p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43.18C2, C42.65Q8, C23.764C24, C4.9(C42.C2), C42⋊8C4.55C2, (C22×C4).269C23, C22.185(C22×Q8), (C2×C42).1018C22, C2.C42.459C22, C23.65C23.92C2, C23.63C23.65C2, C2.51(C23.37C23), C2.117(C23.36C23), (C2×C4).176(C2×Q8), C2.23(C2×C42.C2), (C2×C4).533(C4○D4), (C2×C4⋊C4).567C22, C22.605(C2×C4○D4), SmallGroup(128,1596)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43.18C2
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, ac=ca, dad-1=ac2, bc=cb, dbd-1=b-1c2, dcd-1=a2c >
Subgroups: 308 in 200 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C43, C42⋊8C4, C23.63C23, C23.65C23, C43.18C2
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, C2×C42.C2, C23.36C23, C23.37C23, C43.18C2
►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 31 23 47)(2 32 24 48)(3 29 21 45)(4 30 22 46)(5 42 70 26)(6 43 71 27)(7 44 72 28)(8 41 69 25)(9 57 49 17)(10 58 50 18)(11 59 51 19)(12 60 52 20)(13 61 53 37)(14 62 54 38)(15 63 55 39)(16 64 56 40)(33 107 65 119)(34 108 66 120)(35 105 67 117)(36 106 68 118)(73 98 93 125)(74 99 94 126)(75 100 95 127)(76 97 96 128)(77 109 81 121)(78 110 82 122)(79 111 83 123)(80 112 84 124)(85 113 89 101)(86 114 90 102)(87 115 91 103)(88 116 92 104)
(1 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 46 38 18)(6 47 39 19)(7 48 40 20)(8 45 37 17)(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 89 127 121)(34 90 128 122)(35 91 125 123)(36 92 126 124)(65 85 100 109)(66 86 97 110)(67 87 98 111)(68 88 99 112)(73 79 105 103)(74 80 106 104)(75 77 107 101)(76 78 108 102)(81 119 113 95)(82 120 114 96)(83 117 115 93)(84 118 116 94)
(1 83 51 103)(2 116 52 80)(3 81 49 101)(4 114 50 78)(5 126 62 68)(6 33 63 100)(7 128 64 66)(8 35 61 98)(9 113 21 77)(10 82 22 102)(11 115 23 79)(12 84 24 104)(13 117 25 73)(14 94 26 106)(15 119 27 75)(16 96 28 108)(17 121 29 85)(18 90 30 110)(19 123 31 87)(20 92 32 112)(34 72 97 40)(36 70 99 38)(37 125 69 67)(39 127 71 65)(41 93 53 105)(42 118 54 74)(43 95 55 107)(44 120 56 76)(45 89 57 109)(46 122 58 86)(47 91 59 111)(48 124 60 88)
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), (1,31,23,47)(2,32,24,48)(3,29,21,45)(4,30,22,46)(5,42,70,26)(6,43,71,27)(7,44,72,28)(8,41,69,25)(9,57,49,17)(10,58,50,18)(11,59,51,19)(12,60,52,20)(13,61,53,37)(14,62,54,38)(15,63,55,39)(16,64,56,40)(33,107,65,119)(34,108,66,120)(35,105,67,117)(36,106,68,118)(73,98,93,125)(74,99,94,126)(75,100,95,127)(76,97,96,128)(77,109,81,121)(78,110,82,122)(79,111,83,123)(80,112,84,124)(85,113,89,101)(86,114,90,102)(87,115,91,103)(88,116,92,104), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,46,38,18)(6,47,39,19)(7,48,40,20)(8,45,37,17)(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)(33,89,127,121)(34,90,128,122)(35,91,125,123)(36,92,126,124)(65,85,100,109)(66,86,97,110)(67,87,98,111)(68,88,99,112)(73,79,105,103)(74,80,106,104)(75,77,107,101)(76,78,108,102)(81,119,113,95)(82,120,114,96)(83,117,115,93)(84,118,116,94), (1,83,51,103)(2,116,52,80)(3,81,49,101)(4,114,50,78)(5,126,62,68)(6,33,63,100)(7,128,64,66)(8,35,61,98)(9,113,21,77)(10,82,22,102)(11,115,23,79)(12,84,24,104)(13,117,25,73)(14,94,26,106)(15,119,27,75)(16,96,28,108)(17,121,29,85)(18,90,30,110)(19,123,31,87)(20,92,32,112)(34,72,97,40)(36,70,99,38)(37,125,69,67)(39,127,71,65)(41,93,53,105)(42,118,54,74)(43,95,55,107)(44,120,56,76)(45,89,57,109)(46,122,58,86)(47,91,59,111)(48,124,60,88)>;
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), (1,31,23,47)(2,32,24,48)(3,29,21,45)(4,30,22,46)(5,42,70,26)(6,43,71,27)(7,44,72,28)(8,41,69,25)(9,57,49,17)(10,58,50,18)(11,59,51,19)(12,60,52,20)(13,61,53,37)(14,62,54,38)(15,63,55,39)(16,64,56,40)(33,107,65,119)(34,108,66,120)(35,105,67,117)(36,106,68,118)(73,98,93,125)(74,99,94,126)(75,100,95,127)(76,97,96,128)(77,109,81,121)(78,110,82,122)(79,111,83,123)(80,112,84,124)(85,113,89,101)(86,114,90,102)(87,115,91,103)(88,116,92,104), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,46,38,18)(6,47,39,19)(7,48,40,20)(8,45,37,17)(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)(33,89,127,121)(34,90,128,122)(35,91,125,123)(36,92,126,124)(65,85,100,109)(66,86,97,110)(67,87,98,111)(68,88,99,112)(73,79,105,103)(74,80,106,104)(75,77,107,101)(76,78,108,102)(81,119,113,95)(82,120,114,96)(83,117,115,93)(84,118,116,94), (1,83,51,103)(2,116,52,80)(3,81,49,101)(4,114,50,78)(5,126,62,68)(6,33,63,100)(7,128,64,66)(8,35,61,98)(9,113,21,77)(10,82,22,102)(11,115,23,79)(12,84,24,104)(13,117,25,73)(14,94,26,106)(15,119,27,75)(16,96,28,108)(17,121,29,85)(18,90,30,110)(19,123,31,87)(20,92,32,112)(34,72,97,40)(36,70,99,38)(37,125,69,67)(39,127,71,65)(41,93,53,105)(42,118,54,74)(43,95,55,107)(44,120,56,76)(45,89,57,109)(46,122,58,86)(47,91,59,111)(48,124,60,88) );
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)], [(1,31,23,47),(2,32,24,48),(3,29,21,45),(4,30,22,46),(5,42,70,26),(6,43,71,27),(7,44,72,28),(8,41,69,25),(9,57,49,17),(10,58,50,18),(11,59,51,19),(12,60,52,20),(13,61,53,37),(14,62,54,38),(15,63,55,39),(16,64,56,40),(33,107,65,119),(34,108,66,120),(35,105,67,117),(36,106,68,118),(73,98,93,125),(74,99,94,126),(75,100,95,127),(76,97,96,128),(77,109,81,121),(78,110,82,122),(79,111,83,123),(80,112,84,124),(85,113,89,101),(86,114,90,102),(87,115,91,103),(88,116,92,104)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,46,38,18),(6,47,39,19),(7,48,40,20),(8,45,37,17),(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),(33,89,127,121),(34,90,128,122),(35,91,125,123),(36,92,126,124),(65,85,100,109),(66,86,97,110),(67,87,98,111),(68,88,99,112),(73,79,105,103),(74,80,106,104),(75,77,107,101),(76,78,108,102),(81,119,113,95),(82,120,114,96),(83,117,115,93),(84,118,116,94)], [(1,83,51,103),(2,116,52,80),(3,81,49,101),(4,114,50,78),(5,126,62,68),(6,33,63,100),(7,128,64,66),(8,35,61,98),(9,113,21,77),(10,82,22,102),(11,115,23,79),(12,84,24,104),(13,117,25,73),(14,94,26,106),(15,119,27,75),(16,96,28,108),(17,121,29,85),(18,90,30,110),(19,123,31,87),(20,92,32,112),(34,72,97,40),(36,70,99,38),(37,125,69,67),(39,127,71,65),(41,93,53,105),(42,118,54,74),(43,95,55,107),(44,120,56,76),(45,89,57,109),(46,122,58,86),(47,91,59,111),(48,124,60,88)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4AB | 4AC | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | C4○D4 |
| kernel | C43.18C2 | C43 | C42⋊8C4 | C23.63C23 | C23.65C23 | C42 | C2×C4 |
| # reps | 1 | 1 | 2 | 8 | 4 | 4 | 24 |
Matrix representation of C43.18C2 ►in GL6(𝔽5)
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
G:=sub<GL(6,GF(5))| [0,2,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
C43.18C2 in GAP, Magma, Sage, TeX
C_4^3._{18}C_2 % in TeX
G:=Group("C4^3.18C2"); // GroupNames label
G:=SmallGroup(128,1596);
// by ID
G=gap.SmallGroup(128,1596);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,268,2019,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*c^2,b*c=c*b,d*b*d^-1=b^-1*c^2,d*c*d^-1=a^2*c>;
// generators/relations