p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.180D4, C23.482C24, C22.2642+ 1+4, C22.1972- 1+4, C2.7Q82, (C2×Q8)⋊12Q8, C42⋊9C4.34C2, C2.21(Q8⋊3Q8), C4.105(C22⋊Q8), (C22×C4).112C23, (C2×C42).576C22, C22.323(C22×D4), C22.117(C22×Q8), (C22×Q8).441C22, C2.C42.216C22, C23.65C23.59C2, C23.81C23.18C2, C23.67C23.44C2, C2.23(C22.53C24), C2.16(C22.31C24), (C2×C4×Q8).38C2, (C2×C4).59(C2×Q8), (C2×C4).363(C2×D4), C2.40(C2×C22⋊Q8), (C2×C4).897(C4○D4), (C2×C4⋊C4).328C22, C22.358(C2×C4○D4), SmallGroup(128,1314)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.180D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 388 in 242 conjugacy classes, 124 normal (12 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, C4×Q8, C22×Q8, C42⋊9C4, C23.65C23, C23.67C23, C23.81C23, C2×C4×Q8, C42.180D4
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, 2- 1+4, C2×C22⋊Q8, C22.31C24, Q8⋊3Q8, Q82, C22.53C24, C42.180D4
►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)
(1 125 60 75)(2 128 57 74)(3 127 58 73)(4 126 59 76)(5 55 111 47)(6 54 112 46)(7 53 109 45)(8 56 110 48)(9 96 70 81)(10 95 71 84)(11 94 72 83)(12 93 69 82)(13 34 29 106)(14 33 30 105)(15 36 31 108)(16 35 32 107)(17 38 25 87)(18 37 26 86)(19 40 27 85)(20 39 28 88)(21 91 65 78)(22 90 66 77)(23 89 67 80)(24 92 68 79)(41 118 62 113)(42 117 63 116)(43 120 64 115)(44 119 61 114)(49 121 97 102)(50 124 98 101)(51 123 99 104)(52 122 100 103)
(1 101 45 116)(2 113 46 102)(3 103 47 114)(4 115 48 104)(5 63 127 98)(6 99 128 64)(7 61 125 100)(8 97 126 62)(9 108 23 86)(10 87 24 105)(11 106 21 88)(12 85 22 107)(13 80 20 81)(14 82 17 77)(15 78 18 83)(16 84 19 79)(25 90 30 93)(26 94 31 91)(27 92 32 95)(28 96 29 89)(33 71 38 68)(34 65 39 72)(35 69 40 66)(36 67 37 70)(41 110 49 76)(42 73 50 111)(43 112 51 74)(44 75 52 109)(53 117 60 124)(54 121 57 118)(55 119 58 122)(56 123 59 120)
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), (1,20,45,13)(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), (1,125,60,75)(2,128,57,74)(3,127,58,73)(4,126,59,76)(5,55,111,47)(6,54,112,46)(7,53,109,45)(8,56,110,48)(9,96,70,81)(10,95,71,84)(11,94,72,83)(12,93,69,82)(13,34,29,106)(14,33,30,105)(15,36,31,108)(16,35,32,107)(17,38,25,87)(18,37,26,86)(19,40,27,85)(20,39,28,88)(21,91,65,78)(22,90,66,77)(23,89,67,80)(24,92,68,79)(41,118,62,113)(42,117,63,116)(43,120,64,115)(44,119,61,114)(49,121,97,102)(50,124,98,101)(51,123,99,104)(52,122,100,103), (1,101,45,116)(2,113,46,102)(3,103,47,114)(4,115,48,104)(5,63,127,98)(6,99,128,64)(7,61,125,100)(8,97,126,62)(9,108,23,86)(10,87,24,105)(11,106,21,88)(12,85,22,107)(13,80,20,81)(14,82,17,77)(15,78,18,83)(16,84,19,79)(25,90,30,93)(26,94,31,91)(27,92,32,95)(28,96,29,89)(33,71,38,68)(34,65,39,72)(35,69,40,66)(36,67,37,70)(41,110,49,76)(42,73,50,111)(43,112,51,74)(44,75,52,109)(53,117,60,124)(54,121,57,118)(55,119,58,122)(56,123,59,120)>;
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), (1,20,45,13)(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), (1,125,60,75)(2,128,57,74)(3,127,58,73)(4,126,59,76)(5,55,111,47)(6,54,112,46)(7,53,109,45)(8,56,110,48)(9,96,70,81)(10,95,71,84)(11,94,72,83)(12,93,69,82)(13,34,29,106)(14,33,30,105)(15,36,31,108)(16,35,32,107)(17,38,25,87)(18,37,26,86)(19,40,27,85)(20,39,28,88)(21,91,65,78)(22,90,66,77)(23,89,67,80)(24,92,68,79)(41,118,62,113)(42,117,63,116)(43,120,64,115)(44,119,61,114)(49,121,97,102)(50,124,98,101)(51,123,99,104)(52,122,100,103), (1,101,45,116)(2,113,46,102)(3,103,47,114)(4,115,48,104)(5,63,127,98)(6,99,128,64)(7,61,125,100)(8,97,126,62)(9,108,23,86)(10,87,24,105)(11,106,21,88)(12,85,22,107)(13,80,20,81)(14,82,17,77)(15,78,18,83)(16,84,19,79)(25,90,30,93)(26,94,31,91)(27,92,32,95)(28,96,29,89)(33,71,38,68)(34,65,39,72)(35,69,40,66)(36,67,37,70)(41,110,49,76)(42,73,50,111)(43,112,51,74)(44,75,52,109)(53,117,60,124)(54,121,57,118)(55,119,58,122)(56,123,59,120) );
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)], [(1,20,45,13),(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)], [(1,125,60,75),(2,128,57,74),(3,127,58,73),(4,126,59,76),(5,55,111,47),(6,54,112,46),(7,53,109,45),(8,56,110,48),(9,96,70,81),(10,95,71,84),(11,94,72,83),(12,93,69,82),(13,34,29,106),(14,33,30,105),(15,36,31,108),(16,35,32,107),(17,38,25,87),(18,37,26,86),(19,40,27,85),(20,39,28,88),(21,91,65,78),(22,90,66,77),(23,89,67,80),(24,92,68,79),(41,118,62,113),(42,117,63,116),(43,120,64,115),(44,119,61,114),(49,121,97,102),(50,124,98,101),(51,123,99,104),(52,122,100,103)], [(1,101,45,116),(2,113,46,102),(3,103,47,114),(4,115,48,104),(5,63,127,98),(6,99,128,64),(7,61,125,100),(8,97,126,62),(9,108,23,86),(10,87,24,105),(11,106,21,88),(12,85,22,107),(13,80,20,81),(14,82,17,77),(15,78,18,83),(16,84,19,79),(25,90,30,93),(26,94,31,91),(27,92,32,95),(28,96,29,89),(33,71,38,68),(34,65,39,72),(35,69,40,66),(36,67,37,70),(41,110,49,76),(42,73,50,111),(43,112,51,74),(44,75,52,109),(53,117,60,124),(54,121,57,118),(55,119,58,122),(56,123,59,120)]])
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 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.180D4 | C42⋊9C4 | C23.65C23 | C23.67C23 | C23.81C23 | C2×C4×Q8 | C42 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 8 | 8 | 1 | 1 |
Matrix representation of C42.180D4 ►in GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C42.180D4 in GAP, Magma, Sage, TeX
C_4^2._{180}D_4 % in TeX
G:=Group("C4^2.180D4"); // GroupNames label
G:=SmallGroup(128,1314);
// by ID
G=gap.SmallGroup(128,1314);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,352,675,80]);
// 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,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations