direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8×Q16, C42.523C23, C4.972- (1+4), (Q82).4C2, Q8.9(C2×Q8), (C8×Q8).9C2, C2.40(D4×Q8), C8.36(C2×Q8), C4⋊C4.282D4, (C4×Q16).9C2, C4.31(C2×Q16), C2.67(D4○D8), (C4×C8).96C22, (C2×Q8).272D4, C8⋊2Q8.16C2, C4.40(C22×Q8), C4⋊C4.271C23, C4⋊C8.305C22, (C2×C8).212C23, (C2×C4).574C24, C4.Q16.10C2, C4⋊Q8.203C22, C2.22(C22×Q16), C2.D8.73C22, (C4×Q8).202C22, (C2×Q8).409C23, (C2×Q16).172C22, C22.834(C22×D4), Q8⋊C4.164C22, (C2×C4).1104(C2×D4), SmallGroup(128,2114)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 280 in 168 conjugacy classes, 104 normal (14 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C4 [×13], C22, C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×8], Q8 [×8], C42 [×3], C42 [×6], C4⋊C4 [×9], C4⋊C4 [×12], C2×C8, C2×C8 [×3], Q16 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×6], C4×C8 [×3], Q8⋊C4 [×6], C4⋊C8 [×3], C2.D8 [×9], C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C4⋊Q8 [×6], C4⋊Q8 [×6], C2×Q16, C4×Q16 [×3], C8×Q8, C4.Q16 [×6], C8⋊2Q8 [×3], Q82 [×2], Q8×Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C2×Q8 [×6], C24, C2×Q16 [×6], C22×D4, C22×Q8, 2- (1+4), D4×Q8, C22×Q16, D4○D8, Q8×Q16
Generators and relations
G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 128 points
Generators in S128
(1 25 43 111)(2 26 44 112)(3 27 45 105)(4 28 46 106)(5 29 47 107)(6 30 48 108)(7 31 41 109)(8 32 42 110)(9 84 117 37)(10 85 118 38)(11 86 119 39)(12 87 120 40)(13 88 113 33)(14 81 114 34)(15 82 115 35)(16 83 116 36)(17 54 103 68)(18 55 104 69)(19 56 97 70)(20 49 98 71)(21 50 99 72)(22 51 100 65)(23 52 101 66)(24 53 102 67)(57 73 126 93)(58 74 127 94)(59 75 128 95)(60 76 121 96)(61 77 122 89)(62 78 123 90)(63 79 124 91)(64 80 125 92)
(1 78 43 90)(2 79 44 91)(3 80 45 92)(4 73 46 93)(5 74 47 94)(6 75 48 95)(7 76 41 96)(8 77 42 89)(9 97 117 19)(10 98 118 20)(11 99 119 21)(12 100 120 22)(13 101 113 23)(14 102 114 24)(15 103 115 17)(16 104 116 18)(25 62 111 123)(26 63 112 124)(27 64 105 125)(28 57 106 126)(29 58 107 127)(30 59 108 128)(31 60 109 121)(32 61 110 122)(33 66 88 52)(34 67 81 53)(35 68 82 54)(36 69 83 55)(37 70 84 56)(38 71 85 49)(39 72 86 50)(40 65 87 51)
(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 54 5 50)(2 53 6 49)(3 52 7 56)(4 51 8 55)(9 125 13 121)(10 124 14 128)(11 123 15 127)(12 122 16 126)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 103 29 99)(26 102 30 98)(27 101 31 97)(28 100 32 104)(33 76 37 80)(34 75 38 79)(35 74 39 78)(36 73 40 77)(41 70 45 66)(42 69 46 65)(43 68 47 72)(44 67 48 71)(57 120 61 116)(58 119 62 115)(59 118 63 114)(60 117 64 113)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)
G:=sub<Sym(128)| (1,25,43,111)(2,26,44,112)(3,27,45,105)(4,28,46,106)(5,29,47,107)(6,30,48,108)(7,31,41,109)(8,32,42,110)(9,84,117,37)(10,85,118,38)(11,86,119,39)(12,87,120,40)(13,88,113,33)(14,81,114,34)(15,82,115,35)(16,83,116,36)(17,54,103,68)(18,55,104,69)(19,56,97,70)(20,49,98,71)(21,50,99,72)(22,51,100,65)(23,52,101,66)(24,53,102,67)(57,73,126,93)(58,74,127,94)(59,75,128,95)(60,76,121,96)(61,77,122,89)(62,78,123,90)(63,79,124,91)(64,80,125,92), (1,78,43,90)(2,79,44,91)(3,80,45,92)(4,73,46,93)(5,74,47,94)(6,75,48,95)(7,76,41,96)(8,77,42,89)(9,97,117,19)(10,98,118,20)(11,99,119,21)(12,100,120,22)(13,101,113,23)(14,102,114,24)(15,103,115,17)(16,104,116,18)(25,62,111,123)(26,63,112,124)(27,64,105,125)(28,57,106,126)(29,58,107,127)(30,59,108,128)(31,60,109,121)(32,61,110,122)(33,66,88,52)(34,67,81,53)(35,68,82,54)(36,69,83,55)(37,70,84,56)(38,71,85,49)(39,72,86,50)(40,65,87,51), (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), (1,54,5,50)(2,53,6,49)(3,52,7,56)(4,51,8,55)(9,125,13,121)(10,124,14,128)(11,123,15,127)(12,122,16,126)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,103,29,99)(26,102,30,98)(27,101,31,97)(28,100,32,104)(33,76,37,80)(34,75,38,79)(35,74,39,78)(36,73,40,77)(41,70,45,66)(42,69,46,65)(43,68,47,72)(44,67,48,71)(57,120,61,116)(58,119,62,115)(59,118,63,114)(60,117,64,113)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96)>;
G:=Group( (1,25,43,111)(2,26,44,112)(3,27,45,105)(4,28,46,106)(5,29,47,107)(6,30,48,108)(7,31,41,109)(8,32,42,110)(9,84,117,37)(10,85,118,38)(11,86,119,39)(12,87,120,40)(13,88,113,33)(14,81,114,34)(15,82,115,35)(16,83,116,36)(17,54,103,68)(18,55,104,69)(19,56,97,70)(20,49,98,71)(21,50,99,72)(22,51,100,65)(23,52,101,66)(24,53,102,67)(57,73,126,93)(58,74,127,94)(59,75,128,95)(60,76,121,96)(61,77,122,89)(62,78,123,90)(63,79,124,91)(64,80,125,92), (1,78,43,90)(2,79,44,91)(3,80,45,92)(4,73,46,93)(5,74,47,94)(6,75,48,95)(7,76,41,96)(8,77,42,89)(9,97,117,19)(10,98,118,20)(11,99,119,21)(12,100,120,22)(13,101,113,23)(14,102,114,24)(15,103,115,17)(16,104,116,18)(25,62,111,123)(26,63,112,124)(27,64,105,125)(28,57,106,126)(29,58,107,127)(30,59,108,128)(31,60,109,121)(32,61,110,122)(33,66,88,52)(34,67,81,53)(35,68,82,54)(36,69,83,55)(37,70,84,56)(38,71,85,49)(39,72,86,50)(40,65,87,51), (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), (1,54,5,50)(2,53,6,49)(3,52,7,56)(4,51,8,55)(9,125,13,121)(10,124,14,128)(11,123,15,127)(12,122,16,126)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,103,29,99)(26,102,30,98)(27,101,31,97)(28,100,32,104)(33,76,37,80)(34,75,38,79)(35,74,39,78)(36,73,40,77)(41,70,45,66)(42,69,46,65)(43,68,47,72)(44,67,48,71)(57,120,61,116)(58,119,62,115)(59,118,63,114)(60,117,64,113)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96) );
G=PermutationGroup([(1,25,43,111),(2,26,44,112),(3,27,45,105),(4,28,46,106),(5,29,47,107),(6,30,48,108),(7,31,41,109),(8,32,42,110),(9,84,117,37),(10,85,118,38),(11,86,119,39),(12,87,120,40),(13,88,113,33),(14,81,114,34),(15,82,115,35),(16,83,116,36),(17,54,103,68),(18,55,104,69),(19,56,97,70),(20,49,98,71),(21,50,99,72),(22,51,100,65),(23,52,101,66),(24,53,102,67),(57,73,126,93),(58,74,127,94),(59,75,128,95),(60,76,121,96),(61,77,122,89),(62,78,123,90),(63,79,124,91),(64,80,125,92)], [(1,78,43,90),(2,79,44,91),(3,80,45,92),(4,73,46,93),(5,74,47,94),(6,75,48,95),(7,76,41,96),(8,77,42,89),(9,97,117,19),(10,98,118,20),(11,99,119,21),(12,100,120,22),(13,101,113,23),(14,102,114,24),(15,103,115,17),(16,104,116,18),(25,62,111,123),(26,63,112,124),(27,64,105,125),(28,57,106,126),(29,58,107,127),(30,59,108,128),(31,60,109,121),(32,61,110,122),(33,66,88,52),(34,67,81,53),(35,68,82,54),(36,69,83,55),(37,70,84,56),(38,71,85,49),(39,72,86,50),(40,65,87,51)], [(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)], [(1,54,5,50),(2,53,6,49),(3,52,7,56),(4,51,8,55),(9,125,13,121),(10,124,14,128),(11,123,15,127),(12,122,16,126),(17,107,21,111),(18,106,22,110),(19,105,23,109),(20,112,24,108),(25,103,29,99),(26,102,30,98),(27,101,31,97),(28,100,32,104),(33,76,37,80),(34,75,38,79),(35,74,39,78),(36,73,40,77),(41,70,45,66),(42,69,46,65),(43,68,47,72),(44,67,48,71),(57,120,61,116),(58,119,62,115),(59,118,63,114),(60,117,64,113),(81,95,85,91),(82,94,86,90),(83,93,87,89),(84,92,88,96)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[5,12,0,0,12,12,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,0,12,0,0,7,0] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | ··· | 4O | 4P | ··· | 4U | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | Q16 | 2- (1+4) | D4○D8 |
| kernel | Q8×Q16 | C4×Q16 | C8×Q8 | C4.Q16 | C8⋊2Q8 | Q82 | C4⋊C4 | Q16 | C2×Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 3 | 4 | 1 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_8\times Q_{16} % in TeX
G:=Group("Q8xQ16"); // GroupNames label
G:=SmallGroup(128,2114);
// by ID
G=gap.SmallGroup(128,2114);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=1,b^2=a^2,d^2=c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