p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.176D4, C23.467C24, C22.2512+ 1+4, C2.6Q82, C4⋊C4⋊20Q8, (C2×Q8)⋊9Q8, C4.40(C4⋊Q8), C4.59(C22⋊Q8), C42⋊9C4.31C2, C2.37(D4⋊3Q8), (C22×C4).843C23, (C2×C42).568C22, C22.318(C22×D4), C22.108(C22×Q8), (C22×Q8).437C22, C2.26(C22.29C24), C23.78C23.8C2, C23.67C23.41C2, C2.C42.203C22, C23.65C23.55C2, (C4×C4⋊C4).67C2, C2.16(C2×C4⋊Q8), (C2×C4×Q8).35C2, (C2×C4⋊Q8).34C2, (C2×C4).55(C2×Q8), (C2×C4).361(C2×D4), C2.35(C2×C22⋊Q8), (C2×C4).826(C4○D4), (C2×C4⋊C4).876C22, C22.343(C2×C4○D4), SmallGroup(128,1299)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.176D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 420 in 250 conjugacy classes, 132 normal (18 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, C2×C4⋊C4, C4×Q8, C4⋊Q8, C22×Q8, C22×Q8, C4×C4⋊C4, C42⋊9C4, C23.65C23, C23.67C23, C23.78C23, C2×C4×Q8, C2×C4⋊Q8, C42.176D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C2×C4⋊Q8, C22.29C24, D4⋊3Q8, Q82, C42.176D4
►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 20 45 13)(2 17 46 14)(3 18 47 15)(4 19 48 16)(5 37 127 36)(6 38 128 33)(7 39 125 34)(8 40 126 35)(9 42 23 50)(10 43 24 51)(11 44 21 52)(12 41 22 49)(25 54 30 57)(26 55 31 58)(27 56 32 59)(28 53 29 60)(61 65 100 72)(62 66 97 69)(63 67 98 70)(64 68 99 71)(73 108 111 86)(74 105 112 87)(75 106 109 88)(76 107 110 85)(77 113 82 102)(78 114 83 103)(79 115 84 104)(80 116 81 101)(89 117 96 124)(90 118 93 121)(91 119 94 122)(92 120 95 123)
(1 98 58 52)(2 62 59 43)(3 100 60 50)(4 64 57 41)(5 101 109 122)(6 115 110 118)(7 103 111 124)(8 113 112 120)(9 18 72 28)(10 14 69 32)(11 20 70 26)(12 16 71 30)(13 67 31 21)(15 65 29 23)(17 66 27 24)(19 68 25 22)(33 79 107 90)(34 83 108 96)(35 77 105 92)(36 81 106 94)(37 80 88 91)(38 84 85 93)(39 78 86 89)(40 82 87 95)(42 47 61 53)(44 45 63 55)(46 97 56 51)(48 99 54 49)(73 117 125 114)(74 123 126 102)(75 119 127 116)(76 121 128 104)
(1 125 3 127)(2 128 4 126)(5 45 7 47)(6 48 8 46)(9 80 11 78)(10 79 12 77)(13 34 15 36)(14 33 16 35)(17 38 19 40)(18 37 20 39)(21 83 23 81)(22 82 24 84)(25 87 27 85)(26 86 28 88)(29 106 31 108)(30 105 32 107)(41 102 43 104)(42 101 44 103)(49 113 51 115)(50 116 52 114)(53 109 55 111)(54 112 56 110)(57 74 59 76)(58 73 60 75)(61 122 63 124)(62 121 64 123)(65 94 67 96)(66 93 68 95)(69 90 71 92)(70 89 72 91)(97 118 99 120)(98 117 100 119)
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), (1,20,45,13)(2,17,46,14)(3,18,47,15)(4,19,48,16)(5,37,127,36)(6,38,128,33)(7,39,125,34)(8,40,126,35)(9,42,23,50)(10,43,24,51)(11,44,21,52)(12,41,22,49)(25,54,30,57)(26,55,31,58)(27,56,32,59)(28,53,29,60)(61,65,100,72)(62,66,97,69)(63,67,98,70)(64,68,99,71)(73,108,111,86)(74,105,112,87)(75,106,109,88)(76,107,110,85)(77,113,82,102)(78,114,83,103)(79,115,84,104)(80,116,81,101)(89,117,96,124)(90,118,93,121)(91,119,94,122)(92,120,95,123), (1,98,58,52)(2,62,59,43)(3,100,60,50)(4,64,57,41)(5,101,109,122)(6,115,110,118)(7,103,111,124)(8,113,112,120)(9,18,72,28)(10,14,69,32)(11,20,70,26)(12,16,71,30)(13,67,31,21)(15,65,29,23)(17,66,27,24)(19,68,25,22)(33,79,107,90)(34,83,108,96)(35,77,105,92)(36,81,106,94)(37,80,88,91)(38,84,85,93)(39,78,86,89)(40,82,87,95)(42,47,61,53)(44,45,63,55)(46,97,56,51)(48,99,54,49)(73,117,125,114)(74,123,126,102)(75,119,127,116)(76,121,128,104), (1,125,3,127)(2,128,4,126)(5,45,7,47)(6,48,8,46)(9,80,11,78)(10,79,12,77)(13,34,15,36)(14,33,16,35)(17,38,19,40)(18,37,20,39)(21,83,23,81)(22,82,24,84)(25,87,27,85)(26,86,28,88)(29,106,31,108)(30,105,32,107)(41,102,43,104)(42,101,44,103)(49,113,51,115)(50,116,52,114)(53,109,55,111)(54,112,56,110)(57,74,59,76)(58,73,60,75)(61,122,63,124)(62,121,64,123)(65,94,67,96)(66,93,68,95)(69,90,71,92)(70,89,72,91)(97,118,99,120)(98,117,100,119)>;
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), (1,20,45,13)(2,17,46,14)(3,18,47,15)(4,19,48,16)(5,37,127,36)(6,38,128,33)(7,39,125,34)(8,40,126,35)(9,42,23,50)(10,43,24,51)(11,44,21,52)(12,41,22,49)(25,54,30,57)(26,55,31,58)(27,56,32,59)(28,53,29,60)(61,65,100,72)(62,66,97,69)(63,67,98,70)(64,68,99,71)(73,108,111,86)(74,105,112,87)(75,106,109,88)(76,107,110,85)(77,113,82,102)(78,114,83,103)(79,115,84,104)(80,116,81,101)(89,117,96,124)(90,118,93,121)(91,119,94,122)(92,120,95,123), (1,98,58,52)(2,62,59,43)(3,100,60,50)(4,64,57,41)(5,101,109,122)(6,115,110,118)(7,103,111,124)(8,113,112,120)(9,18,72,28)(10,14,69,32)(11,20,70,26)(12,16,71,30)(13,67,31,21)(15,65,29,23)(17,66,27,24)(19,68,25,22)(33,79,107,90)(34,83,108,96)(35,77,105,92)(36,81,106,94)(37,80,88,91)(38,84,85,93)(39,78,86,89)(40,82,87,95)(42,47,61,53)(44,45,63,55)(46,97,56,51)(48,99,54,49)(73,117,125,114)(74,123,126,102)(75,119,127,116)(76,121,128,104), (1,125,3,127)(2,128,4,126)(5,45,7,47)(6,48,8,46)(9,80,11,78)(10,79,12,77)(13,34,15,36)(14,33,16,35)(17,38,19,40)(18,37,20,39)(21,83,23,81)(22,82,24,84)(25,87,27,85)(26,86,28,88)(29,106,31,108)(30,105,32,107)(41,102,43,104)(42,101,44,103)(49,113,51,115)(50,116,52,114)(53,109,55,111)(54,112,56,110)(57,74,59,76)(58,73,60,75)(61,122,63,124)(62,121,64,123)(65,94,67,96)(66,93,68,95)(69,90,71,92)(70,89,72,91)(97,118,99,120)(98,117,100,119) );
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)], [(1,20,45,13),(2,17,46,14),(3,18,47,15),(4,19,48,16),(5,37,127,36),(6,38,128,33),(7,39,125,34),(8,40,126,35),(9,42,23,50),(10,43,24,51),(11,44,21,52),(12,41,22,49),(25,54,30,57),(26,55,31,58),(27,56,32,59),(28,53,29,60),(61,65,100,72),(62,66,97,69),(63,67,98,70),(64,68,99,71),(73,108,111,86),(74,105,112,87),(75,106,109,88),(76,107,110,85),(77,113,82,102),(78,114,83,103),(79,115,84,104),(80,116,81,101),(89,117,96,124),(90,118,93,121),(91,119,94,122),(92,120,95,123)], [(1,98,58,52),(2,62,59,43),(3,100,60,50),(4,64,57,41),(5,101,109,122),(6,115,110,118),(7,103,111,124),(8,113,112,120),(9,18,72,28),(10,14,69,32),(11,20,70,26),(12,16,71,30),(13,67,31,21),(15,65,29,23),(17,66,27,24),(19,68,25,22),(33,79,107,90),(34,83,108,96),(35,77,105,92),(36,81,106,94),(37,80,88,91),(38,84,85,93),(39,78,86,89),(40,82,87,95),(42,47,61,53),(44,45,63,55),(46,97,56,51),(48,99,54,49),(73,117,125,114),(74,123,126,102),(75,119,127,116),(76,121,128,104)], [(1,125,3,127),(2,128,4,126),(5,45,7,47),(6,48,8,46),(9,80,11,78),(10,79,12,77),(13,34,15,36),(14,33,16,35),(17,38,19,40),(18,37,20,39),(21,83,23,81),(22,82,24,84),(25,87,27,85),(26,86,28,88),(29,106,31,108),(30,105,32,107),(41,102,43,104),(42,101,44,103),(49,113,51,115),(50,116,52,114),(53,109,55,111),(54,112,56,110),(57,74,59,76),(58,73,60,75),(61,122,63,124),(62,121,64,123),(65,94,67,96),(66,93,68,95),(69,90,71,92),(70,89,72,91),(97,118,99,120),(98,117,100,119)]])
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 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | C4○D4 | 2+ 1+4 |
| kernel | C42.176D4 | C4×C4⋊C4 | C42⋊9C4 | C23.65C23 | C23.67C23 | C23.78C23 | C2×C4×Q8 | C2×C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 4 | 8 | 4 | 4 | 2 |
Matrix representation of C42.176D4 ►in GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0] >;
C42.176D4 in GAP, Magma, Sage, TeX
C_4^2._{176}D_4 % in TeX
G:=Group("C4^2.176D4"); // GroupNames label
G:=SmallGroup(128,1299);
// by ID
G=gap.SmallGroup(128,1299);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,184,675,80]);
// 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=a^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations