p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.179D4, C23.481C24, C22.1962- 1+4, (C2×Q8)⋊11Q8, C42⋊8C4.37C2, C2.20(Q8⋊3Q8), C4.104(C22⋊Q8), (C2×C42).575C22, (C22×C4).111C23, C22.322(C22×D4), C22.116(C22×Q8), (C22×Q8).440C22, C2.66(C22.19C24), C23.78C23.9C2, C23.63C23.29C2, C23.65C23.58C2, C23.83C23.17C2, C23.67C23.43C2, C2.C42.215C22, C2.45(C22.50C24), C2.27(C23.38C23), (C2×C4×Q8).37C2, (C2×C4).58(C2×Q8), C2.39(C2×C22⋊Q8), (C2×C4).1197(C2×D4), (C2×C4).156(C4○D4), (C2×C4⋊C4).327C22, C22.357(C2×C4○D4), SmallGroup(128,1313)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.179D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2b2c-1 >
Subgroups: 372 in 234 conjugacy classes, 112 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×Q8, C42⋊8C4, C23.63C23, C23.65C23, C23.67C23, C23.78C23, C23.83C23, C2×C4×Q8, C42.179D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2- 1+4, C2×C22⋊Q8, C22.19C24, C23.38C23, C22.50C24, Q8⋊3Q8, C42.179D4
►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 12 21 43)(2 9 22 44)(3 10 23 41)(4 11 24 42)(5 56 40 26)(6 53 37 27)(7 54 38 28)(8 55 39 25)(13 64 48 70)(14 61 45 71)(15 62 46 72)(16 63 47 69)(17 30 52 60)(18 31 49 57)(19 32 50 58)(20 29 51 59)(33 74 125 108)(34 75 126 105)(35 76 127 106)(36 73 128 107)(65 120 99 94)(66 117 100 95)(67 118 97 96)(68 119 98 93)(77 82 103 116)(78 83 104 113)(79 84 101 114)(80 81 102 115)(85 90 111 124)(86 91 112 121)(87 92 109 122)(88 89 110 123)
(1 74 6 89)(2 107 7 122)(3 76 8 91)(4 105 5 124)(9 128 54 109)(10 35 55 86)(11 126 56 111)(12 33 53 88)(13 114 58 99)(14 83 59 68)(15 116 60 97)(16 81 57 66)(17 118 62 103)(18 95 63 80)(19 120 64 101)(20 93 61 78)(21 108 37 123)(22 73 38 92)(23 106 39 121)(24 75 40 90)(25 112 41 127)(26 85 42 34)(27 110 43 125)(28 87 44 36)(29 98 45 113)(30 67 46 82)(31 100 47 115)(32 65 48 84)(49 117 69 102)(50 94 70 79)(51 119 71 104)(52 96 72 77)
(1 115 21 81)(2 82 22 116)(3 113 23 83)(4 84 24 114)(5 65 40 99)(6 100 37 66)(7 67 38 97)(8 98 39 68)(9 77 44 103)(10 104 41 78)(11 79 42 101)(12 102 43 80)(13 73 48 107)(14 108 45 74)(15 75 46 105)(16 106 47 76)(17 85 52 111)(18 112 49 86)(19 87 50 109)(20 110 51 88)(25 93 55 119)(26 120 56 94)(27 95 53 117)(28 118 54 96)(29 89 59 123)(30 124 60 90)(31 91 57 121)(32 122 58 92)(33 61 125 71)(34 72 126 62)(35 63 127 69)(36 70 128 64)
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,70)(14,61,45,71)(15,62,46,72)(16,63,47,69)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59)(33,74,125,108)(34,75,126,105)(35,76,127,106)(36,73,128,107)(65,120,99,94)(66,117,100,95)(67,118,97,96)(68,119,98,93)(77,82,103,116)(78,83,104,113)(79,84,101,114)(80,81,102,115)(85,90,111,124)(86,91,112,121)(87,92,109,122)(88,89,110,123), (1,74,6,89)(2,107,7,122)(3,76,8,91)(4,105,5,124)(9,128,54,109)(10,35,55,86)(11,126,56,111)(12,33,53,88)(13,114,58,99)(14,83,59,68)(15,116,60,97)(16,81,57,66)(17,118,62,103)(18,95,63,80)(19,120,64,101)(20,93,61,78)(21,108,37,123)(22,73,38,92)(23,106,39,121)(24,75,40,90)(25,112,41,127)(26,85,42,34)(27,110,43,125)(28,87,44,36)(29,98,45,113)(30,67,46,82)(31,100,47,115)(32,65,48,84)(49,117,69,102)(50,94,70,79)(51,119,71,104)(52,96,72,77), (1,115,21,81)(2,82,22,116)(3,113,23,83)(4,84,24,114)(5,65,40,99)(6,100,37,66)(7,67,38,97)(8,98,39,68)(9,77,44,103)(10,104,41,78)(11,79,42,101)(12,102,43,80)(13,73,48,107)(14,108,45,74)(15,75,46,105)(16,106,47,76)(17,85,52,111)(18,112,49,86)(19,87,50,109)(20,110,51,88)(25,93,55,119)(26,120,56,94)(27,95,53,117)(28,118,54,96)(29,89,59,123)(30,124,60,90)(31,91,57,121)(32,122,58,92)(33,61,125,71)(34,72,126,62)(35,63,127,69)(36,70,128,64)>;
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,70)(14,61,45,71)(15,62,46,72)(16,63,47,69)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59)(33,74,125,108)(34,75,126,105)(35,76,127,106)(36,73,128,107)(65,120,99,94)(66,117,100,95)(67,118,97,96)(68,119,98,93)(77,82,103,116)(78,83,104,113)(79,84,101,114)(80,81,102,115)(85,90,111,124)(86,91,112,121)(87,92,109,122)(88,89,110,123), (1,74,6,89)(2,107,7,122)(3,76,8,91)(4,105,5,124)(9,128,54,109)(10,35,55,86)(11,126,56,111)(12,33,53,88)(13,114,58,99)(14,83,59,68)(15,116,60,97)(16,81,57,66)(17,118,62,103)(18,95,63,80)(19,120,64,101)(20,93,61,78)(21,108,37,123)(22,73,38,92)(23,106,39,121)(24,75,40,90)(25,112,41,127)(26,85,42,34)(27,110,43,125)(28,87,44,36)(29,98,45,113)(30,67,46,82)(31,100,47,115)(32,65,48,84)(49,117,69,102)(50,94,70,79)(51,119,71,104)(52,96,72,77), (1,115,21,81)(2,82,22,116)(3,113,23,83)(4,84,24,114)(5,65,40,99)(6,100,37,66)(7,67,38,97)(8,98,39,68)(9,77,44,103)(10,104,41,78)(11,79,42,101)(12,102,43,80)(13,73,48,107)(14,108,45,74)(15,75,46,105)(16,106,47,76)(17,85,52,111)(18,112,49,86)(19,87,50,109)(20,110,51,88)(25,93,55,119)(26,120,56,94)(27,95,53,117)(28,118,54,96)(29,89,59,123)(30,124,60,90)(31,91,57,121)(32,122,58,92)(33,61,125,71)(34,72,126,62)(35,63,127,69)(36,70,128,64) );
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)], [(1,12,21,43),(2,9,22,44),(3,10,23,41),(4,11,24,42),(5,56,40,26),(6,53,37,27),(7,54,38,28),(8,55,39,25),(13,64,48,70),(14,61,45,71),(15,62,46,72),(16,63,47,69),(17,30,52,60),(18,31,49,57),(19,32,50,58),(20,29,51,59),(33,74,125,108),(34,75,126,105),(35,76,127,106),(36,73,128,107),(65,120,99,94),(66,117,100,95),(67,118,97,96),(68,119,98,93),(77,82,103,116),(78,83,104,113),(79,84,101,114),(80,81,102,115),(85,90,111,124),(86,91,112,121),(87,92,109,122),(88,89,110,123)], [(1,74,6,89),(2,107,7,122),(3,76,8,91),(4,105,5,124),(9,128,54,109),(10,35,55,86),(11,126,56,111),(12,33,53,88),(13,114,58,99),(14,83,59,68),(15,116,60,97),(16,81,57,66),(17,118,62,103),(18,95,63,80),(19,120,64,101),(20,93,61,78),(21,108,37,123),(22,73,38,92),(23,106,39,121),(24,75,40,90),(25,112,41,127),(26,85,42,34),(27,110,43,125),(28,87,44,36),(29,98,45,113),(30,67,46,82),(31,100,47,115),(32,65,48,84),(49,117,69,102),(50,94,70,79),(51,119,71,104),(52,96,72,77)], [(1,115,21,81),(2,82,22,116),(3,113,23,83),(4,84,24,114),(5,65,40,99),(6,100,37,66),(7,67,38,97),(8,98,39,68),(9,77,44,103),(10,104,41,78),(11,79,42,101),(12,102,43,80),(13,73,48,107),(14,108,45,74),(15,75,46,105),(16,106,47,76),(17,85,52,111),(18,112,49,86),(19,87,50,109),(20,110,51,88),(25,93,55,119),(26,120,56,94),(27,95,53,117),(28,118,54,96),(29,89,59,123),(30,124,60,90),(31,91,57,121),(32,122,58,92),(33,61,125,71),(34,72,126,62),(35,63,127,69),(36,70,128,64)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC | 4AD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2- 1+4 |
| kernel | C42.179D4 | C42⋊8C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.78C23 | C23.83C23 | C2×C4×Q8 | C42 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 2 |
Matrix representation of C42.179D4 ►in GL6(𝔽5)
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [0,2,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,3,2,0,0,0,0,0,2],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C42.179D4 in GAP, Magma, Sage, TeX
C_4^2._{179}D_4 % in TeX
G:=Group("C4^2.179D4"); // GroupNames label
G:=SmallGroup(128,1313);
// by ID
G=gap.SmallGroup(128,1313);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,352,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations