p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.55Q8, (C4×C8)⋊14C4, (C2×C4).73D8, (C2×C4).35Q16, (C2×C4).70SD16, C4.11(C4.Q8), C4.15(C2.D8), C22.34(C2×D8), C42⋊9C4.4C2, C42.319(C2×C4), C2.1(C4.4D8), (C22×C4).576D4, C23.750(C2×D4), C4.1(C42.C2), C22.27(C2×Q16), C2.7(C42⋊8C4), C22.4Q16.3C2, C2.1(C4.SD16), C22.51(C2×SD16), C4.57(C42⋊C2), (C22×C8).478C22, (C22×C4).1338C23, (C2×C42).1055C22, C22.55(C4.4D4), (C2×C4×C8).17C2, C2.7(C2×C2.D8), C2.7(C2×C4.Q8), (C2×C8).211(C2×C4), C22.96(C2×C4⋊C4), (C2×C4).190(C2×Q8), (C2×C4).130(C4⋊C4), (C2×C4⋊C4).44C22, (C2×C4).556(C4○D4), (C2×C4).536(C22×C4), SmallGroup(128,566)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.55Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c3 >
Subgroups: 236 in 124 conjugacy classes, 76 normal (16 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, C4×C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C42⋊9C4, C2×C4×C8, C42.55Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C2×D8, C2×SD16, C2×Q16, C42⋊8C4, C2×C4.Q8, C2×C2.D8, C4.4D8, C4.SD16, C42.55Q8
►Regular action on 128 points
Generators in S128
(1 75 23 62)(2 76 24 63)(3 77 17 64)(4 78 18 57)(5 79 19 58)(6 80 20 59)(7 73 21 60)(8 74 22 61)(9 51 42 25)(10 52 43 26)(11 53 44 27)(12 54 45 28)(13 55 46 29)(14 56 47 30)(15 49 48 31)(16 50 41 32)(33 91 121 97)(34 92 122 98)(35 93 123 99)(36 94 124 100)(37 95 125 101)(38 96 126 102)(39 89 127 103)(40 90 128 104)(65 115 107 85)(66 116 108 86)(67 117 109 87)(68 118 110 88)(69 119 111 81)(70 120 112 82)(71 113 105 83)(72 114 106 84)
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 60 13 64)(10 61 14 57)(11 62 15 58)(12 63 16 59)(17 25 21 29)(18 26 22 30)(19 27 23 31)(20 28 24 32)(33 106 37 110)(34 107 38 111)(35 108 39 112)(36 109 40 105)(41 80 45 76)(42 73 46 77)(43 74 47 78)(44 75 48 79)(65 126 69 122)(66 127 70 123)(67 128 71 124)(68 121 72 125)(81 92 85 96)(82 93 86 89)(83 94 87 90)(84 95 88 91)(97 114 101 118)(98 115 102 119)(99 116 103 120)(100 117 104 113)
(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 83 55 92)(2 120 56 97)(3 81 49 90)(4 118 50 103)(5 87 51 96)(6 116 52 101)(7 85 53 94)(8 114 54 99)(9 126 58 67)(10 37 59 108)(11 124 60 65)(12 35 61 106)(13 122 62 71)(14 33 63 112)(15 128 64 69)(16 39 57 110)(17 119 31 104)(18 88 32 89)(19 117 25 102)(20 86 26 95)(21 115 27 100)(22 84 28 93)(23 113 29 98)(24 82 30 91)(34 75 105 46)(36 73 107 44)(38 79 109 42)(40 77 111 48)(41 127 78 68)(43 125 80 66)(45 123 74 72)(47 121 76 70)
G:=sub<Sym(128)| (1,75,23,62)(2,76,24,63)(3,77,17,64)(4,78,18,57)(5,79,19,58)(6,80,20,59)(7,73,21,60)(8,74,22,61)(9,51,42,25)(10,52,43,26)(11,53,44,27)(12,54,45,28)(13,55,46,29)(14,56,47,30)(15,49,48,31)(16,50,41,32)(33,91,121,97)(34,92,122,98)(35,93,123,99)(36,94,124,100)(37,95,125,101)(38,96,126,102)(39,89,127,103)(40,90,128,104)(65,115,107,85)(66,116,108,86)(67,117,109,87)(68,118,110,88)(69,119,111,81)(70,120,112,82)(71,113,105,83)(72,114,106,84), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,60,13,64)(10,61,14,57)(11,62,15,58)(12,63,16,59)(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32)(33,106,37,110)(34,107,38,111)(35,108,39,112)(36,109,40,105)(41,80,45,76)(42,73,46,77)(43,74,47,78)(44,75,48,79)(65,126,69,122)(66,127,70,123)(67,128,71,124)(68,121,72,125)(81,92,85,96)(82,93,86,89)(83,94,87,90)(84,95,88,91)(97,114,101,118)(98,115,102,119)(99,116,103,120)(100,117,104,113), (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), (1,83,55,92)(2,120,56,97)(3,81,49,90)(4,118,50,103)(5,87,51,96)(6,116,52,101)(7,85,53,94)(8,114,54,99)(9,126,58,67)(10,37,59,108)(11,124,60,65)(12,35,61,106)(13,122,62,71)(14,33,63,112)(15,128,64,69)(16,39,57,110)(17,119,31,104)(18,88,32,89)(19,117,25,102)(20,86,26,95)(21,115,27,100)(22,84,28,93)(23,113,29,98)(24,82,30,91)(34,75,105,46)(36,73,107,44)(38,79,109,42)(40,77,111,48)(41,127,78,68)(43,125,80,66)(45,123,74,72)(47,121,76,70)>;
G:=Group( (1,75,23,62)(2,76,24,63)(3,77,17,64)(4,78,18,57)(5,79,19,58)(6,80,20,59)(7,73,21,60)(8,74,22,61)(9,51,42,25)(10,52,43,26)(11,53,44,27)(12,54,45,28)(13,55,46,29)(14,56,47,30)(15,49,48,31)(16,50,41,32)(33,91,121,97)(34,92,122,98)(35,93,123,99)(36,94,124,100)(37,95,125,101)(38,96,126,102)(39,89,127,103)(40,90,128,104)(65,115,107,85)(66,116,108,86)(67,117,109,87)(68,118,110,88)(69,119,111,81)(70,120,112,82)(71,113,105,83)(72,114,106,84), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,60,13,64)(10,61,14,57)(11,62,15,58)(12,63,16,59)(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32)(33,106,37,110)(34,107,38,111)(35,108,39,112)(36,109,40,105)(41,80,45,76)(42,73,46,77)(43,74,47,78)(44,75,48,79)(65,126,69,122)(66,127,70,123)(67,128,71,124)(68,121,72,125)(81,92,85,96)(82,93,86,89)(83,94,87,90)(84,95,88,91)(97,114,101,118)(98,115,102,119)(99,116,103,120)(100,117,104,113), (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), (1,83,55,92)(2,120,56,97)(3,81,49,90)(4,118,50,103)(5,87,51,96)(6,116,52,101)(7,85,53,94)(8,114,54,99)(9,126,58,67)(10,37,59,108)(11,124,60,65)(12,35,61,106)(13,122,62,71)(14,33,63,112)(15,128,64,69)(16,39,57,110)(17,119,31,104)(18,88,32,89)(19,117,25,102)(20,86,26,95)(21,115,27,100)(22,84,28,93)(23,113,29,98)(24,82,30,91)(34,75,105,46)(36,73,107,44)(38,79,109,42)(40,77,111,48)(41,127,78,68)(43,125,80,66)(45,123,74,72)(47,121,76,70) );
G=PermutationGroup([[(1,75,23,62),(2,76,24,63),(3,77,17,64),(4,78,18,57),(5,79,19,58),(6,80,20,59),(7,73,21,60),(8,74,22,61),(9,51,42,25),(10,52,43,26),(11,53,44,27),(12,54,45,28),(13,55,46,29),(14,56,47,30),(15,49,48,31),(16,50,41,32),(33,91,121,97),(34,92,122,98),(35,93,123,99),(36,94,124,100),(37,95,125,101),(38,96,126,102),(39,89,127,103),(40,90,128,104),(65,115,107,85),(66,116,108,86),(67,117,109,87),(68,118,110,88),(69,119,111,81),(70,120,112,82),(71,113,105,83),(72,114,106,84)], [(1,49,5,53),(2,50,6,54),(3,51,7,55),(4,52,8,56),(9,60,13,64),(10,61,14,57),(11,62,15,58),(12,63,16,59),(17,25,21,29),(18,26,22,30),(19,27,23,31),(20,28,24,32),(33,106,37,110),(34,107,38,111),(35,108,39,112),(36,109,40,105),(41,80,45,76),(42,73,46,77),(43,74,47,78),(44,75,48,79),(65,126,69,122),(66,127,70,123),(67,128,71,124),(68,121,72,125),(81,92,85,96),(82,93,86,89),(83,94,87,90),(84,95,88,91),(97,114,101,118),(98,115,102,119),(99,116,103,120),(100,117,104,113)], [(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)], [(1,83,55,92),(2,120,56,97),(3,81,49,90),(4,118,50,103),(5,87,51,96),(6,116,52,101),(7,85,53,94),(8,114,54,99),(9,126,58,67),(10,37,59,108),(11,124,60,65),(12,35,61,106),(13,122,62,71),(14,33,63,112),(15,128,64,69),(16,39,57,110),(17,119,31,104),(18,88,32,89),(19,117,25,102),(20,86,26,95),(21,115,27,100),(22,84,28,93),(23,113,29,98),(24,82,30,91),(34,75,105,46),(36,73,107,44),(38,79,109,42),(40,77,111,48),(41,127,78,68),(43,125,80,66),(45,123,74,72),(47,121,76,70)]])
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 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D8 | SD16 | Q16 | C4○D4 |
| kernel | C42.55Q8 | C22.4Q16 | C42⋊9C4 | C2×C4×C8 | C4×C8 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 8 |
Matrix representation of C42.55Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 13 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 15 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 |
| 0 | 14 | 11 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 15 | 5 | 0 | 0 |
| 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 6 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,13,0],[16,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,0,14,0,0,0,6,11,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,15,13,0,0,0,5,2,0,0,0,0,0,1,6,0,0,0,11,16] >;
C42.55Q8 in GAP, Magma, Sage, TeX
C_4^2._{55}Q_8 % in TeX
G:=Group("C4^2.55Q8"); // GroupNames label
G:=SmallGroup(128,566);
// by ID
G=gap.SmallGroup(128,566);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,248,2804,172]);
// 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*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations