p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊18Q8, C43.16C2, C42.347D4, C23.762C24, C4.11(C4⋊Q8), C42⋊8C4.54C2, C4.17(C4.4D4), (C22×C4).267C23, C22.472(C22×D4), C22.183(C22×Q8), (C2×C42).1096C22, (C22×Q8).252C22, C2.C42.457C22, C23.67C23.65C2, C2.49(C23.37C23), C2.23(C2×C4⋊Q8), (C2×C4⋊Q8).40C2, (C2×C4).836(C2×D4), (C2×C4).174(C2×Q8), C2.36(C2×C4.4D4), (C2×C4).676(C4○D4), (C2×C4⋊C4).565C22, C22.603(C2×C4○D4), SmallGroup(128,1594)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊18Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 420 in 252 conjugacy classes, 132 normal (8 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.C42, C2×C42, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C43, C42⋊8C4, C23.67C23, C2×C4⋊Q8, C42⋊18Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C4.4D4, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C4.4D4, C2×C4⋊Q8, C23.37C23, C42⋊18Q8
►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 98 24 26)(2 99 21 27)(3 100 22 28)(4 97 23 25)(5 116 107 83)(6 113 108 84)(7 114 105 81)(8 115 106 82)(9 75 57 13)(10 76 58 14)(11 73 59 15)(12 74 60 16)(17 50 29 63)(18 51 30 64)(19 52 31 61)(20 49 32 62)(33 80 54 40)(34 77 55 37)(35 78 56 38)(36 79 53 39)(41 92 109 122)(42 89 110 123)(43 90 111 124)(44 91 112 121)(45 66 103 120)(46 67 104 117)(47 68 101 118)(48 65 102 119)(69 96 85 126)(70 93 86 127)(71 94 87 128)(72 95 88 125)
(1 76 17 37)(2 73 18 38)(3 74 19 39)(4 75 20 40)(5 65 93 122)(6 66 94 123)(7 67 95 124)(8 68 96 121)(9 62 54 25)(10 63 55 26)(11 64 56 27)(12 61 53 28)(13 32 80 23)(14 29 77 24)(15 30 78 21)(16 31 79 22)(33 97 57 49)(34 98 58 50)(35 99 59 51)(36 100 60 52)(41 116 102 86)(42 113 103 87)(43 114 104 88)(44 115 101 85)(45 71 110 84)(46 72 111 81)(47 69 112 82)(48 70 109 83)(89 108 120 128)(90 105 117 125)(91 106 118 126)(92 107 119 127)
(1 86 17 116)(2 85 18 115)(3 88 19 114)(4 87 20 113)(5 100 93 52)(6 99 94 51)(7 98 95 50)(8 97 96 49)(9 118 54 91)(10 117 55 90)(11 120 56 89)(12 119 53 92)(13 45 80 110)(14 48 77 109)(15 47 78 112)(16 46 79 111)(21 69 30 82)(22 72 31 81)(23 71 32 84)(24 70 29 83)(25 126 62 106)(26 125 63 105)(27 128 64 108)(28 127 61 107)(33 121 57 68)(34 124 58 67)(35 123 59 66)(36 122 60 65)(37 41 76 102)(38 44 73 101)(39 43 74 104)(40 42 75 103)
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), (1,98,24,26)(2,99,21,27)(3,100,22,28)(4,97,23,25)(5,116,107,83)(6,113,108,84)(7,114,105,81)(8,115,106,82)(9,75,57,13)(10,76,58,14)(11,73,59,15)(12,74,60,16)(17,50,29,63)(18,51,30,64)(19,52,31,61)(20,49,32,62)(33,80,54,40)(34,77,55,37)(35,78,56,38)(36,79,53,39)(41,92,109,122)(42,89,110,123)(43,90,111,124)(44,91,112,121)(45,66,103,120)(46,67,104,117)(47,68,101,118)(48,65,102,119)(69,96,85,126)(70,93,86,127)(71,94,87,128)(72,95,88,125), (1,76,17,37)(2,73,18,38)(3,74,19,39)(4,75,20,40)(5,65,93,122)(6,66,94,123)(7,67,95,124)(8,68,96,121)(9,62,54,25)(10,63,55,26)(11,64,56,27)(12,61,53,28)(13,32,80,23)(14,29,77,24)(15,30,78,21)(16,31,79,22)(33,97,57,49)(34,98,58,50)(35,99,59,51)(36,100,60,52)(41,116,102,86)(42,113,103,87)(43,114,104,88)(44,115,101,85)(45,71,110,84)(46,72,111,81)(47,69,112,82)(48,70,109,83)(89,108,120,128)(90,105,117,125)(91,106,118,126)(92,107,119,127), (1,86,17,116)(2,85,18,115)(3,88,19,114)(4,87,20,113)(5,100,93,52)(6,99,94,51)(7,98,95,50)(8,97,96,49)(9,118,54,91)(10,117,55,90)(11,120,56,89)(12,119,53,92)(13,45,80,110)(14,48,77,109)(15,47,78,112)(16,46,79,111)(21,69,30,82)(22,72,31,81)(23,71,32,84)(24,70,29,83)(25,126,62,106)(26,125,63,105)(27,128,64,108)(28,127,61,107)(33,121,57,68)(34,124,58,67)(35,123,59,66)(36,122,60,65)(37,41,76,102)(38,44,73,101)(39,43,74,104)(40,42,75,103)>;
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), (1,98,24,26)(2,99,21,27)(3,100,22,28)(4,97,23,25)(5,116,107,83)(6,113,108,84)(7,114,105,81)(8,115,106,82)(9,75,57,13)(10,76,58,14)(11,73,59,15)(12,74,60,16)(17,50,29,63)(18,51,30,64)(19,52,31,61)(20,49,32,62)(33,80,54,40)(34,77,55,37)(35,78,56,38)(36,79,53,39)(41,92,109,122)(42,89,110,123)(43,90,111,124)(44,91,112,121)(45,66,103,120)(46,67,104,117)(47,68,101,118)(48,65,102,119)(69,96,85,126)(70,93,86,127)(71,94,87,128)(72,95,88,125), (1,76,17,37)(2,73,18,38)(3,74,19,39)(4,75,20,40)(5,65,93,122)(6,66,94,123)(7,67,95,124)(8,68,96,121)(9,62,54,25)(10,63,55,26)(11,64,56,27)(12,61,53,28)(13,32,80,23)(14,29,77,24)(15,30,78,21)(16,31,79,22)(33,97,57,49)(34,98,58,50)(35,99,59,51)(36,100,60,52)(41,116,102,86)(42,113,103,87)(43,114,104,88)(44,115,101,85)(45,71,110,84)(46,72,111,81)(47,69,112,82)(48,70,109,83)(89,108,120,128)(90,105,117,125)(91,106,118,126)(92,107,119,127), (1,86,17,116)(2,85,18,115)(3,88,19,114)(4,87,20,113)(5,100,93,52)(6,99,94,51)(7,98,95,50)(8,97,96,49)(9,118,54,91)(10,117,55,90)(11,120,56,89)(12,119,53,92)(13,45,80,110)(14,48,77,109)(15,47,78,112)(16,46,79,111)(21,69,30,82)(22,72,31,81)(23,71,32,84)(24,70,29,83)(25,126,62,106)(26,125,63,105)(27,128,64,108)(28,127,61,107)(33,121,57,68)(34,124,58,67)(35,123,59,66)(36,122,60,65)(37,41,76,102)(38,44,73,101)(39,43,74,104)(40,42,75,103) );
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)], [(1,98,24,26),(2,99,21,27),(3,100,22,28),(4,97,23,25),(5,116,107,83),(6,113,108,84),(7,114,105,81),(8,115,106,82),(9,75,57,13),(10,76,58,14),(11,73,59,15),(12,74,60,16),(17,50,29,63),(18,51,30,64),(19,52,31,61),(20,49,32,62),(33,80,54,40),(34,77,55,37),(35,78,56,38),(36,79,53,39),(41,92,109,122),(42,89,110,123),(43,90,111,124),(44,91,112,121),(45,66,103,120),(46,67,104,117),(47,68,101,118),(48,65,102,119),(69,96,85,126),(70,93,86,127),(71,94,87,128),(72,95,88,125)], [(1,76,17,37),(2,73,18,38),(3,74,19,39),(4,75,20,40),(5,65,93,122),(6,66,94,123),(7,67,95,124),(8,68,96,121),(9,62,54,25),(10,63,55,26),(11,64,56,27),(12,61,53,28),(13,32,80,23),(14,29,77,24),(15,30,78,21),(16,31,79,22),(33,97,57,49),(34,98,58,50),(35,99,59,51),(36,100,60,52),(41,116,102,86),(42,113,103,87),(43,114,104,88),(44,115,101,85),(45,71,110,84),(46,72,111,81),(47,69,112,82),(48,70,109,83),(89,108,120,128),(90,105,117,125),(91,106,118,126),(92,107,119,127)], [(1,86,17,116),(2,85,18,115),(3,88,19,114),(4,87,20,113),(5,100,93,52),(6,99,94,51),(7,98,95,50),(8,97,96,49),(9,118,54,91),(10,117,55,90),(11,120,56,89),(12,119,53,92),(13,45,80,110),(14,48,77,109),(15,47,78,112),(16,46,79,111),(21,69,30,82),(22,72,31,81),(23,71,32,84),(24,70,29,83),(25,126,62,106),(26,125,63,105),(27,128,64,108),(28,127,61,107),(33,121,57,68),(34,124,58,67),(35,123,59,66),(36,122,60,65),(37,41,76,102),(38,44,73,101),(39,43,74,104),(40,42,75,103)]])
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 | 2 |
| type | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 |
| kernel | C42⋊18Q8 | C43 | C42⋊8C4 | C23.67C23 | C2×C4⋊Q8 | C42 | C42 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 4 | 8 | 16 |
Matrix representation of C42⋊18Q8 ►in GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
C42⋊18Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{18}Q_8 % in TeX
G:=Group("C4^2:18Q8"); // GroupNames label
G:=SmallGroup(128,1594);
// by ID
G=gap.SmallGroup(128,1594);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,184,2019,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations