p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊7(C4⋊C4), C4.6(C4×Q8), C4.Q8⋊12C4, (C2×C8).54Q8, (C2×C8).368D4, C2.6(C8⋊8D4), C2.2(C8⋊3Q8), (C2×C4).63SD16, C2.18(C4×SD16), (C22×C4).555D4, C22.174(C4×D4), C23.788(C2×D4), C4.74(C22⋊Q8), C22.31(C4⋊Q8), C2.2(C8.5Q8), C4.4(C42.C2), C22.71(C4○D8), C22.68(C2×SD16), C22.4Q16.14C2, (C22×C8).486C22, C22.134(C4⋊D4), (C22×C4).1390C23, (C2×C42).1067C22, C23.65C23.10C2, C2.8(C23.65C23), (C2×C4×C8).55C2, C4.39(C2×C4⋊C4), C4⋊C4.87(C2×C4), (C2×C8).183(C2×C4), (C2×C4).202(C2×Q8), (C2×C4.Q8).19C2, (C2×C4).1347(C2×D4), (C2×C4⋊C4).74C22, (C2×C4).585(C4○D4), (C2×C4).408(C22×C4), SmallGroup(128,673)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊7(C4⋊C4)
G = < a,b,c | a8=b4=c4=1, bab-1=a3, ac=ca, cbc-1=b-1 >
Subgroups: 228 in 124 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2.C42, C4×C8, C4.Q8, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C23.65C23, C2×C4×C8, C2×C4.Q8, C8⋊7(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, C4○D8, C23.65C23, C4×SD16, C8⋊8D4, C8⋊3Q8, C8.5Q8, C8⋊7(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)
(1 100 23 114)(2 103 24 117)(3 98 17 120)(4 101 18 115)(5 104 19 118)(6 99 20 113)(7 102 21 116)(8 97 22 119)(9 28 44 124)(10 31 45 127)(11 26 46 122)(12 29 47 125)(13 32 48 128)(14 27 41 123)(15 30 42 126)(16 25 43 121)(33 81 51 67)(34 84 52 70)(35 87 53 65)(36 82 54 68)(37 85 55 71)(38 88 56 66)(39 83 49 69)(40 86 50 72)(57 109 75 91)(58 112 76 94)(59 107 77 89)(60 110 78 92)(61 105 79 95)(62 108 80 90)(63 111 73 93)(64 106 74 96)
(1 54 47 73)(2 55 48 74)(3 56 41 75)(4 49 42 76)(5 50 43 77)(6 51 44 78)(7 52 45 79)(8 53 46 80)(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 89 118 72)(26 90 119 65)(27 91 120 66)(28 92 113 67)(29 93 114 68)(30 94 115 69)(31 95 116 70)(32 96 117 71)(81 124 110 99)(82 125 111 100)(83 126 112 101)(84 127 105 102)(85 128 106 103)(86 121 107 104)(87 122 108 97)(88 123 109 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), (1,100,23,114)(2,103,24,117)(3,98,17,120)(4,101,18,115)(5,104,19,118)(6,99,20,113)(7,102,21,116)(8,97,22,119)(9,28,44,124)(10,31,45,127)(11,26,46,122)(12,29,47,125)(13,32,48,128)(14,27,41,123)(15,30,42,126)(16,25,43,121)(33,81,51,67)(34,84,52,70)(35,87,53,65)(36,82,54,68)(37,85,55,71)(38,88,56,66)(39,83,49,69)(40,86,50,72)(57,109,75,91)(58,112,76,94)(59,107,77,89)(60,110,78,92)(61,105,79,95)(62,108,80,90)(63,111,73,93)(64,106,74,96), (1,54,47,73)(2,55,48,74)(3,56,41,75)(4,49,42,76)(5,50,43,77)(6,51,44,78)(7,52,45,79)(8,53,46,80)(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)(25,89,118,72)(26,90,119,65)(27,91,120,66)(28,92,113,67)(29,93,114,68)(30,94,115,69)(31,95,116,70)(32,96,117,71)(81,124,110,99)(82,125,111,100)(83,126,112,101)(84,127,105,102)(85,128,106,103)(86,121,107,104)(87,122,108,97)(88,123,109,98)>;
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), (1,100,23,114)(2,103,24,117)(3,98,17,120)(4,101,18,115)(5,104,19,118)(6,99,20,113)(7,102,21,116)(8,97,22,119)(9,28,44,124)(10,31,45,127)(11,26,46,122)(12,29,47,125)(13,32,48,128)(14,27,41,123)(15,30,42,126)(16,25,43,121)(33,81,51,67)(34,84,52,70)(35,87,53,65)(36,82,54,68)(37,85,55,71)(38,88,56,66)(39,83,49,69)(40,86,50,72)(57,109,75,91)(58,112,76,94)(59,107,77,89)(60,110,78,92)(61,105,79,95)(62,108,80,90)(63,111,73,93)(64,106,74,96), (1,54,47,73)(2,55,48,74)(3,56,41,75)(4,49,42,76)(5,50,43,77)(6,51,44,78)(7,52,45,79)(8,53,46,80)(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)(25,89,118,72)(26,90,119,65)(27,91,120,66)(28,92,113,67)(29,93,114,68)(30,94,115,69)(31,95,116,70)(32,96,117,71)(81,124,110,99)(82,125,111,100)(83,126,112,101)(84,127,105,102)(85,128,106,103)(86,121,107,104)(87,122,108,97)(88,123,109,98) );
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)], [(1,100,23,114),(2,103,24,117),(3,98,17,120),(4,101,18,115),(5,104,19,118),(6,99,20,113),(7,102,21,116),(8,97,22,119),(9,28,44,124),(10,31,45,127),(11,26,46,122),(12,29,47,125),(13,32,48,128),(14,27,41,123),(15,30,42,126),(16,25,43,121),(33,81,51,67),(34,84,52,70),(35,87,53,65),(36,82,54,68),(37,85,55,71),(38,88,56,66),(39,83,49,69),(40,86,50,72),(57,109,75,91),(58,112,76,94),(59,107,77,89),(60,110,78,92),(61,105,79,95),(62,108,80,90),(63,111,73,93),(64,106,74,96)], [(1,54,47,73),(2,55,48,74),(3,56,41,75),(4,49,42,76),(5,50,43,77),(6,51,44,78),(7,52,45,79),(8,53,46,80),(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),(25,89,118,72),(26,90,119,65),(27,91,120,66),(28,92,113,67),(29,93,114,68),(30,94,115,69),(31,95,116,70),(32,96,117,71),(81,124,110,99),(82,125,111,100),(83,126,112,101),(84,127,105,102),(85,128,106,103),(86,121,107,104),(87,122,108,97),(88,123,109,98)]])
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 |
| type | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | SD16 | C4○D4 | C4○D8 |
| kernel | C8⋊7(C4⋊C4) | C22.4Q16 | C23.65C23 | C2×C4×C8 | C2×C4.Q8 | C4.Q8 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 8 | 4 | 8 |
Matrix representation of C8⋊7(C4⋊C4) ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 13 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,15,0,0,0,0,0,9,0,0,0,0,0,15,0,0,0,0,0,9],[16,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,4,0],[13,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,13] >;
C8⋊7(C4⋊C4) in GAP, Magma, Sage, TeX
C_8\rtimes_7(C_4\rtimes C_4)
% in TeX
G:=Group("C8:7(C4:C4)"); // GroupNames label
G:=SmallGroup(128,673);
// by ID
G=gap.SmallGroup(128,673);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations