p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.101D4, (C4×Q8)⋊14C4, Q8.1(C4⋊C4), (C2×Q8).17Q8, (C2×Q8).158D4, C2.1(Q8.Q8), C42⋊8C4.5C2, C42.139(C2×C4), (C22×C4).269D4, C23.743(C2×D4), C4.96(C22⋊Q8), C4.121(C4⋊D4), C22.42(C4○D8), C22.4Q16.2C2, (C22×C8).13C22, C2.1(Q8.D4), C4.34(C42⋊C2), (C2×C42).250C22, C22.70(C22⋊Q8), C22.108(C4⋊D4), (C22×C4).1327C23, C22.50(C8.C22), C2.14(C23.7Q8), (C22×Q8).384C22, C2.18(C23.38D4), C2.22(C23.24D4), C4.4(C2×C4⋊C4), (C2×C4⋊C8).23C2, (C2×C4×Q8).12C2, C4⋊C4.192(C2×C4), (C2×C4).262(C2×Q8), (C2×C4).1317(C2×D4), (C2×C4⋊C4).35C22, (C2×Q8⋊C4).2C2, (C2×Q8).189(C2×C4), (C2×C4).863(C4○D4), (C2×C4).365(C22×C4), (C2×C4).332(C22⋊C4), C22.250(C2×C22⋊C4), SmallGroup(128,537)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.101D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 260 in 146 conjugacy classes, 68 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×C8, C22×Q8, C22.4Q16, C42⋊8C4, C2×Q8⋊C4, C2×C4⋊C8, C2×C4×Q8, C42.101D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C4○D8, C8.C22, C23.7Q8, C23.24D4, C23.38D4, Q8.D4, Q8.Q8, C42.101D4
►Regular action on 128 points
Generators in S128
(1 83 35 127)(2 124 36 88)(3 85 37 121)(4 126 38 82)(5 87 39 123)(6 128 40 84)(7 81 33 125)(8 122 34 86)(9 68 77 26)(10 31 78 65)(11 70 79 28)(12 25 80 67)(13 72 73 30)(14 27 74 69)(15 66 75 32)(16 29 76 71)(17 50 109 91)(18 96 110 55)(19 52 111 93)(20 90 112 49)(21 54 105 95)(22 92 106 51)(23 56 107 89)(24 94 108 53)(41 64 117 101)(42 98 118 61)(43 58 119 103)(44 100 120 63)(45 60 113 97)(46 102 114 57)(47 62 115 99)(48 104 116 59)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 29 95 98)(2 101 96 32)(3 27 89 104)(4 99 90 30)(5 25 91 102)(6 97 92 28)(7 31 93 100)(8 103 94 26)(9 82 43 20)(10 23 44 85)(11 88 45 18)(12 21 46 83)(13 86 47 24)(14 19 48 81)(15 84 41 22)(16 17 42 87)(33 65 52 63)(34 58 53 68)(35 71 54 61)(36 64 55 66)(37 69 56 59)(38 62 49 72)(39 67 50 57)(40 60 51 70)(73 122 115 108)(74 111 116 125)(75 128 117 106)(76 109 118 123)(77 126 119 112)(78 107 120 121)(79 124 113 110)(80 105 114 127)
(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)
G:=sub<Sym(128)| (1,83,35,127)(2,124,36,88)(3,85,37,121)(4,126,38,82)(5,87,39,123)(6,128,40,84)(7,81,33,125)(8,122,34,86)(9,68,77,26)(10,31,78,65)(11,70,79,28)(12,25,80,67)(13,72,73,30)(14,27,74,69)(15,66,75,32)(16,29,76,71)(17,50,109,91)(18,96,110,55)(19,52,111,93)(20,90,112,49)(21,54,105,95)(22,92,106,51)(23,56,107,89)(24,94,108,53)(41,64,117,101)(42,98,118,61)(43,58,119,103)(44,100,120,63)(45,60,113,97)(46,102,114,57)(47,62,115,99)(48,104,116,59), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,29,95,98)(2,101,96,32)(3,27,89,104)(4,99,90,30)(5,25,91,102)(6,97,92,28)(7,31,93,100)(8,103,94,26)(9,82,43,20)(10,23,44,85)(11,88,45,18)(12,21,46,83)(13,86,47,24)(14,19,48,81)(15,84,41,22)(16,17,42,87)(33,65,52,63)(34,58,53,68)(35,71,54,61)(36,64,55,66)(37,69,56,59)(38,62,49,72)(39,67,50,57)(40,60,51,70)(73,122,115,108)(74,111,116,125)(75,128,117,106)(76,109,118,123)(77,126,119,112)(78,107,120,121)(79,124,113,110)(80,105,114,127), (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)>;
G:=Group( (1,83,35,127)(2,124,36,88)(3,85,37,121)(4,126,38,82)(5,87,39,123)(6,128,40,84)(7,81,33,125)(8,122,34,86)(9,68,77,26)(10,31,78,65)(11,70,79,28)(12,25,80,67)(13,72,73,30)(14,27,74,69)(15,66,75,32)(16,29,76,71)(17,50,109,91)(18,96,110,55)(19,52,111,93)(20,90,112,49)(21,54,105,95)(22,92,106,51)(23,56,107,89)(24,94,108,53)(41,64,117,101)(42,98,118,61)(43,58,119,103)(44,100,120,63)(45,60,113,97)(46,102,114,57)(47,62,115,99)(48,104,116,59), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,29,95,98)(2,101,96,32)(3,27,89,104)(4,99,90,30)(5,25,91,102)(6,97,92,28)(7,31,93,100)(8,103,94,26)(9,82,43,20)(10,23,44,85)(11,88,45,18)(12,21,46,83)(13,86,47,24)(14,19,48,81)(15,84,41,22)(16,17,42,87)(33,65,52,63)(34,58,53,68)(35,71,54,61)(36,64,55,66)(37,69,56,59)(38,62,49,72)(39,67,50,57)(40,60,51,70)(73,122,115,108)(74,111,116,125)(75,128,117,106)(76,109,118,123)(77,126,119,112)(78,107,120,121)(79,124,113,110)(80,105,114,127), (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) );
G=PermutationGroup([[(1,83,35,127),(2,124,36,88),(3,85,37,121),(4,126,38,82),(5,87,39,123),(6,128,40,84),(7,81,33,125),(8,122,34,86),(9,68,77,26),(10,31,78,65),(11,70,79,28),(12,25,80,67),(13,72,73,30),(14,27,74,69),(15,66,75,32),(16,29,76,71),(17,50,109,91),(18,96,110,55),(19,52,111,93),(20,90,112,49),(21,54,105,95),(22,92,106,51),(23,56,107,89),(24,94,108,53),(41,64,117,101),(42,98,118,61),(43,58,119,103),(44,100,120,63),(45,60,113,97),(46,102,114,57),(47,62,115,99),(48,104,116,59)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128)], [(1,29,95,98),(2,101,96,32),(3,27,89,104),(4,99,90,30),(5,25,91,102),(6,97,92,28),(7,31,93,100),(8,103,94,26),(9,82,43,20),(10,23,44,85),(11,88,45,18),(12,21,46,83),(13,86,47,24),(14,19,48,81),(15,84,41,22),(16,17,42,87),(33,65,52,63),(34,58,53,68),(35,71,54,61),(36,64,55,66),(37,69,56,59),(38,62,49,72),(39,67,50,57),(40,60,51,70),(73,122,115,108),(74,111,116,125),(75,128,117,106),(76,109,118,123),(77,126,119,112),(78,107,120,121),(79,124,113,110),(80,105,114,127)], [(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)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | C4○D4 | C4○D8 | C8.C22 |
| kernel | C42.101D4 | C22.4Q16 | C42⋊8C4 | C2×Q8⋊C4 | C2×C4⋊C8 | C2×C4×Q8 | C4×Q8 | C42 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 2 |
Matrix representation of C42.101D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 0 | 3 | 11 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 12 | 10 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,0,0,0,0,6,3,0,0,0,0,11,11],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,0,0,0,0,0,12,0,0,0,0,10,10] >;
C42.101D4 in GAP, Magma, Sage, TeX
C_4^2._{101}D_4 % in TeX
G:=Group("C4^2.101D4"); // GroupNames label
G:=SmallGroup(128,537);
// by ID
G=gap.SmallGroup(128,537);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,352,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations