p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.437D4, (C2×C8).40Q8, C4.13(C4×Q8), C42.C2⋊9C4, C42.270(C2×C4), (C22×C4).561D4, C23.814(C2×D4), C22.36(C4⋊Q8), C4.81(C22⋊Q8), C2.4(C8.5Q8), C42⋊8C4.12C2, C22.80(C4○D8), C22.4Q16.20C2, (C22×C8).497C22, (C22×C4).1425C23, (C2×C42).1080C22, C22.69(C4.4D4), C2.31(C23.24D4), C2.5(C42.78C22), C2.13(C23.67C23), (C2×C4×C8).26C2, C4⋊C4.98(C2×C4), (C2×C4).212(C2×Q8), (C2×C4).1365(C2×D4), (C2×C4⋊C4).96C22, (C2×C42.C2).6C2, (C2×C4).608(C4○D4), (C2×C4).439(C22×C4), (C2×C4).207(C22⋊C4), C22.300(C2×C22⋊C4), SmallGroup(128,723)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.437D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 228 in 126 conjugacy classes, 64 normal (14 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, C2.C42, C4×C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C22×C8, C22.4Q16, C42⋊8C4, C2×C4×C8, C2×C42.C2, C42.437D4
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, C4○D8, C23.67C23, C23.24D4, C42.78C22, C8.5Q8, C42.437D4
►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 39)(2 49 9 40)(3 50 10 37)(4 51 11 38)(5 110 23 119)(6 111 24 120)(7 112 21 117)(8 109 22 118)(13 29 25 34)(14 30 26 35)(15 31 27 36)(16 32 28 33)(17 122 126 105)(18 123 127 106)(19 124 128 107)(20 121 125 108)(41 68 54 81)(42 65 55 82)(43 66 56 83)(44 67 53 84)(45 61 57 70)(46 62 58 71)(47 63 59 72)(48 64 60 69)(73 103 90 94)(74 104 91 95)(75 101 92 96)(76 102 89 93)(77 100 86 113)(78 97 87 114)(79 98 88 115)(80 99 85 116)
(1 95 34 87)(2 103 35 77)(3 93 36 85)(4 101 33 79)(5 67 106 57)(6 83 107 48)(7 65 108 59)(8 81 105 46)(9 94 30 86)(10 102 31 80)(11 96 32 88)(12 104 29 78)(13 97 52 91)(14 113 49 73)(15 99 50 89)(16 115 51 75)(17 71 109 54)(18 61 110 44)(19 69 111 56)(20 63 112 42)(21 82 121 47)(22 68 122 58)(23 84 123 45)(24 66 124 60)(25 114 39 74)(26 100 40 90)(27 116 37 76)(28 98 38 92)(41 126 62 118)(43 128 64 120)(53 127 70 119)(55 125 72 117)
(1 117 3 119)(2 111 4 109)(5 39 7 37)(6 51 8 49)(9 120 11 118)(10 110 12 112)(13 121 15 123)(14 107 16 105)(17 35 19 33)(18 29 20 31)(21 50 23 52)(22 40 24 38)(25 108 27 106)(26 124 28 122)(30 128 32 126)(34 125 36 127)(41 100 43 98)(42 116 44 114)(45 104 47 102)(46 94 48 96)(53 97 55 99)(54 113 56 115)(57 95 59 93)(58 103 60 101)(61 74 63 76)(62 90 64 92)(65 85 67 87)(66 79 68 77)(69 75 71 73)(70 91 72 89)(78 82 80 84)(81 86 83 88)
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), (1,52,12,39)(2,49,9,40)(3,50,10,37)(4,51,11,38)(5,110,23,119)(6,111,24,120)(7,112,21,117)(8,109,22,118)(13,29,25,34)(14,30,26,35)(15,31,27,36)(16,32,28,33)(17,122,126,105)(18,123,127,106)(19,124,128,107)(20,121,125,108)(41,68,54,81)(42,65,55,82)(43,66,56,83)(44,67,53,84)(45,61,57,70)(46,62,58,71)(47,63,59,72)(48,64,60,69)(73,103,90,94)(74,104,91,95)(75,101,92,96)(76,102,89,93)(77,100,86,113)(78,97,87,114)(79,98,88,115)(80,99,85,116), (1,95,34,87)(2,103,35,77)(3,93,36,85)(4,101,33,79)(5,67,106,57)(6,83,107,48)(7,65,108,59)(8,81,105,46)(9,94,30,86)(10,102,31,80)(11,96,32,88)(12,104,29,78)(13,97,52,91)(14,113,49,73)(15,99,50,89)(16,115,51,75)(17,71,109,54)(18,61,110,44)(19,69,111,56)(20,63,112,42)(21,82,121,47)(22,68,122,58)(23,84,123,45)(24,66,124,60)(25,114,39,74)(26,100,40,90)(27,116,37,76)(28,98,38,92)(41,126,62,118)(43,128,64,120)(53,127,70,119)(55,125,72,117), (1,117,3,119)(2,111,4,109)(5,39,7,37)(6,51,8,49)(9,120,11,118)(10,110,12,112)(13,121,15,123)(14,107,16,105)(17,35,19,33)(18,29,20,31)(21,50,23,52)(22,40,24,38)(25,108,27,106)(26,124,28,122)(30,128,32,126)(34,125,36,127)(41,100,43,98)(42,116,44,114)(45,104,47,102)(46,94,48,96)(53,97,55,99)(54,113,56,115)(57,95,59,93)(58,103,60,101)(61,74,63,76)(62,90,64,92)(65,85,67,87)(66,79,68,77)(69,75,71,73)(70,91,72,89)(78,82,80,84)(81,86,83,88)>;
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), (1,52,12,39)(2,49,9,40)(3,50,10,37)(4,51,11,38)(5,110,23,119)(6,111,24,120)(7,112,21,117)(8,109,22,118)(13,29,25,34)(14,30,26,35)(15,31,27,36)(16,32,28,33)(17,122,126,105)(18,123,127,106)(19,124,128,107)(20,121,125,108)(41,68,54,81)(42,65,55,82)(43,66,56,83)(44,67,53,84)(45,61,57,70)(46,62,58,71)(47,63,59,72)(48,64,60,69)(73,103,90,94)(74,104,91,95)(75,101,92,96)(76,102,89,93)(77,100,86,113)(78,97,87,114)(79,98,88,115)(80,99,85,116), (1,95,34,87)(2,103,35,77)(3,93,36,85)(4,101,33,79)(5,67,106,57)(6,83,107,48)(7,65,108,59)(8,81,105,46)(9,94,30,86)(10,102,31,80)(11,96,32,88)(12,104,29,78)(13,97,52,91)(14,113,49,73)(15,99,50,89)(16,115,51,75)(17,71,109,54)(18,61,110,44)(19,69,111,56)(20,63,112,42)(21,82,121,47)(22,68,122,58)(23,84,123,45)(24,66,124,60)(25,114,39,74)(26,100,40,90)(27,116,37,76)(28,98,38,92)(41,126,62,118)(43,128,64,120)(53,127,70,119)(55,125,72,117), (1,117,3,119)(2,111,4,109)(5,39,7,37)(6,51,8,49)(9,120,11,118)(10,110,12,112)(13,121,15,123)(14,107,16,105)(17,35,19,33)(18,29,20,31)(21,50,23,52)(22,40,24,38)(25,108,27,106)(26,124,28,122)(30,128,32,126)(34,125,36,127)(41,100,43,98)(42,116,44,114)(45,104,47,102)(46,94,48,96)(53,97,55,99)(54,113,56,115)(57,95,59,93)(58,103,60,101)(61,74,63,76)(62,90,64,92)(65,85,67,87)(66,79,68,77)(69,75,71,73)(70,91,72,89)(78,82,80,84)(81,86,83,88) );
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)], [(1,52,12,39),(2,49,9,40),(3,50,10,37),(4,51,11,38),(5,110,23,119),(6,111,24,120),(7,112,21,117),(8,109,22,118),(13,29,25,34),(14,30,26,35),(15,31,27,36),(16,32,28,33),(17,122,126,105),(18,123,127,106),(19,124,128,107),(20,121,125,108),(41,68,54,81),(42,65,55,82),(43,66,56,83),(44,67,53,84),(45,61,57,70),(46,62,58,71),(47,63,59,72),(48,64,60,69),(73,103,90,94),(74,104,91,95),(75,101,92,96),(76,102,89,93),(77,100,86,113),(78,97,87,114),(79,98,88,115),(80,99,85,116)], [(1,95,34,87),(2,103,35,77),(3,93,36,85),(4,101,33,79),(5,67,106,57),(6,83,107,48),(7,65,108,59),(8,81,105,46),(9,94,30,86),(10,102,31,80),(11,96,32,88),(12,104,29,78),(13,97,52,91),(14,113,49,73),(15,99,50,89),(16,115,51,75),(17,71,109,54),(18,61,110,44),(19,69,111,56),(20,63,112,42),(21,82,121,47),(22,68,122,58),(23,84,123,45),(24,66,124,60),(25,114,39,74),(26,100,40,90),(27,116,37,76),(28,98,38,92),(41,126,62,118),(43,128,64,120),(53,127,70,119),(55,125,72,117)], [(1,117,3,119),(2,111,4,109),(5,39,7,37),(6,51,8,49),(9,120,11,118),(10,110,12,112),(13,121,15,123),(14,107,16,105),(17,35,19,33),(18,29,20,31),(21,50,23,52),(22,40,24,38),(25,108,27,106),(26,124,28,122),(30,128,32,126),(34,125,36,127),(41,100,43,98),(42,116,44,114),(45,104,47,102),(46,94,48,96),(53,97,55,99),(54,113,56,115),(57,95,59,93),(58,103,60,101),(61,74,63,76),(62,90,64,92),(65,85,67,87),(66,79,68,77),(69,75,71,73),(70,91,72,89),(78,82,80,84),(81,86,83,88)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | C4○D4 | C4○D8 |
| kernel | C42.437D4 | C22.4Q16 | C42⋊8C4 | C2×C4×C8 | C2×C42.C2 | C42.C2 | C42 | C2×C8 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 16 |
Matrix representation of C42.437D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 6 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 6 | 16 | 0 | 0 | 0 | 0 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 4 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[1,6,0,0,0,0,11,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[6,1,0,0,0,0,16,11,0,0,0,0,0,0,15,3,0,0,0,0,4,2,0,0,0,0,0,0,0,8,0,0,0,0,15,0] >;
C42.437D4 in GAP, Magma, Sage, TeX
C_4^2._{437}D_4 % in TeX
G:=Group("C4^2.437D4"); // GroupNames label
G:=SmallGroup(128,723);
// by ID
G=gap.SmallGroup(128,723);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,436,2019,248]);
// 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=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