p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.99D4, Q8⋊1(C4⋊C4), (C4×Q8)⋊13C4, (C2×Q8).16Q8, (C2×C4).60Q16, C4⋊3(Q8⋊C4), (C2×Q8).157D4, C42.137(C2×C4), C42⋊9C4.3C2, C2.1(C4.Q16), C2.1(C4⋊SD16), C2.1(Q8⋊Q8), C2.1(C4⋊2Q16), (C2×C4).119SD16, C23.741(C2×D4), (C22×C4).267D4, C4.94(C22⋊Q8), C22.24(C2×Q16), C4.119(C4⋊D4), C22.4Q16.1C2, (C22×C8).11C22, C22.47(C2×SD16), C4.32(C42⋊C2), C22.60(C8⋊C22), (C2×C42).248C22, C22.68(C22⋊Q8), C22.106(C4⋊D4), (C22×C4).1325C23, C22.49(C8.C22), C2.12(C23.7Q8), (C22×Q8).383C22, C2.22(C23.36D4), C4.2(C2×C4⋊C4), (C2×C4⋊C8).22C2, (C2×C4×Q8).11C2, C4⋊C4.190(C2×C4), (C2×C4).260(C2×Q8), (C2×C4).1315(C2×D4), (C2×C4⋊C4).33C22, (C2×Q8).188(C2×C4), (C2×Q8⋊C4).1C2, C2.18(C2×Q8⋊C4), (C2×C4).861(C4○D4), (C2×C4).363(C22×C4), (C2×C4).359(C22⋊C4), C22.248(C2×C22⋊C4), SmallGroup(128,535)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.99D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 276 in 154 conjugacy classes, 76 normal (36 characteristic)
C1, C2, C4, C4, C4, 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, 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⋊9C4, C2×Q8⋊C4, C2×C4⋊C8, C2×C4×Q8, C42.99D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, Q8⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C2×SD16, C2×Q16, C8⋊C22, C8.C22, C23.7Q8, C2×Q8⋊C4, C23.36D4, C4⋊SD16, C4⋊2Q16, Q8⋊Q8, C4.Q16, C42.99D4
►Regular action on 128 points
Generators in S128
(1 87 39 127)(2 128 40 88)(3 81 33 121)(4 122 34 82)(5 83 35 123)(6 124 36 84)(7 85 37 125)(8 126 38 86)(9 17 52 28)(10 29 53 18)(11 19 54 30)(12 31 55 20)(13 21 56 32)(14 25 49 22)(15 23 50 26)(16 27 51 24)(41 64 113 97)(42 98 114 57)(43 58 115 99)(44 100 116 59)(45 60 117 101)(46 102 118 61)(47 62 119 103)(48 104 120 63)(65 74 106 94)(66 95 107 75)(67 76 108 96)(68 89 109 77)(69 78 110 90)(70 91 111 79)(71 80 112 92)(72 93 105 73)
(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 31 77 98)(2 101 78 26)(3 29 79 104)(4 99 80 32)(5 27 73 102)(6 97 74 30)(7 25 75 100)(8 103 76 28)(9 86 47 67)(10 70 48 81)(11 84 41 65)(12 68 42 87)(13 82 43 71)(14 66 44 85)(15 88 45 69)(16 72 46 83)(17 38 62 96)(18 91 63 33)(19 36 64 94)(20 89 57 39)(21 34 58 92)(22 95 59 37)(23 40 60 90)(24 93 61 35)(49 107 116 125)(50 128 117 110)(51 105 118 123)(52 126 119 108)(53 111 120 121)(54 124 113 106)(55 109 114 127)(56 122 115 112)
(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)| (1,87,39,127)(2,128,40,88)(3,81,33,121)(4,122,34,82)(5,83,35,123)(6,124,36,84)(7,85,37,125)(8,126,38,86)(9,17,52,28)(10,29,53,18)(11,19,54,30)(12,31,55,20)(13,21,56,32)(14,25,49,22)(15,23,50,26)(16,27,51,24)(41,64,113,97)(42,98,114,57)(43,58,115,99)(44,100,116,59)(45,60,117,101)(46,102,118,61)(47,62,119,103)(48,104,120,63)(65,74,106,94)(66,95,107,75)(67,76,108,96)(68,89,109,77)(69,78,110,90)(70,91,111,79)(71,80,112,92)(72,93,105,73), (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), (1,31,77,98)(2,101,78,26)(3,29,79,104)(4,99,80,32)(5,27,73,102)(6,97,74,30)(7,25,75,100)(8,103,76,28)(9,86,47,67)(10,70,48,81)(11,84,41,65)(12,68,42,87)(13,82,43,71)(14,66,44,85)(15,88,45,69)(16,72,46,83)(17,38,62,96)(18,91,63,33)(19,36,64,94)(20,89,57,39)(21,34,58,92)(22,95,59,37)(23,40,60,90)(24,93,61,35)(49,107,116,125)(50,128,117,110)(51,105,118,123)(52,126,119,108)(53,111,120,121)(54,124,113,106)(55,109,114,127)(56,122,115,112), (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( (1,87,39,127)(2,128,40,88)(3,81,33,121)(4,122,34,82)(5,83,35,123)(6,124,36,84)(7,85,37,125)(8,126,38,86)(9,17,52,28)(10,29,53,18)(11,19,54,30)(12,31,55,20)(13,21,56,32)(14,25,49,22)(15,23,50,26)(16,27,51,24)(41,64,113,97)(42,98,114,57)(43,58,115,99)(44,100,116,59)(45,60,117,101)(46,102,118,61)(47,62,119,103)(48,104,120,63)(65,74,106,94)(66,95,107,75)(67,76,108,96)(68,89,109,77)(69,78,110,90)(70,91,111,79)(71,80,112,92)(72,93,105,73), (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), (1,31,77,98)(2,101,78,26)(3,29,79,104)(4,99,80,32)(5,27,73,102)(6,97,74,30)(7,25,75,100)(8,103,76,28)(9,86,47,67)(10,70,48,81)(11,84,41,65)(12,68,42,87)(13,82,43,71)(14,66,44,85)(15,88,45,69)(16,72,46,83)(17,38,62,96)(18,91,63,33)(19,36,64,94)(20,89,57,39)(21,34,58,92)(22,95,59,37)(23,40,60,90)(24,93,61,35)(49,107,116,125)(50,128,117,110)(51,105,118,123)(52,126,119,108)(53,111,120,121)(54,124,113,106)(55,109,114,127)(56,122,115,112), (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([[(1,87,39,127),(2,128,40,88),(3,81,33,121),(4,122,34,82),(5,83,35,123),(6,124,36,84),(7,85,37,125),(8,126,38,86),(9,17,52,28),(10,29,53,18),(11,19,54,30),(12,31,55,20),(13,21,56,32),(14,25,49,22),(15,23,50,26),(16,27,51,24),(41,64,113,97),(42,98,114,57),(43,58,115,99),(44,100,116,59),(45,60,117,101),(46,102,118,61),(47,62,119,103),(48,104,120,63),(65,74,106,94),(66,95,107,75),(67,76,108,96),(68,89,109,77),(69,78,110,90),(70,91,111,79),(71,80,112,92),(72,93,105,73)], [(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)], [(1,31,77,98),(2,101,78,26),(3,29,79,104),(4,99,80,32),(5,27,73,102),(6,97,74,30),(7,25,75,100),(8,103,76,28),(9,86,47,67),(10,70,48,81),(11,84,41,65),(12,68,42,87),(13,82,43,71),(14,66,44,85),(15,88,45,69),(16,72,46,83),(17,38,62,96),(18,91,63,33),(19,36,64,94),(20,89,57,39),(21,34,58,92),(22,95,59,37),(23,40,60,90),(24,93,61,35),(49,107,116,125),(50,128,117,110),(51,105,118,123),(52,126,119,108),(53,111,120,121),(54,124,113,106),(55,109,114,127),(56,122,115,112)], [(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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | SD16 | Q16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.99D4 | C22.4Q16 | C42⋊9C4 | C2×Q8⋊C4 | C2×C4⋊C8 | C2×C4×Q8 | C4×Q8 | C42 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of C42.99D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 12 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 15 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,10,10,0,0,0,0,12,7,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,15,10,0,0,0,0,0,0,12,12,0,0,0,0,5,12] >;
C42.99D4 in GAP, Magma, Sage, TeX
C_4^2._{99}D_4 % in TeX
G:=Group("C4^2.99D4"); // GroupNames label
G:=SmallGroup(128,535);
// by ID
G=gap.SmallGroup(128,535);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,352,2804,718,172]);
// 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,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations