p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.121D4, C4⋊Q8⋊23C4, C4⋊C4⋊10Q8, C4.29(C4×Q8), C4.30(C4⋊Q8), (C2×C4).63Q16, C42.163(C2×C4), C2.5(C4.Q16), C2.5(D4⋊2Q8), (C2×C4).106SD16, (C22×C4).764D4, C23.810(C2×D4), C22.42(C2×Q16), C4.12(Q8⋊C4), C4.49(C4.4D4), (C22×C8).61C22, C22.75(C2×SD16), C22.4Q16.16C2, (C2×C42).333C22, C22.82(C22⋊Q8), C22.100(C8⋊C22), (C22×C4).1421C23, C2.24(C23.37D4), C2.9(C23.67C23), (C2×C4⋊C8).33C2, (C4×C4⋊C4).27C2, C4⋊C4.94(C2×C4), (C2×C4⋊Q8).13C2, (C2×C4).276(C2×Q8), (C2×C4).1361(C2×D4), C2.24(C2×Q8⋊C4), (C2×C4).766(C4○D4), (C2×C4⋊C4).780C22, (C2×C4).435(C22×C4), (C2×C4).262(C22⋊C4), C22.296(C2×C22⋊C4), SmallGroup(128,719)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.121D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >
Subgroups: 284 in 146 conjugacy classes, 72 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2.C42, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C22.4Q16, C4×C4⋊C4, C2×C4⋊C8, C2×C4⋊Q8, C42.121D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, Q8⋊C4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C8⋊C22, C23.67C23, C2×Q8⋊C4, C23.37D4, D4⋊2Q8, C4.Q16, C42.121D4
►Regular action on 128 points
Generators in S128
(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 35 29 9)(2 36 30 10)(3 33 31 11)(4 34 32 12)(5 23 108 124)(6 24 105 121)(7 21 106 122)(8 22 107 123)(13 25 39 49)(14 26 40 50)(15 27 37 51)(16 28 38 52)(17 110 118 126)(18 111 119 127)(19 112 120 128)(20 109 117 125)(41 61 71 53)(42 62 72 54)(43 63 69 55)(44 64 70 56)(45 81 67 57)(46 82 68 58)(47 83 65 59)(48 84 66 60)(73 99 113 89)(74 100 114 90)(75 97 115 91)(76 98 116 92)(77 103 93 85)(78 104 94 86)(79 101 95 87)(80 102 96 88)
(1 47 15 53)(2 48 16 54)(3 45 13 55)(4 46 14 56)(5 96 118 100)(6 93 119 97)(7 94 120 98)(8 95 117 99)(9 83 51 41)(10 84 52 42)(11 81 49 43)(12 82 50 44)(17 90 108 80)(18 91 105 77)(19 92 106 78)(20 89 107 79)(21 104 128 76)(22 101 125 73)(23 102 126 74)(24 103 127 75)(25 69 33 57)(26 70 34 58)(27 71 35 59)(28 72 36 60)(29 65 37 61)(30 66 38 62)(31 67 39 63)(32 68 40 64)(85 111 115 121)(86 112 116 122)(87 109 113 123)(88 110 114 124)
(1 6 31 107)(2 5 32 106)(3 8 29 105)(4 7 30 108)(9 24 33 123)(10 23 34 122)(11 22 35 121)(12 21 36 124)(13 117 37 18)(14 120 38 17)(15 119 39 20)(16 118 40 19)(25 109 51 127)(26 112 52 126)(27 111 49 125)(28 110 50 128)(41 79 69 93)(42 78 70 96)(43 77 71 95)(44 80 72 94)(45 75 65 113)(46 74 66 116)(47 73 67 115)(48 76 68 114)(53 101 63 85)(54 104 64 88)(55 103 61 87)(56 102 62 86)(57 97 83 89)(58 100 84 92)(59 99 81 91)(60 98 82 90)
G:=sub<Sym(128)| (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), (1,35,29,9)(2,36,30,10)(3,33,31,11)(4,34,32,12)(5,23,108,124)(6,24,105,121)(7,21,106,122)(8,22,107,123)(13,25,39,49)(14,26,40,50)(15,27,37,51)(16,28,38,52)(17,110,118,126)(18,111,119,127)(19,112,120,128)(20,109,117,125)(41,61,71,53)(42,62,72,54)(43,63,69,55)(44,64,70,56)(45,81,67,57)(46,82,68,58)(47,83,65,59)(48,84,66,60)(73,99,113,89)(74,100,114,90)(75,97,115,91)(76,98,116,92)(77,103,93,85)(78,104,94,86)(79,101,95,87)(80,102,96,88), (1,47,15,53)(2,48,16,54)(3,45,13,55)(4,46,14,56)(5,96,118,100)(6,93,119,97)(7,94,120,98)(8,95,117,99)(9,83,51,41)(10,84,52,42)(11,81,49,43)(12,82,50,44)(17,90,108,80)(18,91,105,77)(19,92,106,78)(20,89,107,79)(21,104,128,76)(22,101,125,73)(23,102,126,74)(24,103,127,75)(25,69,33,57)(26,70,34,58)(27,71,35,59)(28,72,36,60)(29,65,37,61)(30,66,38,62)(31,67,39,63)(32,68,40,64)(85,111,115,121)(86,112,116,122)(87,109,113,123)(88,110,114,124), (1,6,31,107)(2,5,32,106)(3,8,29,105)(4,7,30,108)(9,24,33,123)(10,23,34,122)(11,22,35,121)(12,21,36,124)(13,117,37,18)(14,120,38,17)(15,119,39,20)(16,118,40,19)(25,109,51,127)(26,112,52,126)(27,111,49,125)(28,110,50,128)(41,79,69,93)(42,78,70,96)(43,77,71,95)(44,80,72,94)(45,75,65,113)(46,74,66,116)(47,73,67,115)(48,76,68,114)(53,101,63,85)(54,104,64,88)(55,103,61,87)(56,102,62,86)(57,97,83,89)(58,100,84,92)(59,99,81,91)(60,98,82,90)>;
G:=Group( (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), (1,35,29,9)(2,36,30,10)(3,33,31,11)(4,34,32,12)(5,23,108,124)(6,24,105,121)(7,21,106,122)(8,22,107,123)(13,25,39,49)(14,26,40,50)(15,27,37,51)(16,28,38,52)(17,110,118,126)(18,111,119,127)(19,112,120,128)(20,109,117,125)(41,61,71,53)(42,62,72,54)(43,63,69,55)(44,64,70,56)(45,81,67,57)(46,82,68,58)(47,83,65,59)(48,84,66,60)(73,99,113,89)(74,100,114,90)(75,97,115,91)(76,98,116,92)(77,103,93,85)(78,104,94,86)(79,101,95,87)(80,102,96,88), (1,47,15,53)(2,48,16,54)(3,45,13,55)(4,46,14,56)(5,96,118,100)(6,93,119,97)(7,94,120,98)(8,95,117,99)(9,83,51,41)(10,84,52,42)(11,81,49,43)(12,82,50,44)(17,90,108,80)(18,91,105,77)(19,92,106,78)(20,89,107,79)(21,104,128,76)(22,101,125,73)(23,102,126,74)(24,103,127,75)(25,69,33,57)(26,70,34,58)(27,71,35,59)(28,72,36,60)(29,65,37,61)(30,66,38,62)(31,67,39,63)(32,68,40,64)(85,111,115,121)(86,112,116,122)(87,109,113,123)(88,110,114,124), (1,6,31,107)(2,5,32,106)(3,8,29,105)(4,7,30,108)(9,24,33,123)(10,23,34,122)(11,22,35,121)(12,21,36,124)(13,117,37,18)(14,120,38,17)(15,119,39,20)(16,118,40,19)(25,109,51,127)(26,112,52,126)(27,111,49,125)(28,110,50,128)(41,79,69,93)(42,78,70,96)(43,77,71,95)(44,80,72,94)(45,75,65,113)(46,74,66,116)(47,73,67,115)(48,76,68,114)(53,101,63,85)(54,104,64,88)(55,103,61,87)(56,102,62,86)(57,97,83,89)(58,100,84,92)(59,99,81,91)(60,98,82,90) );
G=PermutationGroup([[(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)], [(1,35,29,9),(2,36,30,10),(3,33,31,11),(4,34,32,12),(5,23,108,124),(6,24,105,121),(7,21,106,122),(8,22,107,123),(13,25,39,49),(14,26,40,50),(15,27,37,51),(16,28,38,52),(17,110,118,126),(18,111,119,127),(19,112,120,128),(20,109,117,125),(41,61,71,53),(42,62,72,54),(43,63,69,55),(44,64,70,56),(45,81,67,57),(46,82,68,58),(47,83,65,59),(48,84,66,60),(73,99,113,89),(74,100,114,90),(75,97,115,91),(76,98,116,92),(77,103,93,85),(78,104,94,86),(79,101,95,87),(80,102,96,88)], [(1,47,15,53),(2,48,16,54),(3,45,13,55),(4,46,14,56),(5,96,118,100),(6,93,119,97),(7,94,120,98),(8,95,117,99),(9,83,51,41),(10,84,52,42),(11,81,49,43),(12,82,50,44),(17,90,108,80),(18,91,105,77),(19,92,106,78),(20,89,107,79),(21,104,128,76),(22,101,125,73),(23,102,126,74),(24,103,127,75),(25,69,33,57),(26,70,34,58),(27,71,35,59),(28,72,36,60),(29,65,37,61),(30,66,38,62),(31,67,39,63),(32,68,40,64),(85,111,115,121),(86,112,116,122),(87,109,113,123),(88,110,114,124)], [(1,6,31,107),(2,5,32,106),(3,8,29,105),(4,7,30,108),(9,24,33,123),(10,23,34,122),(11,22,35,121),(12,21,36,124),(13,117,37,18),(14,120,38,17),(15,119,39,20),(16,118,40,19),(25,109,51,127),(26,112,52,126),(27,111,49,125),(28,110,50,128),(41,79,69,93),(42,78,70,96),(43,77,71,95),(44,80,72,94),(45,75,65,113),(46,74,66,116),(47,73,67,115),(48,76,68,114),(53,101,63,85),(54,104,64,88),(55,103,61,87),(56,102,62,86),(57,97,83,89),(58,100,84,92),(59,99,81,91),(60,98,82,90)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | SD16 | Q16 | C4○D4 | C8⋊C22 |
| kernel | C42.121D4 | C22.4Q16 | C4×C4⋊C4 | C2×C4⋊C8 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C42.121D4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 15 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 16 | 2 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 | 11 |
| 0 | 0 | 0 | 11 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 13 | 4 | 0 | 0 |
| 0 | 0 | 0 | 7 | 16 |
| 0 | 0 | 0 | 16 | 10 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0],[13,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,13,11,0,0,0,11,4],[16,0,0,0,0,0,13,13,0,0,0,0,4,0,0,0,0,0,7,16,0,0,0,16,10] >;
C42.121D4 in GAP, Magma, Sage, TeX
C_4^2._{121}D_4 % in TeX
G:=Group("C4^2.121D4"); // GroupNames label
G:=SmallGroup(128,719);
// by ID
G=gap.SmallGroup(128,719);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations