p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.124D4, C2.4(C8⋊Q8), (C2×C8).32Q8, C4.14(C4×Q8), C42.C2⋊10C4, C42.166(C2×C4), C23.815(C2×D4), (C22×C4).303D4, C4.82(C22⋊Q8), C22.37(C4⋊Q8), C42⋊9C4.13C2, C22.4Q16.51C2, (C2×C42).336C22, (C22×C8).410C22, C22.102(C8⋊C22), (C22×C4).1426C23, C22.70(C4.4D4), C22.91(C8.C22), C2.31(C23.36D4), C2.4(C42.29C22), C2.4(C42.30C22), C2.14(C23.67C23), C4⋊C4.99(C2×C4), (C2×C4).213(C2×Q8), (C2×C8⋊C4).35C2, (C2×C4).1366(C2×D4), (C2×C4⋊C4).97C22, (C2×C42.C2).7C2, (C2×C4).609(C4○D4), (C2×C4).440(C22×C4), (C2×C4).141(C22⋊C4), C22.301(C2×C22⋊C4), SmallGroup(128,724)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.124D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 244 in 130 conjugacy classes, 64 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C22×C8, C22.4Q16, C42⋊9C4, C2×C8⋊C4, C2×C42.C2, C42.124D4
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, C23.36D4, C42.29C22, C42.30C22, C8⋊Q8, C42.124D4
►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 38 29 13)(2 39 30 14)(3 40 31 15)(4 37 32 16)(5 110 121 19)(6 111 122 20)(7 112 123 17)(8 109 124 18)(9 49 34 25)(10 50 35 26)(11 51 36 27)(12 52 33 28)(21 119 108 126)(22 120 105 127)(23 117 106 128)(24 118 107 125)(41 81 71 60)(42 82 72 57)(43 83 69 58)(44 84 70 59)(45 56 67 61)(46 53 68 62)(47 54 65 63)(48 55 66 64)(73 102 113 85)(74 103 114 86)(75 104 115 87)(76 101 116 88)(77 89 93 98)(78 90 94 99)(79 91 95 100)(80 92 96 97)
(1 87 9 79)(2 86 10 78)(3 85 11 77)(4 88 12 80)(5 84 23 66)(6 83 24 65)(7 82 21 68)(8 81 22 67)(13 75 25 91)(14 74 26 90)(15 73 27 89)(16 76 28 92)(17 72 126 62)(18 71 127 61)(19 70 128 64)(20 69 125 63)(29 104 34 95)(30 103 35 94)(31 102 36 93)(32 101 33 96)(37 116 52 97)(38 115 49 100)(39 114 50 99)(40 113 51 98)(41 120 56 109)(42 119 53 112)(43 118 54 111)(44 117 55 110)(45 124 60 105)(46 123 57 108)(47 122 58 107)(48 121 59 106)
(1 118 31 127)(2 128 32 119)(3 120 29 125)(4 126 30 117)(5 52 123 26)(6 27 124 49)(7 50 121 28)(8 25 122 51)(9 111 36 18)(10 19 33 112)(11 109 34 20)(12 17 35 110)(13 107 40 22)(14 23 37 108)(15 105 38 24)(16 21 39 106)(41 75 69 113)(42 114 70 76)(43 73 71 115)(44 116 72 74)(45 95 65 77)(46 78 66 96)(47 93 67 79)(48 80 68 94)(53 99 64 92)(54 89 61 100)(55 97 62 90)(56 91 63 98)(57 86 84 101)(58 102 81 87)(59 88 82 103)(60 104 83 85)
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), (1,38,29,13)(2,39,30,14)(3,40,31,15)(4,37,32,16)(5,110,121,19)(6,111,122,20)(7,112,123,17)(8,109,124,18)(9,49,34,25)(10,50,35,26)(11,51,36,27)(12,52,33,28)(21,119,108,126)(22,120,105,127)(23,117,106,128)(24,118,107,125)(41,81,71,60)(42,82,72,57)(43,83,69,58)(44,84,70,59)(45,56,67,61)(46,53,68,62)(47,54,65,63)(48,55,66,64)(73,102,113,85)(74,103,114,86)(75,104,115,87)(76,101,116,88)(77,89,93,98)(78,90,94,99)(79,91,95,100)(80,92,96,97), (1,87,9,79)(2,86,10,78)(3,85,11,77)(4,88,12,80)(5,84,23,66)(6,83,24,65)(7,82,21,68)(8,81,22,67)(13,75,25,91)(14,74,26,90)(15,73,27,89)(16,76,28,92)(17,72,126,62)(18,71,127,61)(19,70,128,64)(20,69,125,63)(29,104,34,95)(30,103,35,94)(31,102,36,93)(32,101,33,96)(37,116,52,97)(38,115,49,100)(39,114,50,99)(40,113,51,98)(41,120,56,109)(42,119,53,112)(43,118,54,111)(44,117,55,110)(45,124,60,105)(46,123,57,108)(47,122,58,107)(48,121,59,106), (1,118,31,127)(2,128,32,119)(3,120,29,125)(4,126,30,117)(5,52,123,26)(6,27,124,49)(7,50,121,28)(8,25,122,51)(9,111,36,18)(10,19,33,112)(11,109,34,20)(12,17,35,110)(13,107,40,22)(14,23,37,108)(15,105,38,24)(16,21,39,106)(41,75,69,113)(42,114,70,76)(43,73,71,115)(44,116,72,74)(45,95,65,77)(46,78,66,96)(47,93,67,79)(48,80,68,94)(53,99,64,92)(54,89,61,100)(55,97,62,90)(56,91,63,98)(57,86,84,101)(58,102,81,87)(59,88,82,103)(60,104,83,85)>;
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), (1,38,29,13)(2,39,30,14)(3,40,31,15)(4,37,32,16)(5,110,121,19)(6,111,122,20)(7,112,123,17)(8,109,124,18)(9,49,34,25)(10,50,35,26)(11,51,36,27)(12,52,33,28)(21,119,108,126)(22,120,105,127)(23,117,106,128)(24,118,107,125)(41,81,71,60)(42,82,72,57)(43,83,69,58)(44,84,70,59)(45,56,67,61)(46,53,68,62)(47,54,65,63)(48,55,66,64)(73,102,113,85)(74,103,114,86)(75,104,115,87)(76,101,116,88)(77,89,93,98)(78,90,94,99)(79,91,95,100)(80,92,96,97), (1,87,9,79)(2,86,10,78)(3,85,11,77)(4,88,12,80)(5,84,23,66)(6,83,24,65)(7,82,21,68)(8,81,22,67)(13,75,25,91)(14,74,26,90)(15,73,27,89)(16,76,28,92)(17,72,126,62)(18,71,127,61)(19,70,128,64)(20,69,125,63)(29,104,34,95)(30,103,35,94)(31,102,36,93)(32,101,33,96)(37,116,52,97)(38,115,49,100)(39,114,50,99)(40,113,51,98)(41,120,56,109)(42,119,53,112)(43,118,54,111)(44,117,55,110)(45,124,60,105)(46,123,57,108)(47,122,58,107)(48,121,59,106), (1,118,31,127)(2,128,32,119)(3,120,29,125)(4,126,30,117)(5,52,123,26)(6,27,124,49)(7,50,121,28)(8,25,122,51)(9,111,36,18)(10,19,33,112)(11,109,34,20)(12,17,35,110)(13,107,40,22)(14,23,37,108)(15,105,38,24)(16,21,39,106)(41,75,69,113)(42,114,70,76)(43,73,71,115)(44,116,72,74)(45,95,65,77)(46,78,66,96)(47,93,67,79)(48,80,68,94)(53,99,64,92)(54,89,61,100)(55,97,62,90)(56,91,63,98)(57,86,84,101)(58,102,81,87)(59,88,82,103)(60,104,83,85) );
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)], [(1,38,29,13),(2,39,30,14),(3,40,31,15),(4,37,32,16),(5,110,121,19),(6,111,122,20),(7,112,123,17),(8,109,124,18),(9,49,34,25),(10,50,35,26),(11,51,36,27),(12,52,33,28),(21,119,108,126),(22,120,105,127),(23,117,106,128),(24,118,107,125),(41,81,71,60),(42,82,72,57),(43,83,69,58),(44,84,70,59),(45,56,67,61),(46,53,68,62),(47,54,65,63),(48,55,66,64),(73,102,113,85),(74,103,114,86),(75,104,115,87),(76,101,116,88),(77,89,93,98),(78,90,94,99),(79,91,95,100),(80,92,96,97)], [(1,87,9,79),(2,86,10,78),(3,85,11,77),(4,88,12,80),(5,84,23,66),(6,83,24,65),(7,82,21,68),(8,81,22,67),(13,75,25,91),(14,74,26,90),(15,73,27,89),(16,76,28,92),(17,72,126,62),(18,71,127,61),(19,70,128,64),(20,69,125,63),(29,104,34,95),(30,103,35,94),(31,102,36,93),(32,101,33,96),(37,116,52,97),(38,115,49,100),(39,114,50,99),(40,113,51,98),(41,120,56,109),(42,119,53,112),(43,118,54,111),(44,117,55,110),(45,124,60,105),(46,123,57,108),(47,122,58,107),(48,121,59,106)], [(1,118,31,127),(2,128,32,119),(3,120,29,125),(4,126,30,117),(5,52,123,26),(6,27,124,49),(7,50,121,28),(8,25,122,51),(9,111,36,18),(10,19,33,112),(11,109,34,20),(12,17,35,110),(13,107,40,22),(14,23,37,108),(15,105,38,24),(16,21,39,106),(41,75,69,113),(42,114,70,76),(43,73,71,115),(44,116,72,74),(45,95,65,77),(46,78,66,96),(47,93,67,79),(48,80,68,94),(53,99,64,92),(54,89,61,100),(55,97,62,90),(56,91,63,98),(57,86,84,101),(58,102,81,87),(59,88,82,103),(60,104,83,85)]])
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 | C42.124D4 | C22.4Q16 | C42⋊9C4 | C2×C8⋊C4 | C2×C42.C2 | C42.C2 | 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 C42.124D4 ►in GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 16 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 0 | 16 | 16 |
| 8 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 14 | 2 | 15 |
| 0 | 0 | 0 | 0 | 1 | 9 | 4 | 6 |
| 0 | 0 | 0 | 0 | 1 | 16 | 4 | 13 |
| 0 | 0 | 0 | 0 | 16 | 13 | 8 | 11 |
| 9 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 6 | 15 | 6 | 7 |
| 0 | 0 | 0 | 0 | 11 | 12 | 3 | 12 |
| 0 | 0 | 0 | 0 | 14 | 16 | 8 | 13 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,0,16,0,1,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,16,0,0,0,0,16,0,1,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[8,12,0,0,0,0,0,0,13,9,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,10,1,1,16,0,0,0,0,14,9,16,13,0,0,0,0,2,4,4,8,0,0,0,0,15,6,13,11],[9,14,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,7,12,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,3,6,11,14,0,0,0,0,0,15,12,16,0,0,0,0,1,6,3,8,0,0,0,0,12,7,12,13] >;
C42.124D4 in GAP, Magma, Sage, TeX
C_4^2._{124}D_4 % in TeX
G:=Group("C4^2.124D4"); // GroupNames label
G:=SmallGroup(128,724);
// by ID
G=gap.SmallGroup(128,724);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,723,100,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;
// generators/relations