p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊6Q16, (C2×Q16)⋊7C4, C4.18(C4×D4), (C2×C8).244D4, C2.12(C4×Q16), C4.82(C4⋊D4), C8.19(C22⋊C4), C2.3(C4⋊Q16), C4.2(C4.4D4), C22.182(C4×D4), (C22×C4).559D4, C23.802(C2×D4), (C22×Q16).2C2, C22.40(C2×Q16), C2.5(C8.18D4), C2.4(C8.12D4), C22.76(C4○D8), C22.38(C4⋊1D4), (C22×C8).493C22, (C22×Q8).39C22, C22.144(C4⋊D4), (C2×C42).1076C22, (C22×C4).1409C23, C23.67C23.10C2, C2.21(C24.3C22), (C2×C4×C8).36C2, (C2×C8).172(C2×C4), C4.37(C2×C22⋊C4), (C2×Q8).95(C2×C4), (C2×C2.D8).11C2, (C2×C4).1354(C2×D4), (C2×C4⋊C4).90C22, (C2×Q8⋊C4).8C2, (C2×C4).600(C4○D4), (C2×C4).423(C22×C4), SmallGroup(128,701)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊6Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=ab-1, dcd-1=c-1 >
Subgroups: 292 in 156 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C4×C8, Q8⋊C4, C2.D8, C2×C42, C2×C4⋊C4, C22×C8, C2×Q16, C2×Q16, C22×Q8, C23.67C23, C2×C4×C8, C2×Q8⋊C4, C2×C2.D8, C22×Q16, (C2×C4)⋊6Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, Q16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×Q16, C4○D8, C24.3C22, C4×Q16, C8.18D4, C4⋊Q16, C8.12D4, (C2×C4)⋊6Q16
►Regular action on 128 points
Generators in S128
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 104)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 127)(112 128)
(1 70 37 50)(2 71 38 51)(3 72 39 52)(4 65 40 53)(5 66 33 54)(6 67 34 55)(7 68 35 56)(8 69 36 49)(9 63 26 46)(10 64 27 47)(11 57 28 48)(12 58 29 41)(13 59 30 42)(14 60 31 43)(15 61 32 44)(16 62 25 45)(17 95 106 86)(18 96 107 87)(19 89 108 88)(20 90 109 81)(21 91 110 82)(22 92 111 83)(23 93 112 84)(24 94 105 85)(73 124 103 115)(74 125 104 116)(75 126 97 117)(76 127 98 118)(77 128 99 119)(78 121 100 120)(79 122 101 113)(80 123 102 114)
(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)
(1 112 5 108)(2 111 6 107)(3 110 7 106)(4 109 8 105)(9 116 13 120)(10 115 14 119)(11 114 15 118)(12 113 16 117)(17 39 21 35)(18 38 22 34)(19 37 23 33)(20 36 24 40)(25 126 29 122)(26 125 30 121)(27 124 31 128)(28 123 32 127)(41 95 45 91)(42 94 46 90)(43 93 47 89)(44 92 48 96)(49 100 53 104)(50 99 54 103)(51 98 55 102)(52 97 56 101)(57 87 61 83)(58 86 62 82)(59 85 63 81)(60 84 64 88)(65 74 69 78)(66 73 70 77)(67 80 71 76)(68 79 72 75)
G:=sub<Sym(128)| (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,104)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128), (1,70,37,50)(2,71,38,51)(3,72,39,52)(4,65,40,53)(5,66,33,54)(6,67,34,55)(7,68,35,56)(8,69,36,49)(9,63,26,46)(10,64,27,47)(11,57,28,48)(12,58,29,41)(13,59,30,42)(14,60,31,43)(15,61,32,44)(16,62,25,45)(17,95,106,86)(18,96,107,87)(19,89,108,88)(20,90,109,81)(21,91,110,82)(22,92,111,83)(23,93,112,84)(24,94,105,85)(73,124,103,115)(74,125,104,116)(75,126,97,117)(76,127,98,118)(77,128,99,119)(78,121,100,120)(79,122,101,113)(80,123,102,114), (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), (1,112,5,108)(2,111,6,107)(3,110,7,106)(4,109,8,105)(9,116,13,120)(10,115,14,119)(11,114,15,118)(12,113,16,117)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(25,126,29,122)(26,125,30,121)(27,124,31,128)(28,123,32,127)(41,95,45,91)(42,94,46,90)(43,93,47,89)(44,92,48,96)(49,100,53,104)(50,99,54,103)(51,98,55,102)(52,97,56,101)(57,87,61,83)(58,86,62,82)(59,85,63,81)(60,84,64,88)(65,74,69,78)(66,73,70,77)(67,80,71,76)(68,79,72,75)>;
G:=Group( (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,104)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128), (1,70,37,50)(2,71,38,51)(3,72,39,52)(4,65,40,53)(5,66,33,54)(6,67,34,55)(7,68,35,56)(8,69,36,49)(9,63,26,46)(10,64,27,47)(11,57,28,48)(12,58,29,41)(13,59,30,42)(14,60,31,43)(15,61,32,44)(16,62,25,45)(17,95,106,86)(18,96,107,87)(19,89,108,88)(20,90,109,81)(21,91,110,82)(22,92,111,83)(23,93,112,84)(24,94,105,85)(73,124,103,115)(74,125,104,116)(75,126,97,117)(76,127,98,118)(77,128,99,119)(78,121,100,120)(79,122,101,113)(80,123,102,114), (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), (1,112,5,108)(2,111,6,107)(3,110,7,106)(4,109,8,105)(9,116,13,120)(10,115,14,119)(11,114,15,118)(12,113,16,117)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(25,126,29,122)(26,125,30,121)(27,124,31,128)(28,123,32,127)(41,95,45,91)(42,94,46,90)(43,93,47,89)(44,92,48,96)(49,100,53,104)(50,99,54,103)(51,98,55,102)(52,97,56,101)(57,87,61,83)(58,86,62,82)(59,85,63,81)(60,84,64,88)(65,74,69,78)(66,73,70,77)(67,80,71,76)(68,79,72,75) );
G=PermutationGroup([[(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(41,68),(42,69),(43,70),(44,71),(45,72),(46,65),(47,66),(48,67),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(81,104),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126),(111,127),(112,128)], [(1,70,37,50),(2,71,38,51),(3,72,39,52),(4,65,40,53),(5,66,33,54),(6,67,34,55),(7,68,35,56),(8,69,36,49),(9,63,26,46),(10,64,27,47),(11,57,28,48),(12,58,29,41),(13,59,30,42),(14,60,31,43),(15,61,32,44),(16,62,25,45),(17,95,106,86),(18,96,107,87),(19,89,108,88),(20,90,109,81),(21,91,110,82),(22,92,111,83),(23,93,112,84),(24,94,105,85),(73,124,103,115),(74,125,104,116),(75,126,97,117),(76,127,98,118),(77,128,99,119),(78,121,100,120),(79,122,101,113),(80,123,102,114)], [(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)], [(1,112,5,108),(2,111,6,107),(3,110,7,106),(4,109,8,105),(9,116,13,120),(10,115,14,119),(11,114,15,118),(12,113,16,117),(17,39,21,35),(18,38,22,34),(19,37,23,33),(20,36,24,40),(25,126,29,122),(26,125,30,121),(27,124,31,128),(28,123,32,127),(41,95,45,91),(42,94,46,90),(43,93,47,89),(44,92,48,96),(49,100,53,104),(50,99,54,103),(51,98,55,102),(52,97,56,101),(57,87,61,83),(58,86,62,82),(59,85,63,81),(60,84,64,88),(65,74,69,78),(66,73,70,77),(67,80,71,76),(68,79,72,75)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q16 | C4○D4 | C4○D8 |
| kernel | (C2×C4)⋊6Q16 | C23.67C23 | C2×C4×C8 | C2×Q8⋊C4 | C2×C2.D8 | C22×Q16 | C2×Q16 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 6 | 2 | 8 | 4 | 8 |
Matrix representation of (C2×C4)⋊6Q16 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 14 | 14 | 0 | 0 |
| 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 1 | 7 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,14,14,0,0,0,14,3,0,0,0,0,0,10,1,0,0,0,1,7] >;
(C2×C4)⋊6Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_6Q_{16} % in TeX
G:=Group("(C2xC4):6Q16"); // GroupNames label
G:=SmallGroup(128,701);
// by ID
G=gap.SmallGroup(128,701);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,2019,248,2804,172]);
// 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,b*c=c*b,d*b*d^-1=a*b^-1,d*c*d^-1=c^-1>;
// generators/relations