p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊3C42, C42.94D4, (C4×Q8)⋊12C4, Q8⋊C4⋊9C4, C4.111(C4×D4), C4.6(C2×C42), C22.89(C4×D4), C42.129(C2×C4), C23.733(C2×D4), (C22×C4).672D4, C4.6(C42⋊C2), C2.2(Q16⋊C4), C2.3(SD16⋊C4), C22.55(C8⋊C22), C22.4Q16.45C2, (C22×C8).381C22, (C2×C42).236C22, (C22×C4).1310C23, C2.3(C23.38D4), C2.4(C23.36D4), C22.44(C8.C22), (C22×Q8).377C22, (C4×C4⋊C4).6C2, (C2×C4×Q8).9C2, C4⋊C4.141(C2×C4), (C2×C8).133(C2×C4), C2.21(C4×C22⋊C4), (C2×C8⋊C4).23C2, (C2×C4).1305(C2×D4), (C2×Q8).182(C2×C4), (C2×C4).541(C4○D4), (C2×C4⋊C4).749C22, (C2×C4).353(C22×C4), (C2×Q8⋊C4).31C2, (C2×C4).328(C22⋊C4), C22.122(C2×C22⋊C4), SmallGroup(128,495)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊C42
G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, cd=dc >
Subgroups: 268 in 162 conjugacy classes, 84 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C8⋊C4, Q8⋊C4, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×C8, C22×Q8, C22.4Q16, C4×C4⋊C4, C2×C8⋊C4, C2×Q8⋊C4, C2×C4×Q8, Q8⋊C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C8⋊C22, C8.C22, C4×C22⋊C4, C23.36D4, C23.38D4, SD16⋊C4, Q16⋊C4, Q8⋊C42
►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 98 3 100)(2 97 4 99)(5 96 7 94)(6 95 8 93)(9 41 11 43)(10 44 12 42)(13 90 15 92)(14 89 16 91)(17 88 19 86)(18 87 20 85)(21 81 23 83)(22 84 24 82)(25 77 27 79)(26 80 28 78)(29 70 31 72)(30 69 32 71)(33 76 35 74)(34 75 36 73)(37 126 39 128)(38 125 40 127)(45 124 47 122)(46 123 48 121)(49 118 51 120)(50 117 52 119)(53 115 55 113)(54 114 56 116)(57 111 59 109)(58 110 60 112)(61 104 63 102)(62 103 64 101)(65 105 67 107)(66 108 68 106)
(1 123 14 11)(2 122 15 10)(3 121 16 9)(4 124 13 12)(5 118 17 126)(6 117 18 125)(7 120 19 128)(8 119 20 127)(21 106 33 115)(22 105 34 114)(23 108 35 113)(24 107 36 116)(25 103 30 111)(26 102 31 110)(27 101 32 109)(28 104 29 112)(37 95 49 87)(38 94 50 86)(39 93 51 85)(40 96 52 88)(41 97 46 92)(42 100 47 91)(43 99 48 90)(44 98 45 89)(53 84 68 75)(54 83 65 74)(55 82 66 73)(56 81 67 76)(57 80 62 72)(58 79 63 71)(59 78 64 70)(60 77 61 69)
(1 21 6 28)(2 22 7 25)(3 23 8 26)(4 24 5 27)(9 113 127 110)(10 114 128 111)(11 115 125 112)(12 116 126 109)(13 36 17 32)(14 33 18 29)(15 34 19 30)(16 35 20 31)(37 57 44 54)(38 58 41 55)(39 59 42 56)(40 60 43 53)(45 65 49 62)(46 66 50 63)(47 67 51 64)(48 68 52 61)(69 90 75 88)(70 91 76 85)(71 92 73 86)(72 89 74 87)(77 99 84 96)(78 100 81 93)(79 97 82 94)(80 98 83 95)(101 124 107 118)(102 121 108 119)(103 122 105 120)(104 123 106 117)
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), (1,98,3,100)(2,97,4,99)(5,96,7,94)(6,95,8,93)(9,41,11,43)(10,44,12,42)(13,90,15,92)(14,89,16,91)(17,88,19,86)(18,87,20,85)(21,81,23,83)(22,84,24,82)(25,77,27,79)(26,80,28,78)(29,70,31,72)(30,69,32,71)(33,76,35,74)(34,75,36,73)(37,126,39,128)(38,125,40,127)(45,124,47,122)(46,123,48,121)(49,118,51,120)(50,117,52,119)(53,115,55,113)(54,114,56,116)(57,111,59,109)(58,110,60,112)(61,104,63,102)(62,103,64,101)(65,105,67,107)(66,108,68,106), (1,123,14,11)(2,122,15,10)(3,121,16,9)(4,124,13,12)(5,118,17,126)(6,117,18,125)(7,120,19,128)(8,119,20,127)(21,106,33,115)(22,105,34,114)(23,108,35,113)(24,107,36,116)(25,103,30,111)(26,102,31,110)(27,101,32,109)(28,104,29,112)(37,95,49,87)(38,94,50,86)(39,93,51,85)(40,96,52,88)(41,97,46,92)(42,100,47,91)(43,99,48,90)(44,98,45,89)(53,84,68,75)(54,83,65,74)(55,82,66,73)(56,81,67,76)(57,80,62,72)(58,79,63,71)(59,78,64,70)(60,77,61,69), (1,21,6,28)(2,22,7,25)(3,23,8,26)(4,24,5,27)(9,113,127,110)(10,114,128,111)(11,115,125,112)(12,116,126,109)(13,36,17,32)(14,33,18,29)(15,34,19,30)(16,35,20,31)(37,57,44,54)(38,58,41,55)(39,59,42,56)(40,60,43,53)(45,65,49,62)(46,66,50,63)(47,67,51,64)(48,68,52,61)(69,90,75,88)(70,91,76,85)(71,92,73,86)(72,89,74,87)(77,99,84,96)(78,100,81,93)(79,97,82,94)(80,98,83,95)(101,124,107,118)(102,121,108,119)(103,122,105,120)(104,123,106,117)>;
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), (1,98,3,100)(2,97,4,99)(5,96,7,94)(6,95,8,93)(9,41,11,43)(10,44,12,42)(13,90,15,92)(14,89,16,91)(17,88,19,86)(18,87,20,85)(21,81,23,83)(22,84,24,82)(25,77,27,79)(26,80,28,78)(29,70,31,72)(30,69,32,71)(33,76,35,74)(34,75,36,73)(37,126,39,128)(38,125,40,127)(45,124,47,122)(46,123,48,121)(49,118,51,120)(50,117,52,119)(53,115,55,113)(54,114,56,116)(57,111,59,109)(58,110,60,112)(61,104,63,102)(62,103,64,101)(65,105,67,107)(66,108,68,106), (1,123,14,11)(2,122,15,10)(3,121,16,9)(4,124,13,12)(5,118,17,126)(6,117,18,125)(7,120,19,128)(8,119,20,127)(21,106,33,115)(22,105,34,114)(23,108,35,113)(24,107,36,116)(25,103,30,111)(26,102,31,110)(27,101,32,109)(28,104,29,112)(37,95,49,87)(38,94,50,86)(39,93,51,85)(40,96,52,88)(41,97,46,92)(42,100,47,91)(43,99,48,90)(44,98,45,89)(53,84,68,75)(54,83,65,74)(55,82,66,73)(56,81,67,76)(57,80,62,72)(58,79,63,71)(59,78,64,70)(60,77,61,69), (1,21,6,28)(2,22,7,25)(3,23,8,26)(4,24,5,27)(9,113,127,110)(10,114,128,111)(11,115,125,112)(12,116,126,109)(13,36,17,32)(14,33,18,29)(15,34,19,30)(16,35,20,31)(37,57,44,54)(38,58,41,55)(39,59,42,56)(40,60,43,53)(45,65,49,62)(46,66,50,63)(47,67,51,64)(48,68,52,61)(69,90,75,88)(70,91,76,85)(71,92,73,86)(72,89,74,87)(77,99,84,96)(78,100,81,93)(79,97,82,94)(80,98,83,95)(101,124,107,118)(102,121,108,119)(103,122,105,120)(104,123,106,117) );
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)], [(1,98,3,100),(2,97,4,99),(5,96,7,94),(6,95,8,93),(9,41,11,43),(10,44,12,42),(13,90,15,92),(14,89,16,91),(17,88,19,86),(18,87,20,85),(21,81,23,83),(22,84,24,82),(25,77,27,79),(26,80,28,78),(29,70,31,72),(30,69,32,71),(33,76,35,74),(34,75,36,73),(37,126,39,128),(38,125,40,127),(45,124,47,122),(46,123,48,121),(49,118,51,120),(50,117,52,119),(53,115,55,113),(54,114,56,116),(57,111,59,109),(58,110,60,112),(61,104,63,102),(62,103,64,101),(65,105,67,107),(66,108,68,106)], [(1,123,14,11),(2,122,15,10),(3,121,16,9),(4,124,13,12),(5,118,17,126),(6,117,18,125),(7,120,19,128),(8,119,20,127),(21,106,33,115),(22,105,34,114),(23,108,35,113),(24,107,36,116),(25,103,30,111),(26,102,31,110),(27,101,32,109),(28,104,29,112),(37,95,49,87),(38,94,50,86),(39,93,51,85),(40,96,52,88),(41,97,46,92),(42,100,47,91),(43,99,48,90),(44,98,45,89),(53,84,68,75),(54,83,65,74),(55,82,66,73),(56,81,67,76),(57,80,62,72),(58,79,63,71),(59,78,64,70),(60,77,61,69)], [(1,21,6,28),(2,22,7,25),(3,23,8,26),(4,24,5,27),(9,113,127,110),(10,114,128,111),(11,115,125,112),(12,116,126,109),(13,36,17,32),(14,33,18,29),(15,34,19,30),(16,35,20,31),(37,57,44,54),(38,58,41,55),(39,59,42,56),(40,60,43,53),(45,65,49,62),(46,66,50,63),(47,67,51,64),(48,68,52,61),(69,90,75,88),(70,91,76,85),(71,92,73,86),(72,89,74,87),(77,99,84,96),(78,100,81,93),(79,97,82,94),(80,98,83,95),(101,124,107,118),(102,121,108,119),(103,122,105,120),(104,123,106,117)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4AB | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | Q8⋊C42 | C22.4Q16 | C4×C4⋊C4 | C2×C8⋊C4 | C2×Q8⋊C4 | C2×C4×Q8 | Q8⋊C4 | C4×Q8 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 8 | 2 | 2 | 4 | 1 | 3 |
Matrix representation of Q8⋊C42 ►in GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 10 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 1 |
| 8 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 16 |
| 0 | 0 | 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[10,7,0,0,0,0,0,0,15,7,0,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,16,6,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,0,7,1],[8,4,0,0,0,0,0,0,5,9,0,0,0,0,0,0,0,0,1,11,0,0,0,0,0,0,6,16,0,0,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0] >;
Q8⋊C42 in GAP, Magma, Sage, TeX
Q_8\rtimes C_4^2
% in TeX
G:=Group("Q8:C4^2"); // GroupNames label
G:=SmallGroup(128,495);
// by ID
G=gap.SmallGroup(128,495);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,436,2019,248,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=d^4=1,b^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,c*d=d*c>;
// generators/relations