p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.30Q8, C4⋊C8⋊10C4, C4⋊1(C4.Q8), C4⋊C4.16Q8, C4.36(C4×Q8), C4⋊C4.218D4, C4.144(C4×D4), C4.17(C4⋊Q8), C42.151(C2×C4), C2.4(Q8⋊Q8), C2.4(D4⋊2Q8), C2.4(C4⋊SD16), (C2×C4).104SD16, C23.793(C2×D4), (C22×C4).760D4, C42⋊9C4.10C2, C2.4(D4.D4), C4.16(C42.C2), C22.69(C2×SD16), C22.92(C8⋊C22), C22.4Q16.42C2, (C2×C42).311C22, (C22×C8).315C22, C22.80(C22⋊Q8), C22.139(C4⋊D4), (C22×C4).1397C23, C22.81(C8.C22), C2.14(M4(2)⋊C4), C2.15(C23.65C23), (C4×C4⋊C4).23C2, (C2×C4⋊C8).47C2, C2.10(C2×C4.Q8), (C2×C8).114(C2×C4), (C2×C4).209(C2×Q8), (C2×C4.Q8).20C2, (C2×C4).136(C4⋊C4), (C2×C4).1014(C2×D4), (C2×C4⋊C4).79C22, C22.115(C2×C4⋊C4), (C2×C4).867(C4○D4), (C2×C4).553(C22×C4), SmallGroup(128,680)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.30Q8
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 236 in 128 conjugacy classes, 72 normal (36 characteristic)
C1, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4⋊C8, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C4×C4⋊C4, C42⋊9C4, C2×C4⋊C8, C2×C4.Q8, C42.30Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, C8⋊C22, C8.C22, C23.65C23, C2×C4.Q8, M4(2)⋊C4, C4⋊SD16, D4.D4, Q8⋊Q8, D4⋊2Q8, C42.30Q8
►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 39 11 13)(2 40 12 14)(3 37 9 15)(4 38 10 16)(5 19 118 106)(6 20 119 107)(7 17 120 108)(8 18 117 105)(21 122 110 126)(22 123 111 127)(23 124 112 128)(24 121 109 125)(25 31 51 33)(26 32 52 34)(27 29 49 35)(28 30 50 36)(41 57 69 83)(42 58 70 84)(43 59 71 81)(44 60 72 82)(45 53 67 61)(46 54 68 62)(47 55 65 63)(48 56 66 64)(73 85 101 115)(74 86 102 116)(75 87 103 113)(76 88 104 114)(77 89 99 93)(78 90 100 94)(79 91 97 95)(80 92 98 96)
(1 83 51 61)(2 84 52 62)(3 81 49 63)(4 82 50 64)(5 78 126 74)(6 79 127 75)(7 80 128 76)(8 77 125 73)(9 59 27 55)(10 60 28 56)(11 57 25 53)(12 58 26 54)(13 41 31 45)(14 42 32 46)(15 43 29 47)(16 44 30 48)(17 96 23 114)(18 93 24 115)(19 94 21 116)(20 95 22 113)(33 67 39 69)(34 68 40 70)(35 65 37 71)(36 66 38 72)(85 105 89 109)(86 106 90 110)(87 107 91 111)(88 108 92 112)(97 123 103 119)(98 124 104 120)(99 121 101 117)(100 122 102 118)
(1 87 25 95)(2 86 26 94)(3 85 27 93)(4 88 28 96)(5 84 122 54)(6 83 123 53)(7 82 124 56)(8 81 121 55)(9 115 49 89)(10 114 50 92)(11 113 51 91)(12 116 52 90)(13 103 33 79)(14 102 34 78)(15 101 35 77)(16 104 36 80)(17 72 112 48)(18 71 109 47)(19 70 110 46)(20 69 111 45)(21 68 106 42)(22 67 107 41)(23 66 108 44)(24 65 105 43)(29 99 37 73)(30 98 38 76)(31 97 39 75)(32 100 40 74)(57 127 61 119)(58 126 62 118)(59 125 63 117)(60 128 64 120)
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), (1,39,11,13)(2,40,12,14)(3,37,9,15)(4,38,10,16)(5,19,118,106)(6,20,119,107)(7,17,120,108)(8,18,117,105)(21,122,110,126)(22,123,111,127)(23,124,112,128)(24,121,109,125)(25,31,51,33)(26,32,52,34)(27,29,49,35)(28,30,50,36)(41,57,69,83)(42,58,70,84)(43,59,71,81)(44,60,72,82)(45,53,67,61)(46,54,68,62)(47,55,65,63)(48,56,66,64)(73,85,101,115)(74,86,102,116)(75,87,103,113)(76,88,104,114)(77,89,99,93)(78,90,100,94)(79,91,97,95)(80,92,98,96), (1,83,51,61)(2,84,52,62)(3,81,49,63)(4,82,50,64)(5,78,126,74)(6,79,127,75)(7,80,128,76)(8,77,125,73)(9,59,27,55)(10,60,28,56)(11,57,25,53)(12,58,26,54)(13,41,31,45)(14,42,32,46)(15,43,29,47)(16,44,30,48)(17,96,23,114)(18,93,24,115)(19,94,21,116)(20,95,22,113)(33,67,39,69)(34,68,40,70)(35,65,37,71)(36,66,38,72)(85,105,89,109)(86,106,90,110)(87,107,91,111)(88,108,92,112)(97,123,103,119)(98,124,104,120)(99,121,101,117)(100,122,102,118), (1,87,25,95)(2,86,26,94)(3,85,27,93)(4,88,28,96)(5,84,122,54)(6,83,123,53)(7,82,124,56)(8,81,121,55)(9,115,49,89)(10,114,50,92)(11,113,51,91)(12,116,52,90)(13,103,33,79)(14,102,34,78)(15,101,35,77)(16,104,36,80)(17,72,112,48)(18,71,109,47)(19,70,110,46)(20,69,111,45)(21,68,106,42)(22,67,107,41)(23,66,108,44)(24,65,105,43)(29,99,37,73)(30,98,38,76)(31,97,39,75)(32,100,40,74)(57,127,61,119)(58,126,62,118)(59,125,63,117)(60,128,64,120)>;
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), (1,39,11,13)(2,40,12,14)(3,37,9,15)(4,38,10,16)(5,19,118,106)(6,20,119,107)(7,17,120,108)(8,18,117,105)(21,122,110,126)(22,123,111,127)(23,124,112,128)(24,121,109,125)(25,31,51,33)(26,32,52,34)(27,29,49,35)(28,30,50,36)(41,57,69,83)(42,58,70,84)(43,59,71,81)(44,60,72,82)(45,53,67,61)(46,54,68,62)(47,55,65,63)(48,56,66,64)(73,85,101,115)(74,86,102,116)(75,87,103,113)(76,88,104,114)(77,89,99,93)(78,90,100,94)(79,91,97,95)(80,92,98,96), (1,83,51,61)(2,84,52,62)(3,81,49,63)(4,82,50,64)(5,78,126,74)(6,79,127,75)(7,80,128,76)(8,77,125,73)(9,59,27,55)(10,60,28,56)(11,57,25,53)(12,58,26,54)(13,41,31,45)(14,42,32,46)(15,43,29,47)(16,44,30,48)(17,96,23,114)(18,93,24,115)(19,94,21,116)(20,95,22,113)(33,67,39,69)(34,68,40,70)(35,65,37,71)(36,66,38,72)(85,105,89,109)(86,106,90,110)(87,107,91,111)(88,108,92,112)(97,123,103,119)(98,124,104,120)(99,121,101,117)(100,122,102,118), (1,87,25,95)(2,86,26,94)(3,85,27,93)(4,88,28,96)(5,84,122,54)(6,83,123,53)(7,82,124,56)(8,81,121,55)(9,115,49,89)(10,114,50,92)(11,113,51,91)(12,116,52,90)(13,103,33,79)(14,102,34,78)(15,101,35,77)(16,104,36,80)(17,72,112,48)(18,71,109,47)(19,70,110,46)(20,69,111,45)(21,68,106,42)(22,67,107,41)(23,66,108,44)(24,65,105,43)(29,99,37,73)(30,98,38,76)(31,97,39,75)(32,100,40,74)(57,127,61,119)(58,126,62,118)(59,125,63,117)(60,128,64,120) );
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)], [(1,39,11,13),(2,40,12,14),(3,37,9,15),(4,38,10,16),(5,19,118,106),(6,20,119,107),(7,17,120,108),(8,18,117,105),(21,122,110,126),(22,123,111,127),(23,124,112,128),(24,121,109,125),(25,31,51,33),(26,32,52,34),(27,29,49,35),(28,30,50,36),(41,57,69,83),(42,58,70,84),(43,59,71,81),(44,60,72,82),(45,53,67,61),(46,54,68,62),(47,55,65,63),(48,56,66,64),(73,85,101,115),(74,86,102,116),(75,87,103,113),(76,88,104,114),(77,89,99,93),(78,90,100,94),(79,91,97,95),(80,92,98,96)], [(1,83,51,61),(2,84,52,62),(3,81,49,63),(4,82,50,64),(5,78,126,74),(6,79,127,75),(7,80,128,76),(8,77,125,73),(9,59,27,55),(10,60,28,56),(11,57,25,53),(12,58,26,54),(13,41,31,45),(14,42,32,46),(15,43,29,47),(16,44,30,48),(17,96,23,114),(18,93,24,115),(19,94,21,116),(20,95,22,113),(33,67,39,69),(34,68,40,70),(35,65,37,71),(36,66,38,72),(85,105,89,109),(86,106,90,110),(87,107,91,111),(88,108,92,112),(97,123,103,119),(98,124,104,120),(99,121,101,117),(100,122,102,118)], [(1,87,25,95),(2,86,26,94),(3,85,27,93),(4,88,28,96),(5,84,122,54),(6,83,123,53),(7,82,124,56),(8,81,121,55),(9,115,49,89),(10,114,50,92),(11,113,51,91),(12,116,52,90),(13,103,33,79),(14,102,34,78),(15,101,35,77),(16,104,36,80),(17,72,112,48),(18,71,109,47),(19,70,110,46),(20,69,111,45),(21,68,106,42),(22,67,107,41),(23,66,108,44),(24,65,105,43),(29,99,37,73),(30,98,38,76),(31,97,39,75),(32,100,40,74),(57,127,61,119),(58,126,62,118),(59,125,63,117),(60,128,64,120)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | D4 | SD16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.30Q8 | C22.4Q16 | C4×C4⋊C4 | C42⋊9C4 | C2×C4⋊C8 | C2×C4.Q8 | C4⋊C8 | C42 | C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 8 | 4 | 1 | 1 |
Matrix representation of C42.30Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 6 | 0 | 0 |
| 0 | 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 16 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,6,0,0,0,0,6,4,0,0,0,0,0,0,13,4,0,0,0,0,9,4],[10,1,0,0,0,0,1,7,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,16,1,0,0,0,0,0,1] >;
C42.30Q8 in GAP, Magma, Sage, TeX
C_4^2._{30}Q_8 % in TeX
G:=Group("C4^2.30Q8"); // GroupNames label
G:=SmallGroup(128,680);
// by ID
G=gap.SmallGroup(128,680);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,100,2019,1018,248]);
// 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^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;
// generators/relations