p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.176D4, C23.467C24, C22.2512+ (1+4), C4⋊C4⋊20Q8, (C2×Q8)⋊9Q8, C2.6(Q82), C4.40(C4⋊Q8), C4.59(C22⋊Q8), C42⋊9C4.31C2, C2.37(D4⋊3Q8), (C2×C42).568C22, (C22×C4).843C23, 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, C2.16(C2×C4⋊Q8), (C4×C4⋊C4).67C2, (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
Subgroups: 420 in 250 conjugacy classes, 132 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×26], C2×C4 [×32], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×8], C4⋊C4 [×22], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×12], C2.C42 [×10], C2×C42, C2×C42 [×4], C2×C4⋊C4, C2×C4⋊C4 [×16], C4×Q8 [×4], C4⋊Q8 [×4], C22×Q8, C22×Q8 [×2], C4×C4⋊C4, C42⋊9C4 [×2], C23.65C23 [×4], C23.67C23 [×2], C23.78C23 [×4], C2×C4×Q8, C2×C4⋊Q8, C42.176D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×12], C23 [×15], C2×D4 [×6], C2×Q8 [×18], C4○D4 [×2], C24, C22⋊Q8 [×4], C4⋊Q8 [×4], C22×D4, C22×Q8 [×3], C2×C4○D4, 2+ (1+4) [×2], C2×C22⋊Q8, C2×C4⋊Q8, C22.29C24, D4⋊3Q8 [×2], Q82 [×2], C42.176D4
Generators and relations
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 >
(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 44 50)(2 16 41 51)(3 13 42 52)(4 14 43 49)(5 36 37 127)(6 33 38 128)(7 34 39 125)(8 35 40 126)(9 22 17 46)(10 23 18 47)(11 24 19 48)(12 21 20 45)(25 57 54 30)(26 58 55 31)(27 59 56 32)(28 60 53 29)(61 65 72 100)(62 66 69 97)(63 67 70 98)(64 68 71 99)(73 80 111 114)(74 77 112 115)(75 78 109 116)(76 79 110 113)(81 101 108 88)(82 102 105 85)(83 103 106 86)(84 104 107 87)(89 117 124 96)(90 118 121 93)(91 119 122 94)(92 120 123 95)
(1 70 26 10)(2 62 27 17)(3 72 28 12)(4 64 25 19)(5 106 114 122)(6 82 115 90)(7 108 116 124)(8 84 113 92)(9 41 69 56)(11 43 71 54)(13 100 60 21)(14 68 57 48)(15 98 58 23)(16 66 59 46)(18 44 63 55)(20 42 61 53)(22 51 97 32)(24 49 99 30)(29 45 52 65)(31 47 50 67)(33 102 74 118)(34 88 75 96)(35 104 76 120)(36 86 73 94)(37 83 80 91)(38 105 77 121)(39 81 78 89)(40 107 79 123)(85 112 93 128)(87 110 95 126)(101 109 117 125)(103 111 119 127)
(1 125 3 127)(2 128 4 126)(5 50 7 52)(6 49 8 51)(9 102 11 104)(10 101 12 103)(13 37 15 39)(14 40 16 38)(17 85 19 87)(18 88 20 86)(21 83 23 81)(22 82 24 84)(25 110 27 112)(26 109 28 111)(29 114 31 116)(30 113 32 115)(33 43 35 41)(34 42 36 44)(45 106 47 108)(46 105 48 107)(53 73 55 75)(54 76 56 74)(57 79 59 77)(58 78 60 80)(61 94 63 96)(62 93 64 95)(65 122 67 124)(66 121 68 123)(69 118 71 120)(70 117 72 119)(89 100 91 98)(90 99 92 97)
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), (1,15,44,50)(2,16,41,51)(3,13,42,52)(4,14,43,49)(5,36,37,127)(6,33,38,128)(7,34,39,125)(8,35,40,126)(9,22,17,46)(10,23,18,47)(11,24,19,48)(12,21,20,45)(25,57,54,30)(26,58,55,31)(27,59,56,32)(28,60,53,29)(61,65,72,100)(62,66,69,97)(63,67,70,98)(64,68,71,99)(73,80,111,114)(74,77,112,115)(75,78,109,116)(76,79,110,113)(81,101,108,88)(82,102,105,85)(83,103,106,86)(84,104,107,87)(89,117,124,96)(90,118,121,93)(91,119,122,94)(92,120,123,95), (1,70,26,10)(2,62,27,17)(3,72,28,12)(4,64,25,19)(5,106,114,122)(6,82,115,90)(7,108,116,124)(8,84,113,92)(9,41,69,56)(11,43,71,54)(13,100,60,21)(14,68,57,48)(15,98,58,23)(16,66,59,46)(18,44,63,55)(20,42,61,53)(22,51,97,32)(24,49,99,30)(29,45,52,65)(31,47,50,67)(33,102,74,118)(34,88,75,96)(35,104,76,120)(36,86,73,94)(37,83,80,91)(38,105,77,121)(39,81,78,89)(40,107,79,123)(85,112,93,128)(87,110,95,126)(101,109,117,125)(103,111,119,127), (1,125,3,127)(2,128,4,126)(5,50,7,52)(6,49,8,51)(9,102,11,104)(10,101,12,103)(13,37,15,39)(14,40,16,38)(17,85,19,87)(18,88,20,86)(21,83,23,81)(22,82,24,84)(25,110,27,112)(26,109,28,111)(29,114,31,116)(30,113,32,115)(33,43,35,41)(34,42,36,44)(45,106,47,108)(46,105,48,107)(53,73,55,75)(54,76,56,74)(57,79,59,77)(58,78,60,80)(61,94,63,96)(62,93,64,95)(65,122,67,124)(66,121,68,123)(69,118,71,120)(70,117,72,119)(89,100,91,98)(90,99,92,97)>;
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), (1,15,44,50)(2,16,41,51)(3,13,42,52)(4,14,43,49)(5,36,37,127)(6,33,38,128)(7,34,39,125)(8,35,40,126)(9,22,17,46)(10,23,18,47)(11,24,19,48)(12,21,20,45)(25,57,54,30)(26,58,55,31)(27,59,56,32)(28,60,53,29)(61,65,72,100)(62,66,69,97)(63,67,70,98)(64,68,71,99)(73,80,111,114)(74,77,112,115)(75,78,109,116)(76,79,110,113)(81,101,108,88)(82,102,105,85)(83,103,106,86)(84,104,107,87)(89,117,124,96)(90,118,121,93)(91,119,122,94)(92,120,123,95), (1,70,26,10)(2,62,27,17)(3,72,28,12)(4,64,25,19)(5,106,114,122)(6,82,115,90)(7,108,116,124)(8,84,113,92)(9,41,69,56)(11,43,71,54)(13,100,60,21)(14,68,57,48)(15,98,58,23)(16,66,59,46)(18,44,63,55)(20,42,61,53)(22,51,97,32)(24,49,99,30)(29,45,52,65)(31,47,50,67)(33,102,74,118)(34,88,75,96)(35,104,76,120)(36,86,73,94)(37,83,80,91)(38,105,77,121)(39,81,78,89)(40,107,79,123)(85,112,93,128)(87,110,95,126)(101,109,117,125)(103,111,119,127), (1,125,3,127)(2,128,4,126)(5,50,7,52)(6,49,8,51)(9,102,11,104)(10,101,12,103)(13,37,15,39)(14,40,16,38)(17,85,19,87)(18,88,20,86)(21,83,23,81)(22,82,24,84)(25,110,27,112)(26,109,28,111)(29,114,31,116)(30,113,32,115)(33,43,35,41)(34,42,36,44)(45,106,47,108)(46,105,48,107)(53,73,55,75)(54,76,56,74)(57,79,59,77)(58,78,60,80)(61,94,63,96)(62,93,64,95)(65,122,67,124)(66,121,68,123)(69,118,71,120)(70,117,72,119)(89,100,91,98)(90,99,92,97) );
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)], [(1,15,44,50),(2,16,41,51),(3,13,42,52),(4,14,43,49),(5,36,37,127),(6,33,38,128),(7,34,39,125),(8,35,40,126),(9,22,17,46),(10,23,18,47),(11,24,19,48),(12,21,20,45),(25,57,54,30),(26,58,55,31),(27,59,56,32),(28,60,53,29),(61,65,72,100),(62,66,69,97),(63,67,70,98),(64,68,71,99),(73,80,111,114),(74,77,112,115),(75,78,109,116),(76,79,110,113),(81,101,108,88),(82,102,105,85),(83,103,106,86),(84,104,107,87),(89,117,124,96),(90,118,121,93),(91,119,122,94),(92,120,123,95)], [(1,70,26,10),(2,62,27,17),(3,72,28,12),(4,64,25,19),(5,106,114,122),(6,82,115,90),(7,108,116,124),(8,84,113,92),(9,41,69,56),(11,43,71,54),(13,100,60,21),(14,68,57,48),(15,98,58,23),(16,66,59,46),(18,44,63,55),(20,42,61,53),(22,51,97,32),(24,49,99,30),(29,45,52,65),(31,47,50,67),(33,102,74,118),(34,88,75,96),(35,104,76,120),(36,86,73,94),(37,83,80,91),(38,105,77,121),(39,81,78,89),(40,107,79,123),(85,112,93,128),(87,110,95,126),(101,109,117,125),(103,111,119,127)], [(1,125,3,127),(2,128,4,126),(5,50,7,52),(6,49,8,51),(9,102,11,104),(10,101,12,103),(13,37,15,39),(14,40,16,38),(17,85,19,87),(18,88,20,86),(21,83,23,81),(22,82,24,84),(25,110,27,112),(26,109,28,111),(29,114,31,116),(30,113,32,115),(33,43,35,41),(34,42,36,44),(45,106,47,108),(46,105,48,107),(53,73,55,75),(54,76,56,74),(57,79,59,77),(58,78,60,80),(61,94,63,96),(62,93,64,95),(65,122,67,124),(66,121,68,123),(69,118,71,120),(70,117,72,119),(89,100,91,98),(90,99,92,97)])
Matrix representation ►G ⊆ 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] >;
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 |
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