p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊5(C4⋊C4), C4.56(C4×D4), C4⋊C4.307D4, Q8⋊C4⋊6C4, (C2×Q8).21Q8, C2.9(C4×SD16), (C2×Q8).215D4, C2.3(Q8.Q8), C2.3(Q8⋊D4), C2.2(Q8⋊Q8), (C2×C4).110SD16, C22.147(C4×D4), (C22×C4).686D4, C23.763(C2×D4), C4.28(C22⋊Q8), C2.6(Q16⋊C4), C22.87C22≀C2, C2.4(D4.7D4), C22.52(C4○D8), C22.4Q16.6C2, C22.55(C2×SD16), (C22×C8).314C22, (C2×C42).271C22, C4.9(C22.D4), C22.74(C22⋊Q8), (C22×C4).1356C23, C22.61(C8.C22), C2.21(C23.8Q8), (C22×Q8).386C22, C22.7C42.31C2, C23.65C23.3C2, C4.15(C2×C4⋊C4), (C2×C4×Q8).14C2, C4⋊C4.66(C2×C4), (C2×C8).107(C2×C4), (C2×C4).991(C2×D4), (C2×C4).267(C2×Q8), (C2×C4.Q8).15C2, (C2×Q8).144(C2×C4), (C2×C4).752(C4○D4), (C2×C4⋊C4).760C22, (C2×C4).374(C22×C4), (C2×Q8⋊C4).27C2, SmallGroup(128,597)
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=dad-1=a-1, cbc-1=a-1b, bd=db, dcd-1=ac-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, C4.Q8, 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×C4.Q8, C2×C4×Q8, Q8⋊C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×SD16, C4○D8, C8.C22, C23.8Q8, C4×SD16, Q16⋊C4, Q8⋊D4, D4.7D4, Q8⋊Q8, 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 33 3 35)(2 36 4 34)(5 104 7 102)(6 103 8 101)(9 31 11 29)(10 30 12 32)(13 26 15 28)(14 25 16 27)(17 24 19 22)(18 23 20 21)(37 68 39 66)(38 67 40 65)(41 62 43 64)(42 61 44 63)(45 59 47 57)(46 58 48 60)(49 53 51 55)(50 56 52 54)(69 100 71 98)(70 99 72 97)(73 93 75 95)(74 96 76 94)(77 90 79 92)(78 89 80 91)(81 88 83 86)(82 87 84 85)(105 126 107 128)(106 125 108 127)(109 121 111 123)(110 124 112 122)(113 119 115 117)(114 118 116 120)
(1 125 13 118)(2 128 14 117)(3 127 15 120)(4 126 16 119)(5 18 123 10)(6 17 124 9)(7 20 121 12)(8 19 122 11)(21 112 32 103)(22 111 29 102)(23 110 30 101)(24 109 31 104)(25 114 36 106)(26 113 33 105)(27 116 34 108)(28 115 35 107)(37 99 48 90)(38 98 45 89)(39 97 46 92)(40 100 47 91)(41 96 49 87)(42 95 50 86)(43 94 51 85)(44 93 52 88)(53 81 62 73)(54 84 63 76)(55 83 64 75)(56 82 61 74)(57 79 65 72)(58 78 66 71)(59 77 67 70)(60 80 68 69)
(1 42 9 40)(2 41 10 39)(3 44 11 38)(4 43 12 37)(5 91 128 86)(6 90 125 85)(7 89 126 88)(8 92 127 87)(13 50 17 47)(14 49 18 46)(15 52 19 45)(16 51 20 48)(21 60 27 55)(22 59 28 54)(23 58 25 53)(24 57 26 56)(29 67 35 63)(30 66 36 62)(31 65 33 61)(32 68 34 64)(69 115 75 111)(70 114 76 110)(71 113 73 109)(72 116 74 112)(77 106 84 101)(78 105 81 104)(79 108 82 103)(80 107 83 102)(93 121 98 119)(94 124 99 118)(95 123 100 117)(96 122 97 120)
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), (1,33,3,35)(2,36,4,34)(5,104,7,102)(6,103,8,101)(9,31,11,29)(10,30,12,32)(13,26,15,28)(14,25,16,27)(17,24,19,22)(18,23,20,21)(37,68,39,66)(38,67,40,65)(41,62,43,64)(42,61,44,63)(45,59,47,57)(46,58,48,60)(49,53,51,55)(50,56,52,54)(69,100,71,98)(70,99,72,97)(73,93,75,95)(74,96,76,94)(77,90,79,92)(78,89,80,91)(81,88,83,86)(82,87,84,85)(105,126,107,128)(106,125,108,127)(109,121,111,123)(110,124,112,122)(113,119,115,117)(114,118,116,120), (1,125,13,118)(2,128,14,117)(3,127,15,120)(4,126,16,119)(5,18,123,10)(6,17,124,9)(7,20,121,12)(8,19,122,11)(21,112,32,103)(22,111,29,102)(23,110,30,101)(24,109,31,104)(25,114,36,106)(26,113,33,105)(27,116,34,108)(28,115,35,107)(37,99,48,90)(38,98,45,89)(39,97,46,92)(40,100,47,91)(41,96,49,87)(42,95,50,86)(43,94,51,85)(44,93,52,88)(53,81,62,73)(54,84,63,76)(55,83,64,75)(56,82,61,74)(57,79,65,72)(58,78,66,71)(59,77,67,70)(60,80,68,69), (1,42,9,40)(2,41,10,39)(3,44,11,38)(4,43,12,37)(5,91,128,86)(6,90,125,85)(7,89,126,88)(8,92,127,87)(13,50,17,47)(14,49,18,46)(15,52,19,45)(16,51,20,48)(21,60,27,55)(22,59,28,54)(23,58,25,53)(24,57,26,56)(29,67,35,63)(30,66,36,62)(31,65,33,61)(32,68,34,64)(69,115,75,111)(70,114,76,110)(71,113,73,109)(72,116,74,112)(77,106,84,101)(78,105,81,104)(79,108,82,103)(80,107,83,102)(93,121,98,119)(94,124,99,118)(95,123,100,117)(96,122,97,120)>;
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), (1,33,3,35)(2,36,4,34)(5,104,7,102)(6,103,8,101)(9,31,11,29)(10,30,12,32)(13,26,15,28)(14,25,16,27)(17,24,19,22)(18,23,20,21)(37,68,39,66)(38,67,40,65)(41,62,43,64)(42,61,44,63)(45,59,47,57)(46,58,48,60)(49,53,51,55)(50,56,52,54)(69,100,71,98)(70,99,72,97)(73,93,75,95)(74,96,76,94)(77,90,79,92)(78,89,80,91)(81,88,83,86)(82,87,84,85)(105,126,107,128)(106,125,108,127)(109,121,111,123)(110,124,112,122)(113,119,115,117)(114,118,116,120), (1,125,13,118)(2,128,14,117)(3,127,15,120)(4,126,16,119)(5,18,123,10)(6,17,124,9)(7,20,121,12)(8,19,122,11)(21,112,32,103)(22,111,29,102)(23,110,30,101)(24,109,31,104)(25,114,36,106)(26,113,33,105)(27,116,34,108)(28,115,35,107)(37,99,48,90)(38,98,45,89)(39,97,46,92)(40,100,47,91)(41,96,49,87)(42,95,50,86)(43,94,51,85)(44,93,52,88)(53,81,62,73)(54,84,63,76)(55,83,64,75)(56,82,61,74)(57,79,65,72)(58,78,66,71)(59,77,67,70)(60,80,68,69), (1,42,9,40)(2,41,10,39)(3,44,11,38)(4,43,12,37)(5,91,128,86)(6,90,125,85)(7,89,126,88)(8,92,127,87)(13,50,17,47)(14,49,18,46)(15,52,19,45)(16,51,20,48)(21,60,27,55)(22,59,28,54)(23,58,25,53)(24,57,26,56)(29,67,35,63)(30,66,36,62)(31,65,33,61)(32,68,34,64)(69,115,75,111)(70,114,76,110)(71,113,73,109)(72,116,74,112)(77,106,84,101)(78,105,81,104)(79,108,82,103)(80,107,83,102)(93,121,98,119)(94,124,99,118)(95,123,100,117)(96,122,97,120) );
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)], [(1,33,3,35),(2,36,4,34),(5,104,7,102),(6,103,8,101),(9,31,11,29),(10,30,12,32),(13,26,15,28),(14,25,16,27),(17,24,19,22),(18,23,20,21),(37,68,39,66),(38,67,40,65),(41,62,43,64),(42,61,44,63),(45,59,47,57),(46,58,48,60),(49,53,51,55),(50,56,52,54),(69,100,71,98),(70,99,72,97),(73,93,75,95),(74,96,76,94),(77,90,79,92),(78,89,80,91),(81,88,83,86),(82,87,84,85),(105,126,107,128),(106,125,108,127),(109,121,111,123),(110,124,112,122),(113,119,115,117),(114,118,116,120)], [(1,125,13,118),(2,128,14,117),(3,127,15,120),(4,126,16,119),(5,18,123,10),(6,17,124,9),(7,20,121,12),(8,19,122,11),(21,112,32,103),(22,111,29,102),(23,110,30,101),(24,109,31,104),(25,114,36,106),(26,113,33,105),(27,116,34,108),(28,115,35,107),(37,99,48,90),(38,98,45,89),(39,97,46,92),(40,100,47,91),(41,96,49,87),(42,95,50,86),(43,94,51,85),(44,93,52,88),(53,81,62,73),(54,84,63,76),(55,83,64,75),(56,82,61,74),(57,79,65,72),(58,78,66,71),(59,77,67,70),(60,80,68,69)], [(1,42,9,40),(2,41,10,39),(3,44,11,38),(4,43,12,37),(5,91,128,86),(6,90,125,85),(7,89,126,88),(8,92,127,87),(13,50,17,47),(14,49,18,46),(15,52,19,45),(16,51,20,48),(21,60,27,55),(22,59,28,54),(23,58,25,53),(24,57,26,56),(29,67,35,63),(30,66,36,62),(31,65,33,61),(32,68,34,64),(69,115,75,111),(70,114,76,110),(71,113,73,109),(72,116,74,112),(77,106,84,101),(78,105,81,104),(79,108,82,103),(80,107,83,102),(93,121,98,119),(94,124,99,118),(95,123,100,117),(96,122,97,120)]])
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 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | SD16 | C4○D4 | C4○D8 | C8.C22 |
| kernel | Q8⋊C4⋊C4 | C22.7C42 | C22.4Q16 | C23.65C23 | C2×Q8⋊C4 | C2×C4.Q8 | C2×C4×Q8 | Q8⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of Q8⋊C4⋊C4 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 |
| 0 | 0 | 0 | 0 | 7 | 2 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 15 |
| 0 | 0 | 0 | 0 | 7 | 9 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,5,0,0,0,0,7,7,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,15,7,0,0,0,0,9,2],[7,12,0,0,0,0,10,10,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,8,7,0,0,0,0,15,9] >;
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(128,597);
// by ID
G=gap.SmallGroup(128,597);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,352,2019,1018,521,248,2804,1411,718,172]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations