p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.125D4, (C2×C8)⋊8Q8, C4⋊Q8⋊26C4, C2.5(C8⋊Q8), C4.15(C4×Q8), C42.167(C2×C4), C23.816(C2×D4), (C22×C4).304D4, C4.83(C22⋊Q8), C22.38(C4⋊Q8), C42⋊8C4.13C2, C22.4Q16.52C2, (C22×C8).411C22, (C2×C42).337C22, C22.103(C8⋊C22), (C22×C4).1427C23, C22.71(C4.4D4), C22.92(C8.C22), C2.25(C23.37D4), C2.25(C23.38D4), C2.5(C42.28C22), C2.15(C23.67C23), (C2×C4⋊Q8).16C2, C4⋊C4.100(C2×C4), (C2×C4).214(C2×Q8), (C2×C8⋊C4).36C2, (C2×C4).1367(C2×D4), (C2×C4⋊C4).98C22, (C2×C4).610(C4○D4), (C2×C4).441(C22×C4), (C2×C4).142(C22⋊C4), C22.302(C2×C22⋊C4), SmallGroup(128,725)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.125D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 276 in 138 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2.C42, C8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C22.4Q16, C42⋊8C4, C2×C8⋊C4, C2×C4⋊Q8, C42.125D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C8⋊C22, C8.C22, C23.67C23, C23.37D4, C23.38D4, C42.28C22, C8⋊Q8, C42.125D4
►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 52 12 38)(2 49 9 39)(3 50 10 40)(4 51 11 37)(5 111 23 124)(6 112 24 121)(7 109 21 122)(8 110 22 123)(13 29 25 35)(14 30 26 36)(15 31 27 33)(16 32 28 34)(17 105 126 117)(18 106 127 118)(19 107 128 119)(20 108 125 120)(41 83 45 66)(42 84 46 67)(43 81 47 68)(44 82 48 65)(53 61 60 70)(54 62 57 71)(55 63 58 72)(56 64 59 69)(73 104 90 94)(74 101 91 95)(75 102 92 96)(76 103 89 93)(77 99 86 113)(78 100 87 114)(79 97 88 115)(80 98 85 116)
(1 95 35 87)(2 104 36 77)(3 93 33 85)(4 102 34 79)(5 67 118 60)(6 83 119 56)(7 65 120 58)(8 81 117 54)(9 94 30 86)(10 103 31 80)(11 96 32 88)(12 101 29 78)(13 100 52 91)(14 113 49 73)(15 98 50 89)(16 115 51 75)(17 71 110 43)(18 61 111 46)(19 69 112 41)(20 63 109 48)(21 82 108 55)(22 68 105 57)(23 84 106 53)(24 66 107 59)(25 114 38 74)(26 99 39 90)(27 116 40 76)(28 97 37 92)(42 127 70 124)(44 125 72 122)(45 128 64 121)(47 126 62 123)
(1 109 3 111)(2 112 4 110)(5 52 7 50)(6 51 8 49)(9 121 11 123)(10 124 12 122)(13 120 15 118)(14 119 16 117)(17 36 19 34)(18 35 20 33)(21 40 23 38)(22 39 24 37)(25 108 27 106)(26 107 28 105)(29 125 31 127)(30 128 32 126)(41 115 43 113)(42 114 44 116)(45 97 47 99)(46 100 48 98)(53 95 55 93)(54 94 56 96)(57 104 59 102)(58 103 60 101)(61 91 63 89)(62 90 64 92)(65 80 67 78)(66 79 68 77)(69 75 71 73)(70 74 72 76)(81 86 83 88)(82 85 84 87)
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), (1,52,12,38)(2,49,9,39)(3,50,10,40)(4,51,11,37)(5,111,23,124)(6,112,24,121)(7,109,21,122)(8,110,22,123)(13,29,25,35)(14,30,26,36)(15,31,27,33)(16,32,28,34)(17,105,126,117)(18,106,127,118)(19,107,128,119)(20,108,125,120)(41,83,45,66)(42,84,46,67)(43,81,47,68)(44,82,48,65)(53,61,60,70)(54,62,57,71)(55,63,58,72)(56,64,59,69)(73,104,90,94)(74,101,91,95)(75,102,92,96)(76,103,89,93)(77,99,86,113)(78,100,87,114)(79,97,88,115)(80,98,85,116), (1,95,35,87)(2,104,36,77)(3,93,33,85)(4,102,34,79)(5,67,118,60)(6,83,119,56)(7,65,120,58)(8,81,117,54)(9,94,30,86)(10,103,31,80)(11,96,32,88)(12,101,29,78)(13,100,52,91)(14,113,49,73)(15,98,50,89)(16,115,51,75)(17,71,110,43)(18,61,111,46)(19,69,112,41)(20,63,109,48)(21,82,108,55)(22,68,105,57)(23,84,106,53)(24,66,107,59)(25,114,38,74)(26,99,39,90)(27,116,40,76)(28,97,37,92)(42,127,70,124)(44,125,72,122)(45,128,64,121)(47,126,62,123), (1,109,3,111)(2,112,4,110)(5,52,7,50)(6,51,8,49)(9,121,11,123)(10,124,12,122)(13,120,15,118)(14,119,16,117)(17,36,19,34)(18,35,20,33)(21,40,23,38)(22,39,24,37)(25,108,27,106)(26,107,28,105)(29,125,31,127)(30,128,32,126)(41,115,43,113)(42,114,44,116)(45,97,47,99)(46,100,48,98)(53,95,55,93)(54,94,56,96)(57,104,59,102)(58,103,60,101)(61,91,63,89)(62,90,64,92)(65,80,67,78)(66,79,68,77)(69,75,71,73)(70,74,72,76)(81,86,83,88)(82,85,84,87)>;
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), (1,52,12,38)(2,49,9,39)(3,50,10,40)(4,51,11,37)(5,111,23,124)(6,112,24,121)(7,109,21,122)(8,110,22,123)(13,29,25,35)(14,30,26,36)(15,31,27,33)(16,32,28,34)(17,105,126,117)(18,106,127,118)(19,107,128,119)(20,108,125,120)(41,83,45,66)(42,84,46,67)(43,81,47,68)(44,82,48,65)(53,61,60,70)(54,62,57,71)(55,63,58,72)(56,64,59,69)(73,104,90,94)(74,101,91,95)(75,102,92,96)(76,103,89,93)(77,99,86,113)(78,100,87,114)(79,97,88,115)(80,98,85,116), (1,95,35,87)(2,104,36,77)(3,93,33,85)(4,102,34,79)(5,67,118,60)(6,83,119,56)(7,65,120,58)(8,81,117,54)(9,94,30,86)(10,103,31,80)(11,96,32,88)(12,101,29,78)(13,100,52,91)(14,113,49,73)(15,98,50,89)(16,115,51,75)(17,71,110,43)(18,61,111,46)(19,69,112,41)(20,63,109,48)(21,82,108,55)(22,68,105,57)(23,84,106,53)(24,66,107,59)(25,114,38,74)(26,99,39,90)(27,116,40,76)(28,97,37,92)(42,127,70,124)(44,125,72,122)(45,128,64,121)(47,126,62,123), (1,109,3,111)(2,112,4,110)(5,52,7,50)(6,51,8,49)(9,121,11,123)(10,124,12,122)(13,120,15,118)(14,119,16,117)(17,36,19,34)(18,35,20,33)(21,40,23,38)(22,39,24,37)(25,108,27,106)(26,107,28,105)(29,125,31,127)(30,128,32,126)(41,115,43,113)(42,114,44,116)(45,97,47,99)(46,100,48,98)(53,95,55,93)(54,94,56,96)(57,104,59,102)(58,103,60,101)(61,91,63,89)(62,90,64,92)(65,80,67,78)(66,79,68,77)(69,75,71,73)(70,74,72,76)(81,86,83,88)(82,85,84,87) );
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)], [(1,52,12,38),(2,49,9,39),(3,50,10,40),(4,51,11,37),(5,111,23,124),(6,112,24,121),(7,109,21,122),(8,110,22,123),(13,29,25,35),(14,30,26,36),(15,31,27,33),(16,32,28,34),(17,105,126,117),(18,106,127,118),(19,107,128,119),(20,108,125,120),(41,83,45,66),(42,84,46,67),(43,81,47,68),(44,82,48,65),(53,61,60,70),(54,62,57,71),(55,63,58,72),(56,64,59,69),(73,104,90,94),(74,101,91,95),(75,102,92,96),(76,103,89,93),(77,99,86,113),(78,100,87,114),(79,97,88,115),(80,98,85,116)], [(1,95,35,87),(2,104,36,77),(3,93,33,85),(4,102,34,79),(5,67,118,60),(6,83,119,56),(7,65,120,58),(8,81,117,54),(9,94,30,86),(10,103,31,80),(11,96,32,88),(12,101,29,78),(13,100,52,91),(14,113,49,73),(15,98,50,89),(16,115,51,75),(17,71,110,43),(18,61,111,46),(19,69,112,41),(20,63,109,48),(21,82,108,55),(22,68,105,57),(23,84,106,53),(24,66,107,59),(25,114,38,74),(26,99,39,90),(27,116,40,76),(28,97,37,92),(42,127,70,124),(44,125,72,122),(45,128,64,121),(47,126,62,123)], [(1,109,3,111),(2,112,4,110),(5,52,7,50),(6,51,8,49),(9,121,11,123),(10,124,12,122),(13,120,15,118),(14,119,16,117),(17,36,19,34),(18,35,20,33),(21,40,23,38),(22,39,24,37),(25,108,27,106),(26,107,28,105),(29,125,31,127),(30,128,32,126),(41,115,43,113),(42,114,44,116),(45,97,47,99),(46,100,48,98),(53,95,55,93),(54,94,56,96),(57,104,59,102),(58,103,60,101),(61,91,63,89),(62,90,64,92),(65,80,67,78),(66,79,68,77),(69,75,71,73),(70,74,72,76),(81,86,83,88),(82,85,84,87)]])
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 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.125D4 | C22.4Q16 | C42⋊8C4 | C2×C8⋊C4 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C2×C8 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 2 | 2 |
Matrix representation of C42.125D4 ►in GL8(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 13 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 3 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 16 | 4 | 0 |
| 0 | 0 | 0 | 0 | 6 | 14 | 6 | 13 |
| 7 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 7 | 8 | 7 |
| 0 | 0 | 0 | 0 | 0 | 6 | 6 | 6 |
| 0 | 0 | 0 | 0 | 15 | 11 | 8 | 7 |
| 0 | 0 | 0 | 0 | 2 | 14 | 3 | 4 |
| 8 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 14 | 3 | 4 |
| 0 | 0 | 0 | 0 | 9 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 8 | 11 | 9 | 13 |
| 0 | 0 | 0 | 0 | 0 | 6 | 6 | 6 |
G:=sub<GL(8,GF(17))| [1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,12,0,0,0,0,1,13,0,3,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,13,6,0,0,0,0,16,0,16,14,0,0,0,0,0,0,4,6,0,0,0,0,0,0,0,13],[7,13,0,0,0,0,0,0,12,10,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,15,8,0,0,0,0,0,0,0,0,16,0,15,2,0,0,0,0,7,6,11,14,0,0,0,0,8,6,8,3,0,0,0,0,7,6,7,4],[8,7,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,15,8,0,0,0,0,0,0,0,0,2,9,8,0,0,0,0,0,14,0,11,6,0,0,0,0,3,2,9,6,0,0,0,0,4,1,13,6] >;
C42.125D4 in GAP, Magma, Sage, TeX
C_4^2._{125}D_4 % in TeX
G:=Group("C4^2.125D4"); // GroupNames label
G:=SmallGroup(128,725);
// by ID
G=gap.SmallGroup(128,725);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,723,436,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;
// generators/relations