p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.39Q8, C23.547C24, C22.2392- 1+4, C22.3222+ 1+4, C4.8(C42.C2), C42⋊8C4.39C2, C42⋊4C4.27C2, C42⋊9C4.35C2, (C2×C42).621C22, (C22×C4).157C23, C22.136(C22×Q8), C2.C42.266C22, C23.65C23.68C2, C23.83C23.26C2, C23.81C23.27C2, C2.23(C23.41C23), C2.55(C22.36C24), C2.31(C22.35C24), C2.31(C22.34C24), (C2×C4).133(C2×Q8), C2.20(C2×C42.C2), (C2×C4).667(C4○D4), (C2×C4⋊C4).373C22, C22.419(C2×C4○D4), SmallGroup(128,1379)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.39Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 308 in 180 conjugacy classes, 100 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C42⋊4C4, C42⋊8C4, C42⋊9C4, C23.65C23, C23.81C23, C23.83C23, C42.39Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C42.C2, C22.34C24, C22.35C24, C22.36C24, C23.41C23, C42.39Q8
►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)
(1 124 35 30)(2 121 36 31)(3 122 33 32)(4 123 34 29)(5 44 104 72)(6 41 101 69)(7 42 102 70)(8 43 103 71)(9 62 95 82)(10 63 96 83)(11 64 93 84)(12 61 94 81)(13 80 107 60)(14 77 108 57)(15 78 105 58)(16 79 106 59)(17 90 111 51)(18 91 112 52)(19 92 109 49)(20 89 110 50)(21 55 115 88)(22 56 116 85)(23 53 113 86)(24 54 114 87)(25 67 119 39)(26 68 120 40)(27 65 117 37)(28 66 118 38)(45 125 73 99)(46 126 74 100)(47 127 75 97)(48 128 76 98)
(1 125 117 42)(2 98 118 69)(3 127 119 44)(4 100 120 71)(5 122 47 39)(6 31 48 66)(7 124 45 37)(8 29 46 68)(9 109 59 85)(10 18 60 55)(11 111 57 87)(12 20 58 53)(13 21 63 52)(14 114 64 90)(15 23 61 50)(16 116 62 92)(17 77 54 93)(19 79 56 95)(22 82 49 106)(24 84 51 108)(25 72 33 97)(26 43 34 126)(27 70 35 99)(28 41 36 128)(30 73 65 102)(32 75 67 104)(38 101 121 76)(40 103 123 74)(78 86 94 110)(80 88 96 112)(81 89 105 113)(83 91 107 115)
(1 80 25 12)(2 57 26 95)(3 78 27 10)(4 59 28 93)(5 113 73 52)(6 24 74 92)(7 115 75 50)(8 22 76 90)(9 36 77 120)(11 34 79 118)(13 39 81 30)(14 68 82 121)(15 37 83 32)(16 66 84 123)(17 71 85 128)(18 44 86 99)(19 69 87 126)(20 42 88 97)(21 47 89 102)(23 45 91 104)(29 106 38 64)(31 108 40 62)(33 58 117 96)(35 60 119 94)(41 54 100 109)(43 56 98 111)(46 49 101 114)(48 51 103 116)(53 125 112 72)(55 127 110 70)(61 124 107 67)(63 122 105 65)
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), (1,124,35,30)(2,121,36,31)(3,122,33,32)(4,123,34,29)(5,44,104,72)(6,41,101,69)(7,42,102,70)(8,43,103,71)(9,62,95,82)(10,63,96,83)(11,64,93,84)(12,61,94,81)(13,80,107,60)(14,77,108,57)(15,78,105,58)(16,79,106,59)(17,90,111,51)(18,91,112,52)(19,92,109,49)(20,89,110,50)(21,55,115,88)(22,56,116,85)(23,53,113,86)(24,54,114,87)(25,67,119,39)(26,68,120,40)(27,65,117,37)(28,66,118,38)(45,125,73,99)(46,126,74,100)(47,127,75,97)(48,128,76,98), (1,125,117,42)(2,98,118,69)(3,127,119,44)(4,100,120,71)(5,122,47,39)(6,31,48,66)(7,124,45,37)(8,29,46,68)(9,109,59,85)(10,18,60,55)(11,111,57,87)(12,20,58,53)(13,21,63,52)(14,114,64,90)(15,23,61,50)(16,116,62,92)(17,77,54,93)(19,79,56,95)(22,82,49,106)(24,84,51,108)(25,72,33,97)(26,43,34,126)(27,70,35,99)(28,41,36,128)(30,73,65,102)(32,75,67,104)(38,101,121,76)(40,103,123,74)(78,86,94,110)(80,88,96,112)(81,89,105,113)(83,91,107,115), (1,80,25,12)(2,57,26,95)(3,78,27,10)(4,59,28,93)(5,113,73,52)(6,24,74,92)(7,115,75,50)(8,22,76,90)(9,36,77,120)(11,34,79,118)(13,39,81,30)(14,68,82,121)(15,37,83,32)(16,66,84,123)(17,71,85,128)(18,44,86,99)(19,69,87,126)(20,42,88,97)(21,47,89,102)(23,45,91,104)(29,106,38,64)(31,108,40,62)(33,58,117,96)(35,60,119,94)(41,54,100,109)(43,56,98,111)(46,49,101,114)(48,51,103,116)(53,125,112,72)(55,127,110,70)(61,124,107,67)(63,122,105,65)>;
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), (1,124,35,30)(2,121,36,31)(3,122,33,32)(4,123,34,29)(5,44,104,72)(6,41,101,69)(7,42,102,70)(8,43,103,71)(9,62,95,82)(10,63,96,83)(11,64,93,84)(12,61,94,81)(13,80,107,60)(14,77,108,57)(15,78,105,58)(16,79,106,59)(17,90,111,51)(18,91,112,52)(19,92,109,49)(20,89,110,50)(21,55,115,88)(22,56,116,85)(23,53,113,86)(24,54,114,87)(25,67,119,39)(26,68,120,40)(27,65,117,37)(28,66,118,38)(45,125,73,99)(46,126,74,100)(47,127,75,97)(48,128,76,98), (1,125,117,42)(2,98,118,69)(3,127,119,44)(4,100,120,71)(5,122,47,39)(6,31,48,66)(7,124,45,37)(8,29,46,68)(9,109,59,85)(10,18,60,55)(11,111,57,87)(12,20,58,53)(13,21,63,52)(14,114,64,90)(15,23,61,50)(16,116,62,92)(17,77,54,93)(19,79,56,95)(22,82,49,106)(24,84,51,108)(25,72,33,97)(26,43,34,126)(27,70,35,99)(28,41,36,128)(30,73,65,102)(32,75,67,104)(38,101,121,76)(40,103,123,74)(78,86,94,110)(80,88,96,112)(81,89,105,113)(83,91,107,115), (1,80,25,12)(2,57,26,95)(3,78,27,10)(4,59,28,93)(5,113,73,52)(6,24,74,92)(7,115,75,50)(8,22,76,90)(9,36,77,120)(11,34,79,118)(13,39,81,30)(14,68,82,121)(15,37,83,32)(16,66,84,123)(17,71,85,128)(18,44,86,99)(19,69,87,126)(20,42,88,97)(21,47,89,102)(23,45,91,104)(29,106,38,64)(31,108,40,62)(33,58,117,96)(35,60,119,94)(41,54,100,109)(43,56,98,111)(46,49,101,114)(48,51,103,116)(53,125,112,72)(55,127,110,70)(61,124,107,67)(63,122,105,65) );
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)], [(1,124,35,30),(2,121,36,31),(3,122,33,32),(4,123,34,29),(5,44,104,72),(6,41,101,69),(7,42,102,70),(8,43,103,71),(9,62,95,82),(10,63,96,83),(11,64,93,84),(12,61,94,81),(13,80,107,60),(14,77,108,57),(15,78,105,58),(16,79,106,59),(17,90,111,51),(18,91,112,52),(19,92,109,49),(20,89,110,50),(21,55,115,88),(22,56,116,85),(23,53,113,86),(24,54,114,87),(25,67,119,39),(26,68,120,40),(27,65,117,37),(28,66,118,38),(45,125,73,99),(46,126,74,100),(47,127,75,97),(48,128,76,98)], [(1,125,117,42),(2,98,118,69),(3,127,119,44),(4,100,120,71),(5,122,47,39),(6,31,48,66),(7,124,45,37),(8,29,46,68),(9,109,59,85),(10,18,60,55),(11,111,57,87),(12,20,58,53),(13,21,63,52),(14,114,64,90),(15,23,61,50),(16,116,62,92),(17,77,54,93),(19,79,56,95),(22,82,49,106),(24,84,51,108),(25,72,33,97),(26,43,34,126),(27,70,35,99),(28,41,36,128),(30,73,65,102),(32,75,67,104),(38,101,121,76),(40,103,123,74),(78,86,94,110),(80,88,96,112),(81,89,105,113),(83,91,107,115)], [(1,80,25,12),(2,57,26,95),(3,78,27,10),(4,59,28,93),(5,113,73,52),(6,24,74,92),(7,115,75,50),(8,22,76,90),(9,36,77,120),(11,34,79,118),(13,39,81,30),(14,68,82,121),(15,37,83,32),(16,66,84,123),(17,71,85,128),(18,44,86,99),(19,69,87,126),(20,42,88,97),(21,47,89,102),(23,45,91,104),(29,106,38,64),(31,108,40,62),(33,58,117,96),(35,60,119,94),(41,54,100,109),(43,56,98,111),(46,49,101,114),(48,51,103,116),(53,125,112,72),(55,127,110,70),(61,124,107,67),(63,122,105,65)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.39Q8 | C42⋊4C4 | C42⋊8C4 | C42⋊9C4 | C23.65C23 | C23.81C23 | C23.83C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C42.39Q8 ►in GL8(𝔽5)
| 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 2 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 3 | 2 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 3 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 4 | 4 | 1 |
| 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 3 | 3 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 3 | 3 | 2 |
G:=sub<GL(8,GF(5))| [4,2,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,4,3,0,0,0,0,2,1,0,2,0,0,0,0,2,2,4,0,0,0,0,0,0,4,1,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,2,3,0,0,0,0,0,0,2,1,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2],[2,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,2,4,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1],[2,1,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,2,3,2,0,0,0,0,0,1,1,2,3,0,0,0,0,1,3,0,3,0,0,0,0,0,3,2,2] >;
C42.39Q8 in GAP, Magma, Sage, TeX
C_4^2._{39}Q_8 % in TeX
G:=Group("C4^2.39Q8"); // GroupNames label
G:=SmallGroup(128,1379);
// by ID
G=gap.SmallGroup(128,1379);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,184,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2*c^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations