8, C8⋊5(C4⋊C4)
►Regular action on 128 points
Generators in S128
(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)
(1 104 19 118)(2 103 20 117)(3 102 21 116)(4 101 22 115)(5 100 23 114)(6 99 24 113)(7 98 17 120)(8 97 18 119)(9 27 42 124)(10 26 43 123)(11 25 44 122)(12 32 45 121)(13 31 46 128)(14 30 47 127)(15 29 48 126)(16 28 41 125)(33 85 51 71)(34 84 52 70)(35 83 53 69)(36 82 54 68)(37 81 55 67)(38 88 56 66)(39 87 49 65)(40 86 50 72)(57 105 75 95)(58 112 76 94)(59 111 77 93)(60 110 78 92)(61 109 79 91)(62 108 80 90)(63 107 73 89)(64 106 74 96)
(1 50 41 77)(2 51 42 78)(3 52 43 79)(4 53 44 80)(5 54 45 73)(6 55 46 74)(7 56 47 75)(8 49 48 76)(9 60 20 33)(10 61 21 34)(11 62 22 35)(12 63 23 36)(13 64 24 37)(14 57 17 38)(15 58 18 39)(16 59 19 40)(25 90 115 69)(26 91 116 70)(27 92 117 71)(28 93 118 72)(29 94 119 65)(30 95 120 66)(31 96 113 67)(32 89 114 68)(81 128 106 99)(82 121 107 100)(83 122 108 101)(84 123 109 102)(85 124 110 103)(86 125 111 104)(87 126 112 97)(88 127 105 98)
G:=sub<Sym(128)| (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), (1,104,19,118)(2,103,20,117)(3,102,21,116)(4,101,22,115)(5,100,23,114)(6,99,24,113)(7,98,17,120)(8,97,18,119)(9,27,42,124)(10,26,43,123)(11,25,44,122)(12,32,45,121)(13,31,46,128)(14,30,47,127)(15,29,48,126)(16,28,41,125)(33,85,51,71)(34,84,52,70)(35,83,53,69)(36,82,54,68)(37,81,55,67)(38,88,56,66)(39,87,49,65)(40,86,50,72)(57,105,75,95)(58,112,76,94)(59,111,77,93)(60,110,78,92)(61,109,79,91)(62,108,80,90)(63,107,73,89)(64,106,74,96), (1,50,41,77)(2,51,42,78)(3,52,43,79)(4,53,44,80)(5,54,45,73)(6,55,46,74)(7,56,47,75)(8,49,48,76)(9,60,20,33)(10,61,21,34)(11,62,22,35)(12,63,23,36)(13,64,24,37)(14,57,17,38)(15,58,18,39)(16,59,19,40)(25,90,115,69)(26,91,116,70)(27,92,117,71)(28,93,118,72)(29,94,119,65)(30,95,120,66)(31,96,113,67)(32,89,114,68)(81,128,106,99)(82,121,107,100)(83,122,108,101)(84,123,109,102)(85,124,110,103)(86,125,111,104)(87,126,112,97)(88,127,105,98)>;
G:=Group( (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), (1,104,19,118)(2,103,20,117)(3,102,21,116)(4,101,22,115)(5,100,23,114)(6,99,24,113)(7,98,17,120)(8,97,18,119)(9,27,42,124)(10,26,43,123)(11,25,44,122)(12,32,45,121)(13,31,46,128)(14,30,47,127)(15,29,48,126)(16,28,41,125)(33,85,51,71)(34,84,52,70)(35,83,53,69)(36,82,54,68)(37,81,55,67)(38,88,56,66)(39,87,49,65)(40,86,50,72)(57,105,75,95)(58,112,76,94)(59,111,77,93)(60,110,78,92)(61,109,79,91)(62,108,80,90)(63,107,73,89)(64,106,74,96), (1,50,41,77)(2,51,42,78)(3,52,43,79)(4,53,44,80)(5,54,45,73)(6,55,46,74)(7,56,47,75)(8,49,48,76)(9,60,20,33)(10,61,21,34)(11,62,22,35)(12,63,23,36)(13,64,24,37)(14,57,17,38)(15,58,18,39)(16,59,19,40)(25,90,115,69)(26,91,116,70)(27,92,117,71)(28,93,118,72)(29,94,119,65)(30,95,120,66)(31,96,113,67)(32,89,114,68)(81,128,106,99)(82,121,107,100)(83,122,108,101)(84,123,109,102)(85,124,110,103)(86,125,111,104)(87,126,112,97)(88,127,105,98) );
G=PermutationGroup([[(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)], [(1,104,19,118),(2,103,20,117),(3,102,21,116),(4,101,22,115),(5,100,23,114),(6,99,24,113),(7,98,17,120),(8,97,18,119),(9,27,42,124),(10,26,43,123),(11,25,44,122),(12,32,45,121),(13,31,46,128),(14,30,47,127),(15,29,48,126),(16,28,41,125),(33,85,51,71),(34,84,52,70),(35,83,53,69),(36,82,54,68),(37,81,55,67),(38,88,56,66),(39,87,49,65),(40,86,50,72),(57,105,75,95),(58,112,76,94),(59,111,77,93),(60,110,78,92),(61,109,79,91),(62,108,80,90),(63,107,73,89),(64,106,74,96)], [(1,50,41,77),(2,51,42,78),(3,52,43,79),(4,53,44,80),(5,54,45,73),(6,55,46,74),(7,56,47,75),(8,49,48,76),(9,60,20,33),(10,61,21,34),(11,62,22,35),(12,63,23,36),(13,64,24,37),(14,57,17,38),(15,58,18,39),(16,59,19,40),(25,90,115,69),(26,91,116,70),(27,92,117,71),(28,93,118,72),(29,94,119,65),(30,95,120,66),(31,96,113,67),(32,89,114,68),(81,128,106,99),(82,121,107,100),(83,122,108,101),(84,123,109,102),(85,124,110,103),(86,125,111,104),(87,126,112,97),(88,127,105,98)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D8 | Q16 | C4○D4 | C4○D8 |
| kernel | C8⋊5(C4⋊C4) | C22.4Q16 | C23.65C23 | C2×C4×C8 | C2×C2.D8 | C2.D8 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C8⋊5(C4⋊C4) ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 2 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 14 | 14 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 9 | 0 | 0 |
| 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,14,14,0,0,0,3,14],[16,0,0,0,0,0,13,0,0,0,0,8,4,0,0,0,0,0,0,16,0,0,0,16,0],[13,0,0,0,0,0,4,4,0,0,0,9,13,0,0,0,0,0,1,0,0,0,0,0,1] >;
C8⋊5(C4⋊C4) in GAP, Magma, Sage, TeX
C_8\rtimes_5(C_4\rtimes C_4)
% in TeX
G:=Group("C8:5(C4:C4)"); // GroupNames label
G:=SmallGroup(128,674);
// by ID
G=gap.SmallGroup(128,674);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,436,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations