p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.36Q8, C23.470C24, C22.1912- 1+4, C22.2532+ 1+4, C42⋊9C4.32C2, C42⋊8C4.35C2, (C2×C42).67C22, C4.32(C42.C2), (C22×C4).103C23, C22.111(C22×Q8), C23.83C23.15C2, C23.65C23.56C2, C23.81C23.17C2, C23.63C23.28C2, C2.C42.206C22, C2.29(C23.37C23), C2.41(C22.50C24), C2.60(C22.47C24), C2.14(C23.41C23), C2.66(C22.46C24), C2.19(C22.53C24), (C4×C4⋊C4).69C2, (C2×C4).126(C2×Q8), C2.17(C2×C42.C2), (C2×C4).152(C4○D4), (C2×C4⋊C4).317C22, C22.346(C2×C4○D4), SmallGroup(128,1302)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.36Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 308 in 190 conjugacy classes, 104 normal (42 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×C4⋊C4, C42⋊8C4, C42⋊9C4, C23.63C23, C23.65C23, C23.81C23, C23.83C23, C42.36Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C42.C2, C23.37C23, C23.41C23, C22.46C24, C22.47C24, C22.50C24, C22.53C24, C42.36Q8
►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 12 21 43)(2 9 22 44)(3 10 23 41)(4 11 24 42)(5 56 40 26)(6 53 37 27)(7 54 38 28)(8 55 39 25)(13 64 48 70)(14 61 45 71)(15 62 46 72)(16 63 47 69)(17 30 52 60)(18 31 49 57)(19 32 50 58)(20 29 51 59)(33 74 125 108)(34 75 126 105)(35 76 127 106)(36 73 128 107)(65 120 99 94)(66 117 100 95)(67 118 97 96)(68 119 98 93)(77 82 103 116)(78 83 104 113)(79 84 101 114)(80 81 102 115)(85 90 111 124)(86 91 112 121)(87 92 109 122)(88 89 110 123)
(1 47 39 59)(2 46 40 58)(3 45 37 57)(4 48 38 60)(5 32 22 15)(6 31 23 14)(7 30 24 13)(8 29 21 16)(9 72 26 19)(10 71 27 18)(11 70 28 17)(12 69 25 20)(33 104 112 95)(34 103 109 94)(35 102 110 93)(36 101 111 96)(41 61 53 49)(42 64 54 52)(43 63 55 51)(44 62 56 50)(65 75 116 122)(66 74 113 121)(67 73 114 124)(68 76 115 123)(77 87 120 126)(78 86 117 125)(79 85 118 128)(80 88 119 127)(81 89 98 106)(82 92 99 105)(83 91 100 108)(84 90 97 107)
(1 100 37 81)(2 67 38 116)(3 98 39 83)(4 65 40 114)(5 84 24 99)(6 115 21 66)(7 82 22 97)(8 113 23 68)(9 96 28 103)(10 119 25 78)(11 94 26 101)(12 117 27 80)(13 92 32 107)(14 123 29 74)(15 90 30 105)(16 121 31 76)(17 34 72 111)(18 127 69 86)(19 36 70 109)(20 125 71 88)(33 61 110 51)(35 63 112 49)(41 93 55 104)(42 120 56 79)(43 95 53 102)(44 118 54 77)(45 89 59 108)(46 124 60 75)(47 91 57 106)(48 122 58 73)(50 128 64 87)(52 126 62 85)
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,70)(14,61,45,71)(15,62,46,72)(16,63,47,69)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59)(33,74,125,108)(34,75,126,105)(35,76,127,106)(36,73,128,107)(65,120,99,94)(66,117,100,95)(67,118,97,96)(68,119,98,93)(77,82,103,116)(78,83,104,113)(79,84,101,114)(80,81,102,115)(85,90,111,124)(86,91,112,121)(87,92,109,122)(88,89,110,123), (1,47,39,59)(2,46,40,58)(3,45,37,57)(4,48,38,60)(5,32,22,15)(6,31,23,14)(7,30,24,13)(8,29,21,16)(9,72,26,19)(10,71,27,18)(11,70,28,17)(12,69,25,20)(33,104,112,95)(34,103,109,94)(35,102,110,93)(36,101,111,96)(41,61,53,49)(42,64,54,52)(43,63,55,51)(44,62,56,50)(65,75,116,122)(66,74,113,121)(67,73,114,124)(68,76,115,123)(77,87,120,126)(78,86,117,125)(79,85,118,128)(80,88,119,127)(81,89,98,106)(82,92,99,105)(83,91,100,108)(84,90,97,107), (1,100,37,81)(2,67,38,116)(3,98,39,83)(4,65,40,114)(5,84,24,99)(6,115,21,66)(7,82,22,97)(8,113,23,68)(9,96,28,103)(10,119,25,78)(11,94,26,101)(12,117,27,80)(13,92,32,107)(14,123,29,74)(15,90,30,105)(16,121,31,76)(17,34,72,111)(18,127,69,86)(19,36,70,109)(20,125,71,88)(33,61,110,51)(35,63,112,49)(41,93,55,104)(42,120,56,79)(43,95,53,102)(44,118,54,77)(45,89,59,108)(46,124,60,75)(47,91,57,106)(48,122,58,73)(50,128,64,87)(52,126,62,85)>;
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,70)(14,61,45,71)(15,62,46,72)(16,63,47,69)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59)(33,74,125,108)(34,75,126,105)(35,76,127,106)(36,73,128,107)(65,120,99,94)(66,117,100,95)(67,118,97,96)(68,119,98,93)(77,82,103,116)(78,83,104,113)(79,84,101,114)(80,81,102,115)(85,90,111,124)(86,91,112,121)(87,92,109,122)(88,89,110,123), (1,47,39,59)(2,46,40,58)(3,45,37,57)(4,48,38,60)(5,32,22,15)(6,31,23,14)(7,30,24,13)(8,29,21,16)(9,72,26,19)(10,71,27,18)(11,70,28,17)(12,69,25,20)(33,104,112,95)(34,103,109,94)(35,102,110,93)(36,101,111,96)(41,61,53,49)(42,64,54,52)(43,63,55,51)(44,62,56,50)(65,75,116,122)(66,74,113,121)(67,73,114,124)(68,76,115,123)(77,87,120,126)(78,86,117,125)(79,85,118,128)(80,88,119,127)(81,89,98,106)(82,92,99,105)(83,91,100,108)(84,90,97,107), (1,100,37,81)(2,67,38,116)(3,98,39,83)(4,65,40,114)(5,84,24,99)(6,115,21,66)(7,82,22,97)(8,113,23,68)(9,96,28,103)(10,119,25,78)(11,94,26,101)(12,117,27,80)(13,92,32,107)(14,123,29,74)(15,90,30,105)(16,121,31,76)(17,34,72,111)(18,127,69,86)(19,36,70,109)(20,125,71,88)(33,61,110,51)(35,63,112,49)(41,93,55,104)(42,120,56,79)(43,95,53,102)(44,118,54,77)(45,89,59,108)(46,124,60,75)(47,91,57,106)(48,122,58,73)(50,128,64,87)(52,126,62,85) );
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)], [(1,12,21,43),(2,9,22,44),(3,10,23,41),(4,11,24,42),(5,56,40,26),(6,53,37,27),(7,54,38,28),(8,55,39,25),(13,64,48,70),(14,61,45,71),(15,62,46,72),(16,63,47,69),(17,30,52,60),(18,31,49,57),(19,32,50,58),(20,29,51,59),(33,74,125,108),(34,75,126,105),(35,76,127,106),(36,73,128,107),(65,120,99,94),(66,117,100,95),(67,118,97,96),(68,119,98,93),(77,82,103,116),(78,83,104,113),(79,84,101,114),(80,81,102,115),(85,90,111,124),(86,91,112,121),(87,92,109,122),(88,89,110,123)], [(1,47,39,59),(2,46,40,58),(3,45,37,57),(4,48,38,60),(5,32,22,15),(6,31,23,14),(7,30,24,13),(8,29,21,16),(9,72,26,19),(10,71,27,18),(11,70,28,17),(12,69,25,20),(33,104,112,95),(34,103,109,94),(35,102,110,93),(36,101,111,96),(41,61,53,49),(42,64,54,52),(43,63,55,51),(44,62,56,50),(65,75,116,122),(66,74,113,121),(67,73,114,124),(68,76,115,123),(77,87,120,126),(78,86,117,125),(79,85,118,128),(80,88,119,127),(81,89,98,106),(82,92,99,105),(83,91,100,108),(84,90,97,107)], [(1,100,37,81),(2,67,38,116),(3,98,39,83),(4,65,40,114),(5,84,24,99),(6,115,21,66),(7,82,22,97),(8,113,23,68),(9,96,28,103),(10,119,25,78),(11,94,26,101),(12,117,27,80),(13,92,32,107),(14,123,29,74),(15,90,30,105),(16,121,31,76),(17,34,72,111),(18,127,69,86),(19,36,70,109),(20,125,71,88),(33,61,110,51),(35,63,112,49),(41,93,55,104),(42,120,56,79),(43,95,53,102),(44,118,54,77),(45,89,59,108),(46,124,60,75),(47,91,57,106),(48,122,58,73),(50,128,64,87),(52,126,62,85)]])
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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.36Q8 | C4×C4⋊C4 | C42⋊8C4 | C42⋊9C4 | C23.63C23 | C23.65C23 | C23.81C23 | C23.83C23 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 16 | 1 | 1 |
Matrix representation of C42.36Q8 ►in GL6(𝔽5)
| 2 | 1 | 0 | 0 | 0 | 0 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(5))| [2,2,0,0,0,0,1,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[2,0,0,0,0,0,1,3,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0] >;
C42.36Q8 in GAP, Magma, Sage, TeX
C_4^2._{36}Q_8 % in TeX
G:=Group("C4^2.36Q8"); // GroupNames label
G:=SmallGroup(128,1302);
// by ID
G=gap.SmallGroup(128,1302);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,456,758,723,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*c^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations