p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊9Q16, (C2×Q16)⋊6C4, C4.82(C4×D4), C2.7(C4×Q16), C4⋊C4.311D4, C4.2(C4⋊D4), (C2×Q8).162D4, C2.2(C4⋊2Q16), (C22×C4).688D4, C23.769(C2×D4), C22.152(C4×D4), Q8.2(C22⋊C4), (C22×Q16).1C2, C22.32(C2×Q16), C2.7(Q16⋊C4), C22.94C22≀C2, C2.7(D4.7D4), C22.56(C4○D8), (C22×C8).36C22, C2.5(C22⋊Q16), C2.2(Q8.D4), C22.4Q16.34C2, (C2×C42).278C22, C22.115(C4⋊D4), (C22×C4).1365C23, C4.64(C22.D4), C22.66(C8.C22), (C22×Q8).387C22, C2.17(C23.23D4), C22.7C42.19C2, C23.67C23.5C2, (C2×C4×Q8).15C2, (C2×C8).37(C2×C4), (C2×C4).997(C2×D4), C4.12(C2×C22⋊C4), (C2×Q8).64(C2×C4), (C2×Q8⋊C4).3C2, (C2×C4).562(C4○D4), (C2×C4⋊C4).766C22, (C2×C4).383(C22×C4), SmallGroup(128,610)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊9Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=ab-1, bd=db, dcd-1=c-1 >
Subgroups: 308 in 169 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, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×C8, C2×Q16, C2×Q16, C22×Q8, C22.7C42, C22.4Q16, C23.67C23, C2×Q8⋊C4, C2×C4×Q8, C22×Q16, (C2×C4)⋊9Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, Q16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×Q16, C4○D8, C8.C22, C23.23D4, C4×Q16, Q16⋊C4, C22⋊Q16, D4.7D4, C4⋊2Q16, Q8.D4, (C2×C4)⋊9Q16
►Regular action on 128 points
Generators in S128
(1 125)(2 126)(3 127)(4 128)(5 121)(6 122)(7 123)(8 124)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 65)(16 66)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 81)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 73)(33 95)(34 96)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 102)(42 103)(43 104)(44 97)(45 98)(46 99)(47 100)(48 101)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 105)(56 106)(57 118)(58 119)(59 120)(60 113)(61 114)(62 115)(63 116)(64 117)
(1 35 116 19)(2 85 117 90)(3 37 118 21)(4 87 119 92)(5 39 120 23)(6 81 113 94)(7 33 114 17)(8 83 115 96)(9 100 50 75)(10 27 51 48)(11 102 52 77)(12 29 53 42)(13 104 54 79)(14 31 55 44)(15 98 56 73)(16 25 49 46)(18 62 34 124)(20 64 36 126)(22 58 38 128)(24 60 40 122)(26 67 47 108)(28 69 41 110)(30 71 43 112)(32 65 45 106)(57 86 127 91)(59 88 121 93)(61 82 123 95)(63 84 125 89)(66 74 107 99)(68 76 109 101)(70 78 111 103)(72 80 105 97)
(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 51 5 55)(2 50 6 54)(3 49 7 53)(4 56 8 52)(9 113 13 117)(10 120 14 116)(11 119 15 115)(12 118 16 114)(17 29 21 25)(18 28 22 32)(19 27 23 31)(20 26 24 30)(33 42 37 46)(34 41 38 45)(35 48 39 44)(36 47 40 43)(57 66 61 70)(58 65 62 69)(59 72 63 68)(60 71 64 67)(73 83 77 87)(74 82 78 86)(75 81 79 85)(76 88 80 84)(89 101 93 97)(90 100 94 104)(91 99 95 103)(92 98 96 102)(105 125 109 121)(106 124 110 128)(107 123 111 127)(108 122 112 126)
G:=sub<Sym(128)| (1,125)(2,126)(3,127)(4,128)(5,121)(6,122)(7,123)(8,124)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,95)(34,96)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,102)(42,103)(43,104)(44,97)(45,98)(46,99)(47,100)(48,101)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,105)(56,106)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117), (1,35,116,19)(2,85,117,90)(3,37,118,21)(4,87,119,92)(5,39,120,23)(6,81,113,94)(7,33,114,17)(8,83,115,96)(9,100,50,75)(10,27,51,48)(11,102,52,77)(12,29,53,42)(13,104,54,79)(14,31,55,44)(15,98,56,73)(16,25,49,46)(18,62,34,124)(20,64,36,126)(22,58,38,128)(24,60,40,122)(26,67,47,108)(28,69,41,110)(30,71,43,112)(32,65,45,106)(57,86,127,91)(59,88,121,93)(61,82,123,95)(63,84,125,89)(66,74,107,99)(68,76,109,101)(70,78,111,103)(72,80,105,97), (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), (1,51,5,55)(2,50,6,54)(3,49,7,53)(4,56,8,52)(9,113,13,117)(10,120,14,116)(11,119,15,115)(12,118,16,114)(17,29,21,25)(18,28,22,32)(19,27,23,31)(20,26,24,30)(33,42,37,46)(34,41,38,45)(35,48,39,44)(36,47,40,43)(57,66,61,70)(58,65,62,69)(59,72,63,68)(60,71,64,67)(73,83,77,87)(74,82,78,86)(75,81,79,85)(76,88,80,84)(89,101,93,97)(90,100,94,104)(91,99,95,103)(92,98,96,102)(105,125,109,121)(106,124,110,128)(107,123,111,127)(108,122,112,126)>;
G:=Group( (1,125)(2,126)(3,127)(4,128)(5,121)(6,122)(7,123)(8,124)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,95)(34,96)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,102)(42,103)(43,104)(44,97)(45,98)(46,99)(47,100)(48,101)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,105)(56,106)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117), (1,35,116,19)(2,85,117,90)(3,37,118,21)(4,87,119,92)(5,39,120,23)(6,81,113,94)(7,33,114,17)(8,83,115,96)(9,100,50,75)(10,27,51,48)(11,102,52,77)(12,29,53,42)(13,104,54,79)(14,31,55,44)(15,98,56,73)(16,25,49,46)(18,62,34,124)(20,64,36,126)(22,58,38,128)(24,60,40,122)(26,67,47,108)(28,69,41,110)(30,71,43,112)(32,65,45,106)(57,86,127,91)(59,88,121,93)(61,82,123,95)(63,84,125,89)(66,74,107,99)(68,76,109,101)(70,78,111,103)(72,80,105,97), (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), (1,51,5,55)(2,50,6,54)(3,49,7,53)(4,56,8,52)(9,113,13,117)(10,120,14,116)(11,119,15,115)(12,118,16,114)(17,29,21,25)(18,28,22,32)(19,27,23,31)(20,26,24,30)(33,42,37,46)(34,41,38,45)(35,48,39,44)(36,47,40,43)(57,66,61,70)(58,65,62,69)(59,72,63,68)(60,71,64,67)(73,83,77,87)(74,82,78,86)(75,81,79,85)(76,88,80,84)(89,101,93,97)(90,100,94,104)(91,99,95,103)(92,98,96,102)(105,125,109,121)(106,124,110,128)(107,123,111,127)(108,122,112,126) );
G=PermutationGroup([[(1,125),(2,126),(3,127),(4,128),(5,121),(6,122),(7,123),(8,124),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,65),(16,66),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,81),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,73),(33,95),(34,96),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,102),(42,103),(43,104),(44,97),(45,98),(46,99),(47,100),(48,101),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,105),(56,106),(57,118),(58,119),(59,120),(60,113),(61,114),(62,115),(63,116),(64,117)], [(1,35,116,19),(2,85,117,90),(3,37,118,21),(4,87,119,92),(5,39,120,23),(6,81,113,94),(7,33,114,17),(8,83,115,96),(9,100,50,75),(10,27,51,48),(11,102,52,77),(12,29,53,42),(13,104,54,79),(14,31,55,44),(15,98,56,73),(16,25,49,46),(18,62,34,124),(20,64,36,126),(22,58,38,128),(24,60,40,122),(26,67,47,108),(28,69,41,110),(30,71,43,112),(32,65,45,106),(57,86,127,91),(59,88,121,93),(61,82,123,95),(63,84,125,89),(66,74,107,99),(68,76,109,101),(70,78,111,103),(72,80,105,97)], [(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)], [(1,51,5,55),(2,50,6,54),(3,49,7,53),(4,56,8,52),(9,113,13,117),(10,120,14,116),(11,119,15,115),(12,118,16,114),(17,29,21,25),(18,28,22,32),(19,27,23,31),(20,26,24,30),(33,42,37,46),(34,41,38,45),(35,48,39,44),(36,47,40,43),(57,66,61,70),(58,65,62,69),(59,72,63,68),(60,71,64,67),(73,83,77,87),(74,82,78,86),(75,81,79,85),(76,88,80,84),(89,101,93,97),(90,100,94,104),(91,99,95,103),(92,98,96,102),(105,125,109,121),(106,124,110,128),(107,123,111,127),(108,122,112,126)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q16 | C4○D4 | C4○D8 | C8.C22 |
| kernel | (C2×C4)⋊9Q16 | C22.7C42 | C22.4Q16 | C23.67C23 | C2×Q8⋊C4 | C2×C4×Q8 | C22×Q16 | C2×Q16 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 2 |
Matrix representation of (C2×C4)⋊9Q16 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 11 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 0 | 7 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,10,14,0,0,0,0,11,7,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,0,0,0,1,7,0,0,0,0,7,16] >;
(C2×C4)⋊9Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_9Q_{16} % in TeX
G:=Group("(C2xC4):9Q16"); // GroupNames label
G:=SmallGroup(128,610);
// by ID
G=gap.SmallGroup(128,610);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations