p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.122D4, C4⋊C4⋊11Q8, C4⋊Q8⋊24C4, C4.30(C4×Q8), (C2×C4).136D8, C4.31(C4⋊Q8), C22.49(C2×D8), C42.164(C2×C4), C2.5(D4⋊Q8), C2.5(Q8⋊Q8), (C2×C4).121SD16, (C22×C4).765D4, C23.811(C2×D4), C4.10(D4⋊C4), C4.50(C4.4D4), (C22×C8).62C22, C22.76(C2×SD16), C22.4Q16.17C2, (C2×C42).334C22, C22.83(C22⋊Q8), (C22×C4).1422C23, C22.89(C8.C22), C2.24(C23.38D4), C2.10(C23.67C23), (C4×C4⋊C4).28C2, (C2×C4⋊C8).34C2, C4⋊C4.95(C2×C4), (C2×C4⋊Q8).14C2, (C2×C4).277(C2×Q8), C2.24(C2×D4⋊C4), (C2×C4).1362(C2×D4), (C2×C4).767(C4○D4), (C2×C4⋊C4).781C22, (C2×C4).436(C22×C4), (C2×C4).263(C22⋊C4), C22.297(C2×C22⋊C4), SmallGroup(128,720)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.122D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2b-1c-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, C42.122D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, D4⋊C4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C2×D8, C2×SD16, C8.C22, C23.67C23, C2×D4⋊C4, C23.38D4, D4⋊Q8, Q8⋊Q8, C42.122D4
►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 51 11 25)(2 52 12 26)(3 49 9 27)(4 50 10 28)(5 126 118 122)(6 127 119 123)(7 128 120 124)(8 125 117 121)(13 33 39 31)(14 34 40 32)(15 35 37 29)(16 36 38 30)(17 112 108 23)(18 109 105 24)(19 110 106 21)(20 111 107 22)(41 67 69 45)(42 68 70 46)(43 65 71 47)(44 66 72 48)(53 83 61 57)(54 84 62 58)(55 81 63 59)(56 82 64 60)(73 77 101 99)(74 78 102 100)(75 79 103 97)(76 80 104 98)(85 93 115 89)(86 94 116 90)(87 95 113 91)(88 96 114 92)
(1 65 37 61)(2 66 38 62)(3 67 39 63)(4 68 40 64)(5 78 108 92)(6 79 105 89)(7 80 106 90)(8 77 107 91)(9 45 13 55)(10 46 14 56)(11 47 15 53)(12 48 16 54)(17 96 118 100)(18 93 119 97)(19 94 120 98)(20 95 117 99)(21 116 128 76)(22 113 125 73)(23 114 126 74)(24 115 127 75)(25 71 35 57)(26 72 36 58)(27 69 33 59)(28 70 34 60)(29 83 51 43)(30 84 52 44)(31 81 49 41)(32 82 50 42)(85 123 103 109)(86 124 104 110)(87 121 101 111)(88 122 102 112)
(1 18 3 20)(2 17 4 19)(5 14 7 16)(6 13 8 15)(9 107 11 105)(10 106 12 108)(21 52 23 50)(22 51 24 49)(25 109 27 111)(26 112 28 110)(29 127 31 125)(30 126 32 128)(33 121 35 123)(34 124 36 122)(37 119 39 117)(38 118 40 120)(41 79 43 77)(42 78 44 80)(45 103 47 101)(46 102 48 104)(53 87 55 85)(54 86 56 88)(57 95 59 93)(58 94 60 96)(61 113 63 115)(62 116 64 114)(65 73 67 75)(66 76 68 74)(69 97 71 99)(70 100 72 98)(81 89 83 91)(82 92 84 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), (1,51,11,25)(2,52,12,26)(3,49,9,27)(4,50,10,28)(5,126,118,122)(6,127,119,123)(7,128,120,124)(8,125,117,121)(13,33,39,31)(14,34,40,32)(15,35,37,29)(16,36,38,30)(17,112,108,23)(18,109,105,24)(19,110,106,21)(20,111,107,22)(41,67,69,45)(42,68,70,46)(43,65,71,47)(44,66,72,48)(53,83,61,57)(54,84,62,58)(55,81,63,59)(56,82,64,60)(73,77,101,99)(74,78,102,100)(75,79,103,97)(76,80,104,98)(85,93,115,89)(86,94,116,90)(87,95,113,91)(88,96,114,92), (1,65,37,61)(2,66,38,62)(3,67,39,63)(4,68,40,64)(5,78,108,92)(6,79,105,89)(7,80,106,90)(8,77,107,91)(9,45,13,55)(10,46,14,56)(11,47,15,53)(12,48,16,54)(17,96,118,100)(18,93,119,97)(19,94,120,98)(20,95,117,99)(21,116,128,76)(22,113,125,73)(23,114,126,74)(24,115,127,75)(25,71,35,57)(26,72,36,58)(27,69,33,59)(28,70,34,60)(29,83,51,43)(30,84,52,44)(31,81,49,41)(32,82,50,42)(85,123,103,109)(86,124,104,110)(87,121,101,111)(88,122,102,112), (1,18,3,20)(2,17,4,19)(5,14,7,16)(6,13,8,15)(9,107,11,105)(10,106,12,108)(21,52,23,50)(22,51,24,49)(25,109,27,111)(26,112,28,110)(29,127,31,125)(30,126,32,128)(33,121,35,123)(34,124,36,122)(37,119,39,117)(38,118,40,120)(41,79,43,77)(42,78,44,80)(45,103,47,101)(46,102,48,104)(53,87,55,85)(54,86,56,88)(57,95,59,93)(58,94,60,96)(61,113,63,115)(62,116,64,114)(65,73,67,75)(66,76,68,74)(69,97,71,99)(70,100,72,98)(81,89,83,91)(82,92,84,90)>;
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), (1,51,11,25)(2,52,12,26)(3,49,9,27)(4,50,10,28)(5,126,118,122)(6,127,119,123)(7,128,120,124)(8,125,117,121)(13,33,39,31)(14,34,40,32)(15,35,37,29)(16,36,38,30)(17,112,108,23)(18,109,105,24)(19,110,106,21)(20,111,107,22)(41,67,69,45)(42,68,70,46)(43,65,71,47)(44,66,72,48)(53,83,61,57)(54,84,62,58)(55,81,63,59)(56,82,64,60)(73,77,101,99)(74,78,102,100)(75,79,103,97)(76,80,104,98)(85,93,115,89)(86,94,116,90)(87,95,113,91)(88,96,114,92), (1,65,37,61)(2,66,38,62)(3,67,39,63)(4,68,40,64)(5,78,108,92)(6,79,105,89)(7,80,106,90)(8,77,107,91)(9,45,13,55)(10,46,14,56)(11,47,15,53)(12,48,16,54)(17,96,118,100)(18,93,119,97)(19,94,120,98)(20,95,117,99)(21,116,128,76)(22,113,125,73)(23,114,126,74)(24,115,127,75)(25,71,35,57)(26,72,36,58)(27,69,33,59)(28,70,34,60)(29,83,51,43)(30,84,52,44)(31,81,49,41)(32,82,50,42)(85,123,103,109)(86,124,104,110)(87,121,101,111)(88,122,102,112), (1,18,3,20)(2,17,4,19)(5,14,7,16)(6,13,8,15)(9,107,11,105)(10,106,12,108)(21,52,23,50)(22,51,24,49)(25,109,27,111)(26,112,28,110)(29,127,31,125)(30,126,32,128)(33,121,35,123)(34,124,36,122)(37,119,39,117)(38,118,40,120)(41,79,43,77)(42,78,44,80)(45,103,47,101)(46,102,48,104)(53,87,55,85)(54,86,56,88)(57,95,59,93)(58,94,60,96)(61,113,63,115)(62,116,64,114)(65,73,67,75)(66,76,68,74)(69,97,71,99)(70,100,72,98)(81,89,83,91)(82,92,84,90) );
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)], [(1,51,11,25),(2,52,12,26),(3,49,9,27),(4,50,10,28),(5,126,118,122),(6,127,119,123),(7,128,120,124),(8,125,117,121),(13,33,39,31),(14,34,40,32),(15,35,37,29),(16,36,38,30),(17,112,108,23),(18,109,105,24),(19,110,106,21),(20,111,107,22),(41,67,69,45),(42,68,70,46),(43,65,71,47),(44,66,72,48),(53,83,61,57),(54,84,62,58),(55,81,63,59),(56,82,64,60),(73,77,101,99),(74,78,102,100),(75,79,103,97),(76,80,104,98),(85,93,115,89),(86,94,116,90),(87,95,113,91),(88,96,114,92)], [(1,65,37,61),(2,66,38,62),(3,67,39,63),(4,68,40,64),(5,78,108,92),(6,79,105,89),(7,80,106,90),(8,77,107,91),(9,45,13,55),(10,46,14,56),(11,47,15,53),(12,48,16,54),(17,96,118,100),(18,93,119,97),(19,94,120,98),(20,95,117,99),(21,116,128,76),(22,113,125,73),(23,114,126,74),(24,115,127,75),(25,71,35,57),(26,72,36,58),(27,69,33,59),(28,70,34,60),(29,83,51,43),(30,84,52,44),(31,81,49,41),(32,82,50,42),(85,123,103,109),(86,124,104,110),(87,121,101,111),(88,122,102,112)], [(1,18,3,20),(2,17,4,19),(5,14,7,16),(6,13,8,15),(9,107,11,105),(10,106,12,108),(21,52,23,50),(22,51,24,49),(25,109,27,111),(26,112,28,110),(29,127,31,125),(30,126,32,128),(33,121,35,123),(34,124,36,122),(37,119,39,117),(38,118,40,120),(41,79,43,77),(42,78,44,80),(45,103,47,101),(46,102,48,104),(53,87,55,85),(54,86,56,88),(57,95,59,93),(58,94,60,96),(61,113,63,115),(62,116,64,114),(65,73,67,75),(66,76,68,74),(69,97,71,99),(70,100,72,98),(81,89,83,91),(82,92,84,90)]])
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 | D8 | SD16 | C4○D4 | C8.C22 |
| kernel | C42.122D4 | 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 C42.122D4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 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 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 14 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[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],[4,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,3,14,0,0,0,14,14],[1,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
C42.122D4 in GAP, Magma, Sage, TeX
C_4^2._{122}D_4 % in TeX
G:=Group("C4^2.122D4"); // GroupNames label
G:=SmallGroup(128,720);
// by ID
G=gap.SmallGroup(128,720);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,848,422,100,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^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^-1*c^-1>;
// generators/relations