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), (C2×C42).416C22, (C22×C4).474C23, C22.100(C23×C4), (C22×Q8).400C22, C23.63C23.2C2, C2.C42.45C22, C23.67C23.25C2, C2.7(C23.38C23), C2.9(C23.32C23), C2.4(C22.35C24), 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
Subgroups: 380 in 258 conjugacy classes, 148 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×22], C2×C4 [×34], Q8 [×16], C23, C42 [×8], C42 [×10], C4⋊C4 [×16], C4⋊C4 [×12], C22×C4, C22×C4 [×14], C2×Q8 [×8], C2×Q8 [×8], C2.C42 [×16], C2×C42 [×3], C2×C42 [×4], C2×C4⋊C4 [×10], C4×Q8 [×8], C4⋊Q8 [×8], C22×Q8 [×2], C42⋊4C4, C42⋊8C4, C23.63C23 [×8], C23.67C23 [×2], C2×C4×Q8 [×2], C2×C4⋊Q8, C42.161D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2- (1+4) [×4], C2×C4×D4, C23.32C23 [×2], C23.38C23 [×2], C22.35C24 [×2], C42.161D4
Generators and relations
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 >
(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 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(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 45)(2 60 24 18)(3 29 21 47)(4 58 22 20)(5 16 70 54)(6 41 71 27)(7 14 72 56)(8 43 69 25)(9 59 51 17)(10 32 52 46)(11 57 49 19)(12 30 50 48)(13 63 55 39)(15 61 53 37)(26 38 44 62)(28 40 42 64)(33 105 67 119)(34 74 68 96)(35 107 65 117)(36 76 66 94)(73 98 95 127)(75 100 93 125)(77 109 83 123)(78 86 84 92)(79 111 81 121)(80 88 82 90)(85 115 91 101)(87 113 89 103)(97 118 126 108)(99 120 128 106)(102 110 116 124)(104 112 114 122)
(1 121 11 91)(2 122 12 92)(3 123 9 89)(4 124 10 90)(5 106 40 76)(6 107 37 73)(7 108 38 74)(8 105 39 75)(13 35 43 127)(14 36 44 128)(15 33 41 125)(16 34 42 126)(17 79 47 101)(18 80 48 102)(19 77 45 103)(20 78 46 104)(21 109 51 87)(22 110 52 88)(23 111 49 85)(24 112 50 86)(25 98 55 65)(26 99 56 66)(27 100 53 67)(28 97 54 68)(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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(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), (1,31,23,45)(2,60,24,18)(3,29,21,47)(4,58,22,20)(5,16,70,54)(6,41,71,27)(7,14,72,56)(8,43,69,25)(9,59,51,17)(10,32,52,46)(11,57,49,19)(12,30,50,48)(13,63,55,39)(15,61,53,37)(26,38,44,62)(28,40,42,64)(33,105,67,119)(34,74,68,96)(35,107,65,117)(36,76,66,94)(73,98,95,127)(75,100,93,125)(77,109,83,123)(78,86,84,92)(79,111,81,121)(80,88,82,90)(85,115,91,101)(87,113,89,103)(97,118,126,108)(99,120,128,106)(102,110,116,124)(104,112,114,122), (1,121,11,91)(2,122,12,92)(3,123,9,89)(4,124,10,90)(5,106,40,76)(6,107,37,73)(7,108,38,74)(8,105,39,75)(13,35,43,127)(14,36,44,128)(15,33,41,125)(16,34,42,126)(17,79,47,101)(18,80,48,102)(19,77,45,103)(20,78,46,104)(21,109,51,87)(22,110,52,88)(23,111,49,85)(24,112,50,86)(25,98,55,65)(26,99,56,66)(27,100,53,67)(28,97,54,68)(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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(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), (1,31,23,45)(2,60,24,18)(3,29,21,47)(4,58,22,20)(5,16,70,54)(6,41,71,27)(7,14,72,56)(8,43,69,25)(9,59,51,17)(10,32,52,46)(11,57,49,19)(12,30,50,48)(13,63,55,39)(15,61,53,37)(26,38,44,62)(28,40,42,64)(33,105,67,119)(34,74,68,96)(35,107,65,117)(36,76,66,94)(73,98,95,127)(75,100,93,125)(77,109,83,123)(78,86,84,92)(79,111,81,121)(80,88,82,90)(85,115,91,101)(87,113,89,103)(97,118,126,108)(99,120,128,106)(102,110,116,124)(104,112,114,122), (1,121,11,91)(2,122,12,92)(3,123,9,89)(4,124,10,90)(5,106,40,76)(6,107,37,73)(7,108,38,74)(8,105,39,75)(13,35,43,127)(14,36,44,128)(15,33,41,125)(16,34,42,126)(17,79,47,101)(18,80,48,102)(19,77,45,103)(20,78,46,104)(21,109,51,87)(22,110,52,88)(23,111,49,85)(24,112,50,86)(25,98,55,65)(26,99,56,66)(27,100,53,67)(28,97,54,68)(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)], [(1,13,9,41),(2,14,10,42),(3,15,11,43),(4,16,12,44),(5,48,38,20),(6,45,39,17),(7,46,40,18),(8,47,37,19),(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)], [(1,31,23,45),(2,60,24,18),(3,29,21,47),(4,58,22,20),(5,16,70,54),(6,41,71,27),(7,14,72,56),(8,43,69,25),(9,59,51,17),(10,32,52,46),(11,57,49,19),(12,30,50,48),(13,63,55,39),(15,61,53,37),(26,38,44,62),(28,40,42,64),(33,105,67,119),(34,74,68,96),(35,107,65,117),(36,76,66,94),(73,98,95,127),(75,100,93,125),(77,109,83,123),(78,86,84,92),(79,111,81,121),(80,88,82,90),(85,115,91,101),(87,113,89,103),(97,118,126,108),(99,120,128,106),(102,110,116,124),(104,112,114,122)], [(1,121,11,91),(2,122,12,92),(3,123,9,89),(4,124,10,90),(5,106,40,76),(6,107,37,73),(7,108,38,74),(8,105,39,75),(13,35,43,127),(14,36,44,128),(15,33,41,125),(16,34,42,126),(17,79,47,101),(18,80,48,102),(19,77,45,103),(20,78,46,104),(21,109,51,87),(22,110,52,88),(23,111,49,85),(24,112,50,86),(25,98,55,65),(26,99,56,66),(27,100,53,67),(28,97,54,68),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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