p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×Q8).109D4, C23.929(C2×D4), (C22×C4).159D4, C42⋊8C4.15C2, (C22×C8).87C22, C4.40(C42⋊2C2), C22.121(C4○D8), C22.4Q16.26C2, (C2×C42).383C22, C2.22(Q8.D4), (C22×Q8).79C22, C22.250(C4⋊D4), (C22×C4).1463C23, C2.9(C23.11D4), C4.29(C22.D4), C22.96(C4.4D4), C2.10(C23.20D4), C22.139(C8.C22), C22.7C42.14C2, C23.67C23.20C2, C2.7(C42.30C22), C2.10(C42.78C22), C22.119(C22.D4), (C2×C4).1062(C2×D4), (C2×C4).625(C4○D4), (C2×C4⋊C4).148C22, (C2×Q8⋊C4).17C2, SmallGroup(128,806)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×Q8).109D4
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=ab-1, ebe-1=b-1c, cd=dc, ece-1=b2c, ede-1=b2d-1 >
Subgroups: 232 in 111 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C42⋊8C4, C23.67C23, C2×Q8⋊C4, (C2×Q8).109D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4○D8, C8.C22, C23.11D4, Q8.D4, C23.20D4, C42.78C22, C42.30C22, (C2×Q8).109D4
►Regular action on 128 points
Generators in S128
(1 16)(2 13)(3 14)(4 15)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(17 71)(18 72)(19 69)(20 70)(21 27)(22 28)(23 25)(24 26)(37 43)(38 44)(39 41)(40 42)(45 89)(46 90)(47 91)(48 92)(49 55)(50 56)(51 53)(52 54)(57 63)(58 64)(59 61)(60 62)(65 103)(66 104)(67 101)(68 102)(73 79)(74 80)(75 77)(76 78)(81 95)(82 96)(83 93)(84 94)(85 99)(86 100)(87 97)(88 98)(105 111)(106 112)(107 109)(108 110)(113 119)(114 120)(115 117)(116 118)(121 127)(122 128)(123 125)(124 126)
(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 26 3 28)(2 25 4 27)(5 69 7 71)(6 72 8 70)(9 53 11 55)(10 56 12 54)(13 23 15 21)(14 22 16 24)(17 29 19 31)(18 32 20 30)(33 51 35 49)(34 50 36 52)(37 59 39 57)(38 58 40 60)(41 63 43 61)(42 62 44 64)(45 85 47 87)(46 88 48 86)(65 117 67 119)(66 120 68 118)(73 83 75 81)(74 82 76 84)(77 95 79 93)(78 94 80 96)(89 99 91 97)(90 98 92 100)(101 113 103 115)(102 116 104 114)(105 127 107 125)(106 126 108 128)(109 123 111 121)(110 122 112 124)
(1 65 35 78)(2 102 36 75)(3 67 33 80)(4 104 34 73)(5 108 44 85)(6 109 41 98)(7 106 42 87)(8 111 43 100)(9 74 14 101)(10 79 15 66)(11 76 16 103)(12 77 13 68)(17 124 60 89)(18 125 57 48)(19 122 58 91)(20 127 59 46)(21 120 56 93)(22 113 53 82)(23 118 54 95)(24 115 55 84)(25 116 52 81)(26 117 49 94)(27 114 50 83)(28 119 51 96)(29 110 38 99)(30 107 39 88)(31 112 40 97)(32 105 37 86)(45 71 126 62)(47 69 128 64)(61 90 70 121)(63 92 72 123)
(1 100 16 86)(2 91 13 47)(3 98 14 88)(4 89 15 45)(5 83 29 93)(6 76 30 78)(7 81 31 95)(8 74 32 80)(9 107 33 109)(10 126 34 124)(11 105 35 111)(12 128 36 122)(17 77 71 75)(18 96 72 82)(19 79 69 73)(20 94 70 84)(21 87 27 97)(22 46 28 90)(23 85 25 99)(24 48 26 92)(37 67 43 101)(38 120 44 114)(39 65 41 103)(40 118 42 116)(49 123 55 125)(50 112 56 106)(51 121 53 127)(52 110 54 108)(57 119 63 113)(58 66 64 104)(59 117 61 115)(60 68 62 102)
G:=sub<Sym(128)| (1,16)(2,13)(3,14)(4,15)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(17,71)(18,72)(19,69)(20,70)(21,27)(22,28)(23,25)(24,26)(37,43)(38,44)(39,41)(40,42)(45,89)(46,90)(47,91)(48,92)(49,55)(50,56)(51,53)(52,54)(57,63)(58,64)(59,61)(60,62)(65,103)(66,104)(67,101)(68,102)(73,79)(74,80)(75,77)(76,78)(81,95)(82,96)(83,93)(84,94)(85,99)(86,100)(87,97)(88,98)(105,111)(106,112)(107,109)(108,110)(113,119)(114,120)(115,117)(116,118)(121,127)(122,128)(123,125)(124,126), (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), (1,26,3,28)(2,25,4,27)(5,69,7,71)(6,72,8,70)(9,53,11,55)(10,56,12,54)(13,23,15,21)(14,22,16,24)(17,29,19,31)(18,32,20,30)(33,51,35,49)(34,50,36,52)(37,59,39,57)(38,58,40,60)(41,63,43,61)(42,62,44,64)(45,85,47,87)(46,88,48,86)(65,117,67,119)(66,120,68,118)(73,83,75,81)(74,82,76,84)(77,95,79,93)(78,94,80,96)(89,99,91,97)(90,98,92,100)(101,113,103,115)(102,116,104,114)(105,127,107,125)(106,126,108,128)(109,123,111,121)(110,122,112,124), (1,65,35,78)(2,102,36,75)(3,67,33,80)(4,104,34,73)(5,108,44,85)(6,109,41,98)(7,106,42,87)(8,111,43,100)(9,74,14,101)(10,79,15,66)(11,76,16,103)(12,77,13,68)(17,124,60,89)(18,125,57,48)(19,122,58,91)(20,127,59,46)(21,120,56,93)(22,113,53,82)(23,118,54,95)(24,115,55,84)(25,116,52,81)(26,117,49,94)(27,114,50,83)(28,119,51,96)(29,110,38,99)(30,107,39,88)(31,112,40,97)(32,105,37,86)(45,71,126,62)(47,69,128,64)(61,90,70,121)(63,92,72,123), (1,100,16,86)(2,91,13,47)(3,98,14,88)(4,89,15,45)(5,83,29,93)(6,76,30,78)(7,81,31,95)(8,74,32,80)(9,107,33,109)(10,126,34,124)(11,105,35,111)(12,128,36,122)(17,77,71,75)(18,96,72,82)(19,79,69,73)(20,94,70,84)(21,87,27,97)(22,46,28,90)(23,85,25,99)(24,48,26,92)(37,67,43,101)(38,120,44,114)(39,65,41,103)(40,118,42,116)(49,123,55,125)(50,112,56,106)(51,121,53,127)(52,110,54,108)(57,119,63,113)(58,66,64,104)(59,117,61,115)(60,68,62,102)>;
G:=Group( (1,16)(2,13)(3,14)(4,15)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(17,71)(18,72)(19,69)(20,70)(21,27)(22,28)(23,25)(24,26)(37,43)(38,44)(39,41)(40,42)(45,89)(46,90)(47,91)(48,92)(49,55)(50,56)(51,53)(52,54)(57,63)(58,64)(59,61)(60,62)(65,103)(66,104)(67,101)(68,102)(73,79)(74,80)(75,77)(76,78)(81,95)(82,96)(83,93)(84,94)(85,99)(86,100)(87,97)(88,98)(105,111)(106,112)(107,109)(108,110)(113,119)(114,120)(115,117)(116,118)(121,127)(122,128)(123,125)(124,126), (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), (1,26,3,28)(2,25,4,27)(5,69,7,71)(6,72,8,70)(9,53,11,55)(10,56,12,54)(13,23,15,21)(14,22,16,24)(17,29,19,31)(18,32,20,30)(33,51,35,49)(34,50,36,52)(37,59,39,57)(38,58,40,60)(41,63,43,61)(42,62,44,64)(45,85,47,87)(46,88,48,86)(65,117,67,119)(66,120,68,118)(73,83,75,81)(74,82,76,84)(77,95,79,93)(78,94,80,96)(89,99,91,97)(90,98,92,100)(101,113,103,115)(102,116,104,114)(105,127,107,125)(106,126,108,128)(109,123,111,121)(110,122,112,124), (1,65,35,78)(2,102,36,75)(3,67,33,80)(4,104,34,73)(5,108,44,85)(6,109,41,98)(7,106,42,87)(8,111,43,100)(9,74,14,101)(10,79,15,66)(11,76,16,103)(12,77,13,68)(17,124,60,89)(18,125,57,48)(19,122,58,91)(20,127,59,46)(21,120,56,93)(22,113,53,82)(23,118,54,95)(24,115,55,84)(25,116,52,81)(26,117,49,94)(27,114,50,83)(28,119,51,96)(29,110,38,99)(30,107,39,88)(31,112,40,97)(32,105,37,86)(45,71,126,62)(47,69,128,64)(61,90,70,121)(63,92,72,123), (1,100,16,86)(2,91,13,47)(3,98,14,88)(4,89,15,45)(5,83,29,93)(6,76,30,78)(7,81,31,95)(8,74,32,80)(9,107,33,109)(10,126,34,124)(11,105,35,111)(12,128,36,122)(17,77,71,75)(18,96,72,82)(19,79,69,73)(20,94,70,84)(21,87,27,97)(22,46,28,90)(23,85,25,99)(24,48,26,92)(37,67,43,101)(38,120,44,114)(39,65,41,103)(40,118,42,116)(49,123,55,125)(50,112,56,106)(51,121,53,127)(52,110,54,108)(57,119,63,113)(58,66,64,104)(59,117,61,115)(60,68,62,102) );
G=PermutationGroup([[(1,16),(2,13),(3,14),(4,15),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(17,71),(18,72),(19,69),(20,70),(21,27),(22,28),(23,25),(24,26),(37,43),(38,44),(39,41),(40,42),(45,89),(46,90),(47,91),(48,92),(49,55),(50,56),(51,53),(52,54),(57,63),(58,64),(59,61),(60,62),(65,103),(66,104),(67,101),(68,102),(73,79),(74,80),(75,77),(76,78),(81,95),(82,96),(83,93),(84,94),(85,99),(86,100),(87,97),(88,98),(105,111),(106,112),(107,109),(108,110),(113,119),(114,120),(115,117),(116,118),(121,127),(122,128),(123,125),(124,126)], [(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)], [(1,26,3,28),(2,25,4,27),(5,69,7,71),(6,72,8,70),(9,53,11,55),(10,56,12,54),(13,23,15,21),(14,22,16,24),(17,29,19,31),(18,32,20,30),(33,51,35,49),(34,50,36,52),(37,59,39,57),(38,58,40,60),(41,63,43,61),(42,62,44,64),(45,85,47,87),(46,88,48,86),(65,117,67,119),(66,120,68,118),(73,83,75,81),(74,82,76,84),(77,95,79,93),(78,94,80,96),(89,99,91,97),(90,98,92,100),(101,113,103,115),(102,116,104,114),(105,127,107,125),(106,126,108,128),(109,123,111,121),(110,122,112,124)], [(1,65,35,78),(2,102,36,75),(3,67,33,80),(4,104,34,73),(5,108,44,85),(6,109,41,98),(7,106,42,87),(8,111,43,100),(9,74,14,101),(10,79,15,66),(11,76,16,103),(12,77,13,68),(17,124,60,89),(18,125,57,48),(19,122,58,91),(20,127,59,46),(21,120,56,93),(22,113,53,82),(23,118,54,95),(24,115,55,84),(25,116,52,81),(26,117,49,94),(27,114,50,83),(28,119,51,96),(29,110,38,99),(30,107,39,88),(31,112,40,97),(32,105,37,86),(45,71,126,62),(47,69,128,64),(61,90,70,121),(63,92,72,123)], [(1,100,16,86),(2,91,13,47),(3,98,14,88),(4,89,15,45),(5,83,29,93),(6,76,30,78),(7,81,31,95),(8,74,32,80),(9,107,33,109),(10,126,34,124),(11,105,35,111),(12,128,36,122),(17,77,71,75),(18,96,72,82),(19,79,69,73),(20,94,70,84),(21,87,27,97),(22,46,28,90),(23,85,25,99),(24,48,26,92),(37,67,43,101),(38,120,44,114),(39,65,41,103),(40,118,42,116),(49,123,55,125),(50,112,56,106),(51,121,53,127),(52,110,54,108),(57,119,63,113),(58,66,64,104),(59,117,61,115),(60,68,62,102)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | C8.C22 |
| kernel | (C2×Q8).109D4 | C22.7C42 | C22.4Q16 | C42⋊8C4 | C23.67C23 | C2×Q8⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 10 | 8 | 2 |
Matrix representation of (C2×Q8).109D4 ►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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 10 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 12 |
| 0 | 0 | 0 | 0 | 5 | 3 |
| 15 | 5 | 0 | 0 | 0 | 0 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 14 |
| 0 | 0 | 0 | 0 | 3 | 12 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,10,0,0,0,0,10,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[2,4,0,0,0,0,3,15,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,14,5,0,0,0,0,12,3],[15,13,0,0,0,0,5,2,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,0,0,0,0,5,3,0,0,0,0,14,12] >;
(C2×Q8).109D4 in GAP, Magma, Sage, TeX
(C_2\times Q_8)._{109}D_4 % in TeX
G:=Group("(C2xQ8).109D4"); // GroupNames label
G:=SmallGroup(128,806);
// by ID
G=gap.SmallGroup(128,806);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,456,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^4=1,c^2=b^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1,e*b*e^-1=b^-1*c,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=b^2*d^-1>;
// generators/relations