p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.56Q8, (C4×C8)⋊15C4, C42.320(C2×C4), C42⋊8C4.6C2, (C22×C4).547D4, C23.751(C2×D4), C4.2(C42.C2), C2.8(C42⋊8C4), C22.46(C4○D8), C22.4Q16.4C2, C4.58(C42⋊C2), (C22×C8).479C22, (C22×C4).1339C23, (C2×C42).1056C22, C2.9(C23.25D4), C22.56(C4.4D4), C2.1(C42.78C22), (C2×C4×C8).18C2, (C2×C4).86(C4⋊C4), (C2×C8).212(C2×C4), C22.97(C2×C4⋊C4), (C2×C4).191(C2×Q8), (C2×C4⋊C4).45C22, (C2×C4).557(C4○D4), (C2×C4).537(C22×C4), SmallGroup(128,567)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.56Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2b2c3 >
Subgroups: 204 in 108 conjugacy classes, 60 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2.C42, C4×C8, C2×C42, C2×C4⋊C4, C22×C8, C22.4Q16, C42⋊8C4, C2×C4×C8, C42.56Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C4○D8, C42⋊8C4, C23.25D4, C42.78C22, C42.56Q8
►Regular action on 128 points
Generators in S128
(1 79 12 58)(2 80 13 59)(3 73 14 60)(4 74 15 61)(5 75 16 62)(6 76 9 63)(7 77 10 64)(8 78 11 57)(17 31 48 49)(18 32 41 50)(19 25 42 51)(20 26 43 52)(21 27 44 53)(22 28 45 54)(23 29 46 55)(24 30 47 56)(33 92 122 105)(34 93 123 106)(35 94 124 107)(36 95 125 108)(37 96 126 109)(38 89 127 110)(39 90 128 111)(40 91 121 112)(65 115 97 85)(66 116 98 86)(67 117 99 87)(68 118 100 88)(69 119 101 81)(70 120 102 82)(71 113 103 83)(72 114 104 84)
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)(17 79 21 75)(18 80 22 76)(19 73 23 77)(20 74 24 78)(33 101 37 97)(34 102 38 98)(35 103 39 99)(36 104 40 100)(41 59 45 63)(42 60 46 64)(43 61 47 57)(44 62 48 58)(65 122 69 126)(66 123 70 127)(67 124 71 128)(68 125 72 121)(81 96 85 92)(82 89 86 93)(83 90 87 94)(84 91 88 95)(105 119 109 115)(106 120 110 116)(107 113 111 117)(108 114 112 118)
(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 87 51 92)(2 116 52 112)(3 85 53 90)(4 114 54 110)(5 83 55 96)(6 120 56 108)(7 81 49 94)(8 118 50 106)(9 82 30 95)(10 119 31 107)(11 88 32 93)(12 117 25 105)(13 86 26 91)(14 115 27 111)(15 84 28 89)(16 113 29 109)(17 39 77 97)(18 127 78 72)(19 37 79 103)(20 125 80 70)(21 35 73 101)(22 123 74 68)(23 33 75 99)(24 121 76 66)(34 61 100 45)(36 59 102 43)(38 57 104 41)(40 63 98 47)(42 126 58 71)(44 124 60 69)(46 122 62 67)(48 128 64 65)
G:=sub<Sym(128)| (1,79,12,58)(2,80,13,59)(3,73,14,60)(4,74,15,61)(5,75,16,62)(6,76,9,63)(7,77,10,64)(8,78,11,57)(17,31,48,49)(18,32,41,50)(19,25,42,51)(20,26,43,52)(21,27,44,53)(22,28,45,54)(23,29,46,55)(24,30,47,56)(33,92,122,105)(34,93,123,106)(35,94,124,107)(36,95,125,108)(37,96,126,109)(38,89,127,110)(39,90,128,111)(40,91,121,112)(65,115,97,85)(66,116,98,86)(67,117,99,87)(68,118,100,88)(69,119,101,81)(70,120,102,82)(71,113,103,83)(72,114,104,84), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,79,21,75)(18,80,22,76)(19,73,23,77)(20,74,24,78)(33,101,37,97)(34,102,38,98)(35,103,39,99)(36,104,40,100)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58)(65,122,69,126)(66,123,70,127)(67,124,71,128)(68,125,72,121)(81,96,85,92)(82,89,86,93)(83,90,87,94)(84,91,88,95)(105,119,109,115)(106,120,110,116)(107,113,111,117)(108,114,112,118), (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), (1,87,51,92)(2,116,52,112)(3,85,53,90)(4,114,54,110)(5,83,55,96)(6,120,56,108)(7,81,49,94)(8,118,50,106)(9,82,30,95)(10,119,31,107)(11,88,32,93)(12,117,25,105)(13,86,26,91)(14,115,27,111)(15,84,28,89)(16,113,29,109)(17,39,77,97)(18,127,78,72)(19,37,79,103)(20,125,80,70)(21,35,73,101)(22,123,74,68)(23,33,75,99)(24,121,76,66)(34,61,100,45)(36,59,102,43)(38,57,104,41)(40,63,98,47)(42,126,58,71)(44,124,60,69)(46,122,62,67)(48,128,64,65)>;
G:=Group( (1,79,12,58)(2,80,13,59)(3,73,14,60)(4,74,15,61)(5,75,16,62)(6,76,9,63)(7,77,10,64)(8,78,11,57)(17,31,48,49)(18,32,41,50)(19,25,42,51)(20,26,43,52)(21,27,44,53)(22,28,45,54)(23,29,46,55)(24,30,47,56)(33,92,122,105)(34,93,123,106)(35,94,124,107)(36,95,125,108)(37,96,126,109)(38,89,127,110)(39,90,128,111)(40,91,121,112)(65,115,97,85)(66,116,98,86)(67,117,99,87)(68,118,100,88)(69,119,101,81)(70,120,102,82)(71,113,103,83)(72,114,104,84), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,79,21,75)(18,80,22,76)(19,73,23,77)(20,74,24,78)(33,101,37,97)(34,102,38,98)(35,103,39,99)(36,104,40,100)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58)(65,122,69,126)(66,123,70,127)(67,124,71,128)(68,125,72,121)(81,96,85,92)(82,89,86,93)(83,90,87,94)(84,91,88,95)(105,119,109,115)(106,120,110,116)(107,113,111,117)(108,114,112,118), (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), (1,87,51,92)(2,116,52,112)(3,85,53,90)(4,114,54,110)(5,83,55,96)(6,120,56,108)(7,81,49,94)(8,118,50,106)(9,82,30,95)(10,119,31,107)(11,88,32,93)(12,117,25,105)(13,86,26,91)(14,115,27,111)(15,84,28,89)(16,113,29,109)(17,39,77,97)(18,127,78,72)(19,37,79,103)(20,125,80,70)(21,35,73,101)(22,123,74,68)(23,33,75,99)(24,121,76,66)(34,61,100,45)(36,59,102,43)(38,57,104,41)(40,63,98,47)(42,126,58,71)(44,124,60,69)(46,122,62,67)(48,128,64,65) );
G=PermutationGroup([[(1,79,12,58),(2,80,13,59),(3,73,14,60),(4,74,15,61),(5,75,16,62),(6,76,9,63),(7,77,10,64),(8,78,11,57),(17,31,48,49),(18,32,41,50),(19,25,42,51),(20,26,43,52),(21,27,44,53),(22,28,45,54),(23,29,46,55),(24,30,47,56),(33,92,122,105),(34,93,123,106),(35,94,124,107),(36,95,125,108),(37,96,126,109),(38,89,127,110),(39,90,128,111),(40,91,121,112),(65,115,97,85),(66,116,98,86),(67,117,99,87),(68,118,100,88),(69,119,101,81),(70,120,102,82),(71,113,103,83),(72,114,104,84)], [(1,53,5,49),(2,54,6,50),(3,55,7,51),(4,56,8,52),(9,32,13,28),(10,25,14,29),(11,26,15,30),(12,27,16,31),(17,79,21,75),(18,80,22,76),(19,73,23,77),(20,74,24,78),(33,101,37,97),(34,102,38,98),(35,103,39,99),(36,104,40,100),(41,59,45,63),(42,60,46,64),(43,61,47,57),(44,62,48,58),(65,122,69,126),(66,123,70,127),(67,124,71,128),(68,125,72,121),(81,96,85,92),(82,89,86,93),(83,90,87,94),(84,91,88,95),(105,119,109,115),(106,120,110,116),(107,113,111,117),(108,114,112,118)], [(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)], [(1,87,51,92),(2,116,52,112),(3,85,53,90),(4,114,54,110),(5,83,55,96),(6,120,56,108),(7,81,49,94),(8,118,50,106),(9,82,30,95),(10,119,31,107),(11,88,32,93),(12,117,25,105),(13,86,26,91),(14,115,27,111),(15,84,28,89),(16,113,29,109),(17,39,77,97),(18,127,78,72),(19,37,79,103),(20,125,80,70),(21,35,73,101),(22,123,74,68),(23,33,75,99),(24,121,76,66),(34,61,100,45),(36,59,102,43),(38,57,104,41),(40,63,98,47),(42,126,58,71),(44,124,60,69),(46,122,62,67),(48,128,64,65)]])
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | C4○D4 | C4○D8 |
| kernel | C42.56Q8 | C22.4Q16 | C42⋊8C4 | C2×C4×C8 | C4×C8 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 2 | 8 | 16 |
Matrix representation of C42.56Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 0 | 13 | 6 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,7,0,0,0,0,7,16,0,0,0,0,0,0,11,13,0,0,0,0,13,6] >;
C42.56Q8 in GAP, Magma, Sage, TeX
C_4^2._{56}Q_8 % in TeX
G:=Group("C4^2.56Q8"); // GroupNames label
G:=SmallGroup(128,567);
// by ID
G=gap.SmallGroup(128,567);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,568,422,58,2019,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations