p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊4(C4⋊C4), C4.54(C4×D4), C2.6(C4×Q16), C4⋊C4.305D4, Q8⋊C4⋊3C4, (C2×Q8).20Q8, (C2×C4).47Q16, (C2×Q8).214D4, C2.2(Q8.Q8), C2.3(D4⋊D4), C2.2(C4.Q16), C23.761(C2×D4), (C22×C4).684D4, C22.145(C4×D4), C4.26(C22⋊Q8), C22.30(C2×Q16), C22.85C22≀C2, C22.50(C4○D8), C22.4Q16.5C2, C2.3(C22⋊Q16), C2.9(SD16⋊C4), C22.71(C8⋊C22), (C2×C42).269C22, (C22×C8).103C22, C4.7(C22.D4), C22.72(C22⋊Q8), (C22×C4).1354C23, C22.60(C8.C22), C2.19(C23.8Q8), (C22×Q8).385C22, C23.65C23.2C2, C22.7C42.18C2, C4.13(C2×C4⋊C4), (C2×C4×Q8).13C2, C4⋊C4.65(C2×C4), (C2×C8).35(C2×C4), (C2×C2.D8).4C2, (C2×C4).989(C2×D4), (C2×C4).265(C2×Q8), (C2×Q8).143(C2×C4), (C2×C4).750(C4○D4), (C2×C4⋊C4).758C22, (C2×C4).372(C22×C4), (C2×Q8⋊C4).19C2, SmallGroup(128,595)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊(C4⋊C4)
G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 268 in 149 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, Q8⋊C4, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C23.65C23, C2×Q8⋊C4, C2×C2.D8, C2×C4×Q8, Q8⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×Q16, C4○D8, C8⋊C22, C8.C22, C23.8Q8, C4×Q16, SD16⋊C4, D4⋊D4, C22⋊Q16, C4.Q16, Q8.Q8, Q8⋊(C4⋊C4)
►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 24 3 22)(2 23 4 21)(5 115 7 113)(6 114 8 116)(9 28 11 26)(10 27 12 25)(13 29 15 31)(14 32 16 30)(17 35 19 33)(18 34 20 36)(37 53 39 55)(38 56 40 54)(41 60 43 58)(42 59 44 57)(45 61 47 63)(46 64 48 62)(49 68 51 66)(50 67 52 65)(69 88 71 86)(70 87 72 85)(73 91 75 89)(74 90 76 92)(77 96 79 94)(78 95 80 93)(81 100 83 98)(82 99 84 97)(101 120 103 118)(102 119 104 117)(105 123 107 121)(106 122 108 124)(109 128 111 126)(110 127 112 125)
(1 5 13 121)(2 8 14 124)(3 7 15 123)(4 6 16 122)(9 126 19 117)(10 125 20 120)(11 128 17 119)(12 127 18 118)(21 113 30 107)(22 116 31 106)(23 115 32 105)(24 114 29 108)(25 111 34 104)(26 110 35 103)(27 109 36 102)(28 112 33 101)(37 95 46 88)(38 94 47 87)(39 93 48 86)(40 96 45 85)(41 98 51 91)(42 97 52 90)(43 100 49 89)(44 99 50 92)(53 79 64 70)(54 78 61 69)(55 77 62 72)(56 80 63 71)(57 83 67 73)(58 82 68 76)(59 81 65 75)(60 84 66 74)
(1 58 11 53)(2 59 12 54)(3 60 9 55)(4 57 10 56)(5 76 128 70)(6 73 125 71)(7 74 126 72)(8 75 127 69)(13 68 17 64)(14 65 18 61)(15 66 19 62)(16 67 20 63)(21 44 27 38)(22 41 28 39)(23 42 25 40)(24 43 26 37)(29 49 35 46)(30 50 36 47)(31 51 33 48)(32 52 34 45)(77 123 84 117)(78 124 81 118)(79 121 82 119)(80 122 83 120)(85 115 90 111)(86 116 91 112)(87 113 92 109)(88 114 89 110)(93 106 98 101)(94 107 99 102)(95 108 100 103)(96 105 97 104)
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), (1,24,3,22)(2,23,4,21)(5,115,7,113)(6,114,8,116)(9,28,11,26)(10,27,12,25)(13,29,15,31)(14,32,16,30)(17,35,19,33)(18,34,20,36)(37,53,39,55)(38,56,40,54)(41,60,43,58)(42,59,44,57)(45,61,47,63)(46,64,48,62)(49,68,51,66)(50,67,52,65)(69,88,71,86)(70,87,72,85)(73,91,75,89)(74,90,76,92)(77,96,79,94)(78,95,80,93)(81,100,83,98)(82,99,84,97)(101,120,103,118)(102,119,104,117)(105,123,107,121)(106,122,108,124)(109,128,111,126)(110,127,112,125), (1,5,13,121)(2,8,14,124)(3,7,15,123)(4,6,16,122)(9,126,19,117)(10,125,20,120)(11,128,17,119)(12,127,18,118)(21,113,30,107)(22,116,31,106)(23,115,32,105)(24,114,29,108)(25,111,34,104)(26,110,35,103)(27,109,36,102)(28,112,33,101)(37,95,46,88)(38,94,47,87)(39,93,48,86)(40,96,45,85)(41,98,51,91)(42,97,52,90)(43,100,49,89)(44,99,50,92)(53,79,64,70)(54,78,61,69)(55,77,62,72)(56,80,63,71)(57,83,67,73)(58,82,68,76)(59,81,65,75)(60,84,66,74), (1,58,11,53)(2,59,12,54)(3,60,9,55)(4,57,10,56)(5,76,128,70)(6,73,125,71)(7,74,126,72)(8,75,127,69)(13,68,17,64)(14,65,18,61)(15,66,19,62)(16,67,20,63)(21,44,27,38)(22,41,28,39)(23,42,25,40)(24,43,26,37)(29,49,35,46)(30,50,36,47)(31,51,33,48)(32,52,34,45)(77,123,84,117)(78,124,81,118)(79,121,82,119)(80,122,83,120)(85,115,90,111)(86,116,91,112)(87,113,92,109)(88,114,89,110)(93,106,98,101)(94,107,99,102)(95,108,100,103)(96,105,97,104)>;
