p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊4C4.2C2, (C2×C42).4C22, C2.4(C42⋊5C4), C22.49(C8○D4), (C22×C8).22C22, C4.45(C42⋊2C2), C2.C42.10C4, C23.310(C22×C4), (C22×C4).1624C23, C22.7C42.6C2, C22.81(C42⋊C2), C2.9(C42.7C22), (C2×C4).932(C4○D4), (C22×C4).115(C2×C4), SmallGroup(128,572)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊4C4.C2
G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, cac-1=ab2, dad-1=a-1b2c2, bc=cb, bd=db, dcd-1=a2b2c >
Subgroups: 164 in 98 conjugacy classes, 52 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C2.C42, C2×C42, C22×C8, C22.7C42, C42⋊4C4, C42⋊4C4.C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C42⋊C2, C42⋊2C2, C8○D4, C42⋊5C4, C42.7C22, C42⋊4C4.C2
►Regular action on 128 points
Generators in S128
(1 71 99 14)(2 76 100 128)(3 65 101 16)(4 78 102 122)(5 67 103 10)(6 80 104 124)(7 69 97 12)(8 74 98 126)(9 42 66 22)(11 44 68 24)(13 46 70 18)(15 48 72 20)(17 125 45 73)(19 127 47 75)(21 121 41 77)(23 123 43 79)(25 61 53 33)(26 94 54 114)(27 63 55 35)(28 96 56 116)(29 57 49 37)(30 90 50 118)(31 59 51 39)(32 92 52 120)(34 106 62 86)(36 108 64 88)(38 110 58 82)(40 112 60 84)(81 117 109 89)(83 119 111 91)(85 113 105 93)(87 115 107 95)
(1 17 5 21)(2 18 6 22)(3 19 7 23)(4 20 8 24)(9 76 13 80)(10 77 14 73)(11 78 15 74)(12 79 16 75)(25 83 29 87)(26 84 30 88)(27 85 31 81)(28 86 32 82)(33 91 37 95)(34 92 38 96)(35 93 39 89)(36 94 40 90)(41 99 45 103)(42 100 46 104)(43 101 47 97)(44 102 48 98)(49 107 53 111)(50 108 54 112)(51 109 55 105)(52 110 56 106)(57 115 61 119)(58 116 62 120)(59 117 63 113)(60 118 64 114)(65 127 69 123)(66 128 70 124)(67 121 71 125)(68 122 72 126)
(1 31 23 87)(2 56 24 112)(3 25 17 81)(4 50 18 106)(5 27 19 83)(6 52 20 108)(7 29 21 85)(8 54 22 110)(9 62 74 118)(10 39 75 95)(11 64 76 120)(12 33 77 89)(13 58 78 114)(14 35 79 91)(15 60 80 116)(16 37 73 93)(26 42 82 98)(28 44 84 100)(30 46 86 102)(32 48 88 104)(34 126 90 66)(36 128 92 68)(38 122 94 70)(40 124 96 72)(41 105 97 49)(43 107 99 51)(45 109 101 53)(47 111 103 55)(57 125 113 65)(59 127 115 67)(61 121 117 69)(63 123 119 71)
(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)
G:=sub<Sym(128)| (1,71,99,14)(2,76,100,128)(3,65,101,16)(4,78,102,122)(5,67,103,10)(6,80,104,124)(7,69,97,12)(8,74,98,126)(9,42,66,22)(11,44,68,24)(13,46,70,18)(15,48,72,20)(17,125,45,73)(19,127,47,75)(21,121,41,77)(23,123,43,79)(25,61,53,33)(26,94,54,114)(27,63,55,35)(28,96,56,116)(29,57,49,37)(30,90,50,118)(31,59,51,39)(32,92,52,120)(34,106,62,86)(36,108,64,88)(38,110,58,82)(40,112,60,84)(81,117,109,89)(83,119,111,91)(85,113,105,93)(87,115,107,95), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,76,13,80)(10,77,14,73)(11,78,15,74)(12,79,16,75)(25,83,29,87)(26,84,30,88)(27,85,31,81)(28,86,32,82)(33,91,37,95)(34,92,38,96)(35,93,39,89)(36,94,40,90)(41,99,45,103)(42,100,46,104)(43,101,47,97)(44,102,48,98)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,115,61,119)(58,116,62,120)(59,117,63,113)(60,118,64,114)(65,127,69,123)(66,128,70,124)(67,121,71,125)(68,122,72,126), (1,31,23,87)(2,56,24,112)(3,25,17,81)(4,50,18,106)(5,27,19,83)(6,52,20,108)(7,29,21,85)(8,54,22,110)(9,62,74,118)(10,39,75,95)(11,64,76,120)(12,33,77,89)(13,58,78,114)(14,35,79,91)(15,60,80,116)(16,37,73,93)(26,42,82,98)(28,44,84,100)(30,46,86,102)(32,48,88,104)(34,126,90,66)(36,128,92,68)(38,122,94,70)(40,124,96,72)(41,105,97,49)(43,107,99,51)(45,109,101,53)(47,111,103,55)(57,125,113,65)(59,127,115,67)(61,121,117,69)(63,123,119,71), (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)>;
G:=Group( (1,71,99,14)(2,76,100,128)(3,65,101,16)(4,78,102,122)(5,67,103,10)(6,80,104,124)(7,69,97,12)(8,74,98,126)(9,42,66,22)(11,44,68,24)(13,46,70,18)(15,48,72,20)(17,125,45,73)(19,127,47,75)(21,121,41,77)(23,123,43,79)(25,61,53,33)(26,94,54,114)(27,63,55,35)(28,96,56,116)(29,57,49,37)(30,90,50,118)(31,59,51,39)(32,92,52,120)(34,106,62,86)(36,108,64,88)(38,110,58,82)(40,112,60,84)(81,117,109,89)(83,119,111,91)(85,113,105,93)(87,115,107,95), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,76,13,80)(10,77,14,73)(11,78,15,74)(12,79,16,75)(25,83,29,87)(26,84,30,88)(27,85,31,81)(28,86,32,82)(33,91,37,95)(34,92,38,96)(35,93,39,89)(36,94,40,90)(41,99,45,103)(42,100,46,104)(43,101,47,97)(44,102,48,98)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,115,61,119)(58,116,62,120)(59,117,63,113)(60,118,64,114)(65,127,69,123)(66,128,70,124)(67,121,71,125)(68,122,72,126), (1,31,23,87)(2,56,24,112)(3,25,17,81)(4,50,18,106)(5,27,19,83)(6,52,20,108)(7,29,21,85)(8,54,22,110)(9,62,74,118)(10,39,75,95)(11,64,76,120)(12,33,77,89)(13,58,78,114)(14,35,79,91)(15,60,80,116)(16,37,73,93)(26,42,82,98)(28,44,84,100)(30,46,86,102)(32,48,88,104)(34,126,90,66)(36,128,92,68)(38,122,94,70)(40,124,96,72)(41,105,97,49)(43,107,99,51)(45,109,101,53)(47,111,103,55)(57,125,113,65)(59,127,115,67)(61,121,117,69)(63,123,119,71), (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) );
G=PermutationGroup([[(1,71,99,14),(2,76,100,128),(3,65,101,16),(4,78,102,122),(5,67,103,10),(6,80,104,124),(7,69,97,12),(8,74,98,126),(9,42,66,22),(11,44,68,24),(13,46,70,18),(15,48,72,20),(17,125,45,73),(19,127,47,75),(21,121,41,77),(23,123,43,79),(25,61,53,33),(26,94,54,114),(27,63,55,35),(28,96,56,116),(29,57,49,37),(30,90,50,118),(31,59,51,39),(32,92,52,120),(34,106,62,86),(36,108,64,88),(38,110,58,82),(40,112,60,84),(81,117,109,89),(83,119,111,91),(85,113,105,93),(87,115,107,95)], [(1,17,5,21),(2,18,6,22),(3,19,7,23),(4,20,8,24),(9,76,13,80),(10,77,14,73),(11,78,15,74),(12,79,16,75),(25,83,29,87),(26,84,30,88),(27,85,31,81),(28,86,32,82),(33,91,37,95),(34,92,38,96),(35,93,39,89),(36,94,40,90),(41,99,45,103),(42,100,46,104),(43,101,47,97),(44,102,48,98),(49,107,53,111),(50,108,54,112),(51,109,55,105),(52,110,56,106),(57,115,61,119),(58,116,62,120),(59,117,63,113),(60,118,64,114),(65,127,69,123),(66,128,70,124),(67,121,71,125),(68,122,72,126)], [(1,31,23,87),(2,56,24,112),(3,25,17,81),(4,50,18,106),(5,27,19,83),(6,52,20,108),(7,29,21,85),(8,54,22,110),(9,62,74,118),(10,39,75,95),(11,64,76,120),(12,33,77,89),(13,58,78,114),(14,35,79,91),(15,60,80,116),(16,37,73,93),(26,42,82,98),(28,44,84,100),(30,46,86,102),(32,48,88,104),(34,126,90,66),(36,128,92,68),(38,122,94,70),(40,124,96,72),(41,105,97,49),(43,107,99,51),(45,109,101,53),(47,111,103,55),(57,125,113,65),(59,127,115,67),(61,121,117,69),(63,123,119,71)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C2 | C4 | C4○D4 | C8○D4 |
| kernel | C42⋊4C4.C2 | C22.7C42 | C42⋊4C4 | C2.C42 | C2×C4 | C22 |
| # reps | 1 | 6 | 1 | 8 | 12 | 16 |
Matrix representation of C42⋊4C4.C2 ►in GL6(𝔽17)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 11 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 12 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 10 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,2,16,0,0,0,0,0,0,2,11,0,0,0,0,9,15,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,2,0,0,0,0,16,1],[9,8,0,0,0,0,1,8,0,0,0,0,0,0,13,12,0,0,0,0,16,4,0,0,0,0,0,0,0,10,0,0,0,0,5,0] >;
C42⋊4C4.C2 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4C_4.C_2
% in TeX
G:=Group("C4^2:4C4.C2"); // GroupNames label
G:=SmallGroup(128,572);
// by ID
G=gap.SmallGroup(128,572);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,176,422,723,58,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b*c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2*c^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c>;
// generators/relations