p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.95D4, (C2×Q8).97D4, (C2×C4).16Q16, (C2×C4).35SD16, C4.24(C4⋊D4), (C22×C4).314D4, C23.915(C2×D4), C22.56(C2×Q16), C2.22(Q8⋊D4), C2.13(C4⋊2Q16), (C22×C8).75C22, C2.6(C4.SD16), C22.98(C2×SD16), C22.225C22≀C2, C2.13(D4.D4), (C2×C42).366C22, C2.22(C22⋊Q16), (C22×Q8).67C22, C22.232(C4⋊D4), (C22×C4).1449C23, C22.92(C4.4D4), C4.73(C22.D4), C2.16(C23.10D4), C22.128(C8.C22), C22.7C42.12C2, C23.65C23.14C2, C2.6(C42.30C22), (C2×C4⋊Q8).20C2, (C2×C4).1041(C2×D4), (C2×C4).880(C4○D4), (C2×C4⋊C4).128C22, (C2×Q8⋊C4).12C2, SmallGroup(128,775)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4.95D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=b-1, dbd-1=ab, dcd-1=c-1 >
Subgroups: 296 in 149 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×C8, C22×Q8, C22.7C42, C23.65C23, C2×Q8⋊C4, C2×C4⋊Q8, C4⋊C4.95D4
Quotients: C1, C2, C22, D4, C23, SD16, Q16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×SD16, C2×Q16, C8.C22, C23.10D4, Q8⋊D4, C22⋊Q16, D4.D4, C4⋊2Q16, C4.SD16, C42.30C22, C4⋊C4.95D4
►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)
(1 113 11 15)(2 116 12 14)(3 115 9 13)(4 114 10 16)(5 124 23 127)(6 123 24 126)(7 122 21 125)(8 121 22 128)(17 99 37 120)(18 98 38 119)(19 97 39 118)(20 100 40 117)(25 106 32 109)(26 105 29 112)(27 108 30 111)(28 107 31 110)(33 104 56 84)(34 103 53 83)(35 102 54 82)(36 101 55 81)(41 94 48 91)(42 93 45 90)(43 96 46 89)(44 95 47 92)(49 77 70 74)(50 80 71 73)(51 79 72 76)(52 78 69 75)(57 88 64 65)(58 87 61 68)(59 86 62 67)(60 85 63 66)
(1 99 6 108)(2 100 7 105)(3 97 8 106)(4 98 5 107)(9 118 22 109)(10 119 23 110)(11 120 24 111)(12 117 21 112)(13 39 128 32)(14 40 125 29)(15 37 126 30)(16 38 127 31)(17 123 27 113)(18 124 28 114)(19 121 25 115)(20 122 26 116)(33 73 43 67)(34 74 44 68)(35 75 41 65)(36 76 42 66)(45 85 55 79)(46 86 56 80)(47 87 53 77)(48 88 54 78)(49 92 58 83)(50 89 59 84)(51 90 60 81)(52 91 57 82)(61 103 70 95)(62 104 71 96)(63 101 72 93)(64 102 69 94)
(1 76 3 74)(2 75 4 73)(5 67 7 65)(6 66 8 68)(9 77 11 79)(10 80 12 78)(13 69 15 71)(14 72 16 70)(17 84 19 82)(18 83 20 81)(21 88 23 86)(22 87 24 85)(25 91 27 89)(26 90 28 92)(29 93 31 95)(30 96 32 94)(33 100 35 98)(34 99 36 97)(37 104 39 102)(38 103 40 101)(41 107 43 105)(42 106 44 108)(45 109 47 111)(46 112 48 110)(49 116 51 114)(50 115 52 113)(53 120 55 118)(54 119 56 117)(57 123 59 121)(58 122 60 124)(61 125 63 127)(62 128 64 126)
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), (1,113,11,15)(2,116,12,14)(3,115,9,13)(4,114,10,16)(5,124,23,127)(6,123,24,126)(7,122,21,125)(8,121,22,128)(17,99,37,120)(18,98,38,119)(19,97,39,118)(20,100,40,117)(25,106,32,109)(26,105,29,112)(27,108,30,111)(28,107,31,110)(33,104,56,84)(34,103,53,83)(35,102,54,82)(36,101,55,81)(41,94,48,91)(42,93,45,90)(43,96,46,89)(44,95,47,92)(49,77,70,74)(50,80,71,73)(51,79,72,76)(52,78,69,75)(57,88,64,65)(58,87,61,68)(59,86,62,67)(60,85,63,66), (1,99,6,108)(2,100,7,105)(3,97,8,106)(4,98,5,107)(9,118,22,109)(10,119,23,110)(11,120,24,111)(12,117,21,112)(13,39,128,32)(14,40,125,29)(15,37,126,30)(16,38,127,31)(17,123,27,113)(18,124,28,114)(19,121,25,115)(20,122,26,116)(33,73,43,67)(34,74,44,68)(35,75,41,65)(36,76,42,66)(45,85,55,79)(46,86,56,80)(47,87,53,77)(48,88,54,78)(49,92,58,83)(50,89,59,84)(51,90,60,81)(52,91,57,82)(61,103,70,95)(62,104,71,96)(63,101,72,93)(64,102,69,94), (1,76,3,74)(2,75,4,73)(5,67,7,65)(6,66,8,68)(9,77,11,79)(10,80,12,78)(13,69,15,71)(14,72,16,70)(17,84,19,82)(18,83,20,81)(21,88,23,86)(22,87,24,85)(25,91,27,89)(26,90,28,92)(29,93,31,95)(30,96,32,94)(33,100,35,98)(34,99,36,97)(37,104,39,102)(38,103,40,101)(41,107,43,105)(42,106,44,108)(45,109,47,111)(46,112,48,110)(49,116,51,114)(50,115,52,113)(53,120,55,118)(54,119,56,117)(57,123,59,121)(58,122,60,124)(61,125,63,127)(62,128,64,126)>;
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), (1,113,11,15)(2,116,12,14)(3,115,9,13)(4,114,10,16)(5,124,23,127)(6,123,24,126)(7,122,21,125)(8,121,22,128)(17,99,37,120)(18,98,38,119)(19,97,39,118)(20,100,40,117)(25,106,32,109)(26,105,29,112)(27,108,30,111)(28,107,31,110)(33,104,56,84)(34,103,53,83)(35,102,54,82)(36,101,55,81)(41,94,48,91)(42,93,45,90)(43,96,46,89)(44,95,47,92)(49,77,70,74)(50,80,71,73)(51,79,72,76)(52,78,69,75)(57,88,64,65)(58,87,61,68)(59,86,62,67)(60,85,63,66), (1,99,6,108)(2,100,7,105)(3,97,8,106)(4,98,5,107)(9,118,22,109)(10,119,23,110)(11,120,24,111)(12,117,21,112)(13,39,128,32)(14,40,125,29)(15,37,126,30)(16,38,127,31)(17,123,27,113)(18,124,28,114)(19,121,25,115)(20,122,26,116)(33,73,43,67)(34,74,44,68)(35,75,41,65)(36,76,42,66)(45,85,55,79)(46,86,56,80)(47,87,53,77)(48,88,54,78)(49,92,58,83)(50,89,59,84)(51,90,60,81)(52,91,57,82)(61,103,70,95)(62,104,71,96)(63,101,72,93)(64,102,69,94), (1,76,3,74)(2,75,4,73)(5,67,7,65)(6,66,8,68)(9,77,11,79)(10,80,12,78)(13,69,15,71)(14,72,16,70)(17,84,19,82)(18,83,20,81)(21,88,23,86)(22,87,24,85)(25,91,27,89)(26,90,28,92)(29,93,31,95)(30,96,32,94)(33,100,35,98)(34,99,36,97)(37,104,39,102)(38,103,40,101)(41,107,43,105)(42,106,44,108)(45,109,47,111)(46,112,48,110)(49,116,51,114)(50,115,52,113)(53,120,55,118)(54,119,56,117)(57,123,59,121)(58,122,60,124)(61,125,63,127)(62,128,64,126) );
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)], [(1,113,11,15),(2,116,12,14),(3,115,9,13),(4,114,10,16),(5,124,23,127),(6,123,24,126),(7,122,21,125),(8,121,22,128),(17,99,37,120),(18,98,38,119),(19,97,39,118),(20,100,40,117),(25,106,32,109),(26,105,29,112),(27,108,30,111),(28,107,31,110),(33,104,56,84),(34,103,53,83),(35,102,54,82),(36,101,55,81),(41,94,48,91),(42,93,45,90),(43,96,46,89),(44,95,47,92),(49,77,70,74),(50,80,71,73),(51,79,72,76),(52,78,69,75),(57,88,64,65),(58,87,61,68),(59,86,62,67),(60,85,63,66)], [(1,99,6,108),(2,100,7,105),(3,97,8,106),(4,98,5,107),(9,118,22,109),(10,119,23,110),(11,120,24,111),(12,117,21,112),(13,39,128,32),(14,40,125,29),(15,37,126,30),(16,38,127,31),(17,123,27,113),(18,124,28,114),(19,121,25,115),(20,122,26,116),(33,73,43,67),(34,74,44,68),(35,75,41,65),(36,76,42,66),(45,85,55,79),(46,86,56,80),(47,87,53,77),(48,88,54,78),(49,92,58,83),(50,89,59,84),(51,90,60,81),(52,91,57,82),(61,103,70,95),(62,104,71,96),(63,101,72,93),(64,102,69,94)], [(1,76,3,74),(2,75,4,73),(5,67,7,65),(6,66,8,68),(9,77,11,79),(10,80,12,78),(13,69,15,71),(14,72,16,70),(17,84,19,82),(18,83,20,81),(21,88,23,86),(22,87,24,85),(25,91,27,89),(26,90,28,92),(29,93,31,95),(30,96,32,94),(33,100,35,98),(34,99,36,97),(37,104,39,102),(38,103,40,101),(41,107,43,105),(42,106,44,108),(45,109,47,111),(46,112,48,110),(49,116,51,114),(50,115,52,113),(53,120,55,118),(54,119,56,117),(57,123,59,121),(58,122,60,124),(61,125,63,127),(62,128,64,126)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | SD16 | Q16 | C4○D4 | C8.C22 |
| kernel | C4⋊C4.95D4 | C22.7C42 | C23.65C23 | C2×Q8⋊C4 | C2×C4⋊Q8 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 4 | 4 | 4 | 6 | 2 |
Matrix representation of C4⋊C4.95D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 10 | 13 | 0 | 0 | 0 | 0 |
| 4 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 |
| 4 | 7 | 0 | 0 | 0 | 0 |
| 10 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 13 | 10 | 0 | 0 | 0 | 0 |
| 7 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 12 | 10 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[10,4,0,0,0,0,13,7,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[4,10,0,0,0,0,7,13,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,7,0,0,0,0,10,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,7,12,0,0,0,0,10,10] >;
C4⋊C4.95D4 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{95}D_4 % in TeX
G:=Group("C4:C4.95D4"); // GroupNames label
G:=SmallGroup(128,775);
// by ID
G=gap.SmallGroup(128,775);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,58,2804,1411,718,172,4037,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations