p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.31Q8, C4⋊C8⋊9C4, C4⋊C4.17Q8, C4.37(C4×Q8), C4⋊C4.219D4, C4.145(C4×D4), C4.18(C4⋊Q8), C2.6(D4.Q8), C2.6(Q8.Q8), C42.152(C2×C4), C23.794(C2×D4), (C22×C4).699D4, C42⋊8C4.10C2, C2.6(D4.2D4), C22.73(C4○D8), C2.6(Q8.D4), C4.17(C42.C2), C22.93(C8⋊C22), C22.4Q16.36C2, (C2×C42).312C22, (C22×C8).316C22, C22.81(C22⋊Q8), C22.140(C4⋊D4), (C22×C4).1398C23, C22.82(C8.C22), C2.15(M4(2)⋊C4), C2.12(C23.25D4), C2.16(C23.65C23), (C2×C4⋊C8).44C2, (C4×C4⋊C4).24C2, (C2×C4).55(C4⋊C4), (C2×C8).115(C2×C4), (C2×C4).210(C2×Q8), (C2×C4.Q8).21C2, (C2×C2.D8).10C2, (C2×C4).1015(C2×D4), (C2×C4⋊C4).80C22, C22.116(C2×C4⋊C4), (C2×C4).762(C4○D4), (C2×C4).554(C22×C4), SmallGroup(128,681)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.31Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=b-1c-1 >
Subgroups: 220 in 120 conjugacy classes, 64 normal (44 characteristic)
C1, C2, 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.D8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22.4Q16, C4×C4⋊C4, C42⋊8C4, C2×C4⋊C8, C2×C4.Q8, C2×C2.D8, C42.31Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C4○D8, C8⋊C22, C8.C22, C23.65C23, C23.25D4, M4(2)⋊C4, D4.2D4, Q8.D4, D4.Q8, Q8.Q8, C42.31Q8
►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 52 12 39)(2 49 9 40)(3 50 10 37)(4 51 11 38)(5 119 23 110)(6 120 24 111)(7 117 21 112)(8 118 22 109)(13 29 25 34)(14 30 26 35)(15 31 27 36)(16 32 28 33)(17 105 126 122)(18 106 127 123)(19 107 128 124)(20 108 125 121)(41 81 54 68)(42 82 55 65)(43 83 56 66)(44 84 53 67)(45 70 57 61)(46 71 58 62)(47 72 59 63)(48 69 60 64)(73 103 90 94)(74 104 91 95)(75 101 92 96)(76 102 89 93)(77 100 86 113)(78 97 87 114)(79 98 88 115)(80 99 85 116)
(1 70 34 53)(2 71 35 54)(3 72 36 55)(4 69 33 56)(5 95 106 87)(6 96 107 88)(7 93 108 85)(8 94 105 86)(9 62 30 41)(10 63 31 42)(11 64 32 43)(12 61 29 44)(13 84 52 45)(14 81 49 46)(15 82 50 47)(16 83 51 48)(17 113 109 73)(18 114 110 74)(19 115 111 75)(20 116 112 76)(21 102 121 80)(22 103 122 77)(23 104 123 78)(24 101 124 79)(25 67 39 57)(26 68 40 58)(27 65 37 59)(28 66 38 60)(89 125 99 117)(90 126 100 118)(91 127 97 119)(92 128 98 120)
(1 95 34 87)(2 103 35 77)(3 93 36 85)(4 101 33 79)(5 67 106 57)(6 83 107 48)(7 65 108 59)(8 81 105 46)(9 94 30 86)(10 102 31 80)(11 96 32 88)(12 104 29 78)(13 97 52 91)(14 113 49 73)(15 99 50 89)(16 115 51 75)(17 71 109 54)(18 61 110 44)(19 69 111 56)(20 63 112 42)(21 82 121 47)(22 68 122 58)(23 84 123 45)(24 66 124 60)(25 114 39 74)(26 100 40 90)(27 116 37 76)(28 98 38 92)(41 126 62 118)(43 128 64 120)(53 127 70 119)(55 125 72 117)
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), (1,52,12,39)(2,49,9,40)(3,50,10,37)(4,51,11,38)(5,119,23,110)(6,120,24,111)(7,117,21,112)(8,118,22,109)(13,29,25,34)(14,30,26,35)(15,31,27,36)(16,32,28,33)(17,105,126,122)(18,106,127,123)(19,107,128,124)(20,108,125,121)(41,81,54,68)(42,82,55,65)(43,83,56,66)(44,84,53,67)(45,70,57,61)(46,71,58,62)(47,72,59,63)(48,69,60,64)(73,103,90,94)(74,104,91,95)(75,101,92,96)(76,102,89,93)(77,100,86,113)(78,97,87,114)(79,98,88,115)(80,99,85,116), (1,70,34,53)(2,71,35,54)(3,72,36,55)(4,69,33,56)(5,95,106,87)(6,96,107,88)(7,93,108,85)(8,94,105,86)(9,62,30,41)(10,63,31,42)(11,64,32,43)(12,61,29,44)(13,84,52,45)(14,81,49,46)(15,82,50,47)(16,83,51,48)(17,113,109,73)(18,114,110,74)(19,115,111,75)(20,116,112,76)(21,102,121,80)(22,103,122,77)(23,104,123,78)(24,101,124,79)(25,67,39,57)(26,68,40,58)(27,65,37,59)(28,66,38,60)(89,125,99,117)(90,126,100,118)(91,127,97,119)(92,128,98,120), (1,95,34,87)(2,103,35,77)(3,93,36,85)(4,101,33,79)(5,67,106,57)(6,83,107,48)(7,65,108,59)(8,81,105,46)(9,94,30,86)(10,102,31,80)(11,96,32,88)(12,104,29,78)(13,97,52,91)(14,113,49,73)(15,99,50,89)(16,115,51,75)(17,71,109,54)(18,61,110,44)(19,69,111,56)(20,63,112,42)(21,82,121,47)(22,68,122,58)(23,84,123,45)(24,66,124,60)(25,114,39,74)(26,100,40,90)(27,116,37,76)(28,98,38,92)(41,126,62,118)(43,128,64,120)(53,127,70,119)(55,125,72,117)>;
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), (1,52,12,39)(2,49,9,40)(3,50,10,37)(4,51,11,38)(5,119,23,110)(6,120,24,111)(7,117,21,112)(8,118,22,109)(13,29,25,34)(14,30,26,35)(15,31,27,36)(16,32,28,33)(17,105,126,122)(18,106,127,123)(19,107,128,124)(20,108,125,121)(41,81,54,68)(42,82,55,65)(43,83,56,66)(44,84,53,67)(45,70,57,61)(46,71,58,62)(47,72,59,63)(48,69,60,64)(73,103,90,94)(74,104,91,95)(75,101,92,96)(76,102,89,93)(77,100,86,113)(78,97,87,114)(79,98,88,115)(80,99,85,116), (1,70,34,53)(2,71,35,54)(3,72,36,55)(4,69,33,56)(5,95,106,87)(6,96,107,88)(7,93,108,85)(8,94,105,86)(9,62,30,41)(10,63,31,42)(11,64,32,43)(12,61,29,44)(13,84,52,45)(14,81,49,46)(15,82,50,47)(16,83,51,48)(17,113,109,73)(18,114,110,74)(19,115,111,75)(20,116,112,76)(21,102,121,80)(22,103,122,77)(23,104,123,78)(24,101,124,79)(25,67,39,57)(26,68,40,58)(27,65,37,59)(28,66,38,60)(89,125,99,117)(90,126,100,118)(91,127,97,119)(92,128,98,120), (1,95,34,87)(2,103,35,77)(3,93,36,85)(4,101,33,79)(5,67,106,57)(6,83,107,48)(7,65,108,59)(8,81,105,46)(9,94,30,86)(10,102,31,80)(11,96,32,88)(12,104,29,78)(13,97,52,91)(14,113,49,73)(15,99,50,89)(16,115,51,75)(17,71,109,54)(18,61,110,44)(19,69,111,56)(20,63,112,42)(21,82,121,47)(22,68,122,58)(23,84,123,45)(24,66,124,60)(25,114,39,74)(26,100,40,90)(27,116,37,76)(28,98,38,92)(41,126,62,118)(43,128,64,120)(53,127,70,119)(55,125,72,117) );
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)], [(1,52,12,39),(2,49,9,40),(3,50,10,37),(4,51,11,38),(5,119,23,110),(6,120,24,111),(7,117,21,112),(8,118,22,109),(13,29,25,34),(14,30,26,35),(15,31,27,36),(16,32,28,33),(17,105,126,122),(18,106,127,123),(19,107,128,124),(20,108,125,121),(41,81,54,68),(42,82,55,65),(43,83,56,66),(44,84,53,67),(45,70,57,61),(46,71,58,62),(47,72,59,63),(48,69,60,64),(73,103,90,94),(74,104,91,95),(75,101,92,96),(76,102,89,93),(77,100,86,113),(78,97,87,114),(79,98,88,115),(80,99,85,116)], [(1,70,34,53),(2,71,35,54),(3,72,36,55),(4,69,33,56),(5,95,106,87),(6,96,107,88),(7,93,108,85),(8,94,105,86),(9,62,30,41),(10,63,31,42),(11,64,32,43),(12,61,29,44),(13,84,52,45),(14,81,49,46),(15,82,50,47),(16,83,51,48),(17,113,109,73),(18,114,110,74),(19,115,111,75),(20,116,112,76),(21,102,121,80),(22,103,122,77),(23,104,123,78),(24,101,124,79),(25,67,39,57),(26,68,40,58),(27,65,37,59),(28,66,38,60),(89,125,99,117),(90,126,100,118),(91,127,97,119),(92,128,98,120)], [(1,95,34,87),(2,103,35,77),(3,93,36,85),(4,101,33,79),(5,67,106,57),(6,83,107,48),(7,65,108,59),(8,81,105,46),(9,94,30,86),(10,102,31,80),(11,96,32,88),(12,104,29,78),(13,97,52,91),(14,113,49,73),(15,99,50,89),(16,115,51,75),(17,71,109,54),(18,61,110,44),(19,69,111,56),(20,63,112,42),(21,82,121,47),(22,68,122,58),(23,84,123,45),(24,66,124,60),(25,114,39,74),(26,100,40,90),(27,116,37,76),(28,98,38,92),(41,126,62,118),(43,128,64,120),(53,127,70,119),(55,125,72,117)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | D4 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C42.31Q8 | C22.4Q16 | C4×C4⋊C4 | C42⋊8C4 | C2×C4⋊C8 | C2×C4.Q8 | C2×C2.D8 | C4⋊C8 | C42 | C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 |
Matrix representation of C42.31Q8 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 13 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 4 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 12 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 5 |
| 0 | 0 | 0 | 0 | 14 | 13 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,5,0,0,0,0,13,11],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,4,0,0,0,0,4,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,0,0,10,7,0,0,0,0,0,0,4,14,0,0,0,0,5,13] >;
C42.31Q8 in GAP, Magma, Sage, TeX
C_4^2._{31}Q_8 % in TeX
G:=Group("C4^2.31Q8"); // GroupNames label
G:=SmallGroup(128,681);
// by ID
G=gap.SmallGroup(128,681);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,436,2019,1018,248]);
// 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^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^-1>;
// generators/relations