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), (1,24,3,22)(2,23,4,21)(5,115,7,113)(6,114,8,116)(9,28,11,26)(10,27,12,25)(13,29,15,31)(14,32,16,30)(17,35,19,33)(18,34,20,36)(37,53,39,55)(38,56,40,54)(41,60,43,58)(42,59,44,57)(45,61,47,63)(46,64,48,62)(49,68,51,66)(50,67,52,65)(69,88,71,86)(70,87,72,85)(73,91,75,89)(74,90,76,92)(77,96,79,94)(78,95,80,93)(81,100,83,98)(82,99,84,97)(101,120,103,118)(102,119,104,117)(105,123,107,121)(106,122,108,124)(109,128,111,126)(110,127,112,125), (1,5,13,121)(2,8,14,124)(3,7,15,123)(4,6,16,122)(9,126,19,117)(10,125,20,120)(11,128,17,119)(12,127,18,118)(21,113,30,107)(22,116,31,106)(23,115,32,105)(24,114,29,108)(25,111,34,104)(26,110,35,103)(27,109,36,102)(28,112,33,101)(37,95,46,88)(38,94,47,87)(39,93,48,86)(40,96,45,85)(41,98,51,91)(42,97,52,90)(43,100,49,89)(44,99,50,92)(53,79,64,70)(54,78,61,69)(55,77,62,72)(56,80,63,71)(57,83,67,73)(58,82,68,76)(59,81,65,75)(60,84,66,74), (1,58,11,53)(2,59,12,54)(3,60,9,55)(4,57,10,56)(5,76,128,70)(6,73,125,71)(7,74,126,72)(8,75,127,69)(13,68,17,64)(14,65,18,61)(15,66,19,62)(16,67,20,63)(21,44,27,38)(22,41,28,39)(23,42,25,40)(24,43,26,37)(29,49,35,46)(30,50,36,47)(31,51,33,48)(32,52,34,45)(77,123,84,117)(78,124,81,118)(79,121,82,119)(80,122,83,120)(85,115,90,111)(86,116,91,112)(87,113,92,109)(88,114,89,110)(93,106,98,101)(94,107,99,102)(95,108,100,103)(96,105,97,104) );
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)], [(1,24,3,22),(2,23,4,21),(5,115,7,113),(6,114,8,116),(9,28,11,26),(10,27,12,25),(13,29,15,31),(14,32,16,30),(17,35,19,33),(18,34,20,36),(37,53,39,55),(38,56,40,54),(41,60,43,58),(42,59,44,57),(45,61,47,63),(46,64,48,62),(49,68,51,66),(50,67,52,65),(69,88,71,86),(70,87,72,85),(73,91,75,89),(74,90,76,92),(77,96,79,94),(78,95,80,93),(81,100,83,98),(82,99,84,97),(101,120,103,118),(102,119,104,117),(105,123,107,121),(106,122,108,124),(109,128,111,126),(110,127,112,125)], [(1,5,13,121),(2,8,14,124),(3,7,15,123),(4,6,16,122),(9,126,19,117),(10,125,20,120),(11,128,17,119),(12,127,18,118),(21,113,30,107),(22,116,31,106),(23,115,32,105),(24,114,29,108),(25,111,34,104),(26,110,35,103),(27,109,36,102),(28,112,33,101),(37,95,46,88),(38,94,47,87),(39,93,48,86),(40,96,45,85),(41,98,51,91),(42,97,52,90),(43,100,49,89),(44,99,50,92),(53,79,64,70),(54,78,61,69),(55,77,62,72),(56,80,63,71),(57,83,67,73),(58,82,68,76),(59,81,65,75),(60,84,66,74)], [(1,58,11,53),(2,59,12,54),(3,60,9,55),(4,57,10,56),(5,76,128,70),(6,73,125,71),(7,74,126,72),(8,75,127,69),(13,68,17,64),(14,65,18,61),(15,66,19,62),(16,67,20,63),(21,44,27,38),(22,41,28,39),(23,42,25,40),(24,43,26,37),(29,49,35,46),(30,50,36,47),(31,51,33,48),(32,52,34,45),(77,123,84,117),(78,124,81,118),(79,121,82,119),(80,122,83,120),(85,115,90,111),(86,116,91,112),(87,113,92,109),(88,114,89,110),(93,106,98,101),(94,107,99,102),(95,108,100,103),(96,105,97,104)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | Q16 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | Q8⋊(C4⋊C4) | C22.7C42 | C22.4Q16 | C23.65C23 | C2×Q8⋊C4 | C2×C2.D8 | C2×C4×Q8 | Q8⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of Q8⋊(C4⋊C4) ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 12 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 16 | 2 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,13,4,0,0,0,0,4],[16,0,0,0,0,0,13,0,0,0,0,8,4,0,0,0,0,0,0,12,0,0,0,7,0],[13,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,13,4,0,0,0,9,4] >;
Q8⋊(C4⋊C4) in GAP, Magma, Sage, TeX
Q_8\rtimes (C_4\rtimes C_4)
% in TeX
G:=Group("Q8:(C4:C4)"); // GroupNames label
G:=SmallGroup(128,595);
// by ID
G=gap.SmallGroup(128,595);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,352,2019,1018,521,248,2804,1411,718,172]);
// 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*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations