p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.161D4, C23.209C24, C22.322- 1+4, C4⋊Q8⋊30C4, C4.38(C4×D4), C42.180(C2×C4), C42⋊8C4.20C2, C42⋊4C4.13C2, C22.97(C22×D4), (C22×C4).474C23, (C2×C42).416C22, C22.100(C23×C4), (C22×Q8).400C22, C2.C42.45C22, C23.63C23.2C2, C23.67C23.25C2, C2.4(C22.35C24), C2.9(C23.32C23), C2.7(C23.38C23), C2.26(C2×C4×D4), (C2×C4×Q8).22C2, (C2×C4⋊Q8).24C2, C4⋊C4.104(C2×C4), (C2×C4).1189(C2×D4), (C2×C4).30(C22×C4), (C2×Q8).149(C2×C4), C22.94(C2×C4○D4), (C2×C4).649(C4○D4), (C2×C4⋊C4).808C22, SmallGroup(128,1059)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.161D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 380 in 258 conjugacy classes, 148 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C4⋊Q8, C22×Q8, C42⋊4C4, C42⋊8C4, C23.63C23, C23.67C23, C2×C4×Q8, C2×C4⋊Q8, C42.161D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4×D4, C23.32C23, C23.38C23, C22.35C24, C42.161D4
►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 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 46 38 18)(6 47 39 19)(7 48 40 20)(8 45 37 17)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 123 127 91)(34 124 128 92)(35 121 125 89)(36 122 126 90)(65 111 100 87)(66 112 97 88)(67 109 98 85)(68 110 99 86)(73 101 105 77)(74 102 106 78)(75 103 107 79)(76 104 108 80)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)
(1 31 23 47)(2 60 24 20)(3 29 21 45)(4 58 22 18)(5 14 70 54)(6 43 71 27)(7 16 72 56)(8 41 69 25)(9 57 49 17)(10 30 50 46)(11 59 51 19)(12 32 52 48)(13 61 53 37)(15 63 55 39)(26 38 42 62)(28 40 44 64)(33 107 65 119)(34 76 66 96)(35 105 67 117)(36 74 68 94)(73 98 93 125)(75 100 95 127)(77 109 81 121)(78 86 82 90)(79 111 83 123)(80 88 84 92)(85 113 89 101)(87 115 91 103)(97 120 128 108)(99 118 126 106)(102 110 114 122)(104 112 116 124)
(1 121 9 91)(2 122 10 92)(3 123 11 89)(4 124 12 90)(5 108 40 74)(6 105 37 75)(7 106 38 76)(8 107 39 73)(13 33 43 125)(14 34 44 126)(15 35 41 127)(16 36 42 128)(17 79 47 101)(18 80 48 102)(19 77 45 103)(20 78 46 104)(21 111 51 85)(22 112 52 86)(23 109 49 87)(24 110 50 88)(25 100 55 67)(26 97 56 68)(27 98 53 65)(28 99 54 66)(29 115 59 81)(30 116 60 82)(31 113 57 83)(32 114 58 84)(61 95 71 117)(62 96 72 118)(63 93 69 119)(64 94 70 120)
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,46,38,18)(6,47,39,19)(7,48,40,20)(8,45,37,17)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,31,23,47)(2,60,24,20)(3,29,21,45)(4,58,22,18)(5,14,70,54)(6,43,71,27)(7,16,72,56)(8,41,69,25)(9,57,49,17)(10,30,50,46)(11,59,51,19)(12,32,52,48)(13,61,53,37)(15,63,55,39)(26,38,42,62)(28,40,44,64)(33,107,65,119)(34,76,66,96)(35,105,67,117)(36,74,68,94)(73,98,93,125)(75,100,95,127)(77,109,81,121)(78,86,82,90)(79,111,83,123)(80,88,84,92)(85,113,89,101)(87,115,91,103)(97,120,128,108)(99,118,126,106)(102,110,114,122)(104,112,116,124), (1,121,9,91)(2,122,10,92)(3,123,11,89)(4,124,12,90)(5,108,40,74)(6,105,37,75)(7,106,38,76)(8,107,39,73)(13,33,43,125)(14,34,44,126)(15,35,41,127)(16,36,42,128)(17,79,47,101)(18,80,48,102)(19,77,45,103)(20,78,46,104)(21,111,51,85)(22,112,52,86)(23,109,49,87)(24,110,50,88)(25,100,55,67)(26,97,56,68)(27,98,53,65)(28,99,54,66)(29,115,59,81)(30,116,60,82)(31,113,57,83)(32,114,58,84)(61,95,71,117)(62,96,72,118)(63,93,69,119)(64,94,70,120)>;
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,46,38,18)(6,47,39,19)(7,48,40,20)(8,45,37,17)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,31,23,47)(2,60,24,20)(3,29,21,45)(4,58,22,18)(5,14,70,54)(6,43,71,27)(7,16,72,56)(8,41,69,25)(9,57,49,17)(10,30,50,46)(11,59,51,19)(12,32,52,48)(13,61,53,37)(15,63,55,39)(26,38,42,62)(28,40,44,64)(33,107,65,119)(34,76,66,96)(35,105,67,117)(36,74,68,94)(73,98,93,125)(75,100,95,127)(77,109,81,121)(78,86,82,90)(79,111,83,123)(80,88,84,92)(85,113,89,101)(87,115,91,103)(97,120,128,108)(99,118,126,106)(102,110,114,122)(104,112,116,124), (1,121,9,91)(2,122,10,92)(3,123,11,89)(4,124,12,90)(5,108,40,74)(6,105,37,75)(7,106,38,76)(8,107,39,73)(13,33,43,125)(14,34,44,126)(15,35,41,127)(16,36,42,128)(17,79,47,101)(18,80,48,102)(19,77,45,103)(20,78,46,104)(21,111,51,85)(22,112,52,86)(23,109,49,87)(24,110,50,88)(25,100,55,67)(26,97,56,68)(27,98,53,65)(28,99,54,66)(29,115,59,81)(30,116,60,82)(31,113,57,83)(32,114,58,84)(61,95,71,117)(62,96,72,118)(63,93,69,119)(64,94,70,120) );
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)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,46,38,18),(6,47,39,19),(7,48,40,20),(8,45,37,17),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,69),(30,62,58,70),(31,63,59,71),(32,64,60,72),(33,123,127,91),(34,124,128,92),(35,121,125,89),(36,122,126,90),(65,111,100,87),(66,112,97,88),(67,109,98,85),(68,110,99,86),(73,101,105,77),(74,102,106,78),(75,103,107,79),(76,104,108,80),(81,93,113,117),(82,94,114,118),(83,95,115,119),(84,96,116,120)], [(1,31,23,47),(2,60,24,20),(3,29,21,45),(4,58,22,18),(5,14,70,54),(6,43,71,27),(7,16,72,56),(8,41,69,25),(9,57,49,17),(10,30,50,46),(11,59,51,19),(12,32,52,48),(13,61,53,37),(15,63,55,39),(26,38,42,62),(28,40,44,64),(33,107,65,119),(34,76,66,96),(35,105,67,117),(36,74,68,94),(73,98,93,125),(75,100,95,127),(77,109,81,121),(78,86,82,90),(79,111,83,123),(80,88,84,92),(85,113,89,101),(87,115,91,103),(97,120,128,108),(99,118,126,106),(102,110,114,122),(104,112,116,124)], [(1,121,9,91),(2,122,10,92),(3,123,11,89),(4,124,12,90),(5,108,40,74),(6,105,37,75),(7,106,38,76),(8,107,39,73),(13,33,43,125),(14,34,44,126),(15,35,41,127),(16,36,42,128),(17,79,47,101),(18,80,48,102),(19,77,45,103),(20,78,46,104),(21,111,51,85),(22,112,52,86),(23,109,49,87),(24,110,50,88),(25,100,55,67),(26,97,56,68),(27,98,53,65),(28,99,54,66),(29,115,59,81),(30,116,60,82),(31,113,57,83),(32,114,58,84),(61,95,71,117),(62,96,72,118),(63,93,69,119),(64,94,70,120)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | 2- 1+4 |
| kernel | C42.161D4 | C42⋊4C4 | C42⋊8C4 | C23.63C23 | C23.67C23 | C2×C4×Q8 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 16 | 4 | 4 | 4 |
Matrix representation of C42.161D4 ►in GL8(𝔽5)
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 | 0 | 4 |
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 | 1 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 | 1 | 4 |
| 3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 3 |
| 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 2 | 3 |
G:=sub<GL(8,GF(5))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,3,3,0,0,0,0,1,0,2,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4],[3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,3,1,1,0,0,0,0,0,2,0,4,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,3],[1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,3] >;
C42.161D4 in GAP, Magma, Sage, TeX
C_4^2._{161}D_4 % in TeX
G:=Group("C4^2.161D4"); // GroupNames label
G:=SmallGroup(128,1059);
// by ID
G=gap.SmallGroup(128,1059);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,268,675,80]);
// 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*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations