p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8⋊4C42, C23.158C24, C22.172- 1+4, C22.292+ 1+4, (C4×Q8)⋊16C4, C4.11(C2×C42), C42.174(C2×C4), Q8○(C2.C42), C42⋊4C4.10C2, C2.11(C22×C42), C22.30(C23×C4), (C2×C42).401C22, (C22×C4).1234C23, (C22×Q8).506C22, C2.C42.569C22, C2.3(C23.33C23), C2.2(C23.32C23), (C4×C4⋊C4).30C2, (C2×C4×Q8).18C2, C4⋊C4.237(C2×C4), (C2×Q8).217(C2×C4), (C2×C4⋊C4).972C22, C4⋊C4○2(C2.C42), (C2×C4).482(C22×C4), SmallGroup(128,1008)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊4C42
G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, bd=db, cd=dc >
Subgroups: 380 in 314 conjugacy classes, 260 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C42⋊4C4, C4×C4⋊C4, C2×C4×Q8, Q8⋊4C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, 2+ 1+4, 2- 1+4, C22×C42, C23.32C23, C23.33C23, Q8⋊4C42
►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)
(1 101 3 103)(2 104 4 102)(5 45 7 47)(6 48 8 46)(9 55 11 53)(10 54 12 56)(13 49 15 51)(14 52 16 50)(17 96 19 94)(18 95 20 93)(21 108 23 106)(22 107 24 105)(25 98 27 100)(26 97 28 99)(29 81 31 83)(30 84 32 82)(33 89 35 91)(34 92 36 90)(37 79 39 77)(38 78 40 80)(41 87 43 85)(42 86 44 88)(57 128 59 126)(58 127 60 125)(61 115 63 113)(62 114 64 116)(65 109 67 111)(66 112 68 110)(69 123 71 121)(70 122 72 124)(73 117 75 119)(74 120 76 118)
(1 45 22 55)(2 46 23 56)(3 47 24 53)(4 48 21 54)(5 107 9 101)(6 108 10 102)(7 105 11 103)(8 106 12 104)(13 97 126 93)(14 98 127 94)(15 99 128 95)(16 100 125 96)(17 50 27 58)(18 51 28 59)(19 52 25 60)(20 49 26 57)(29 66 39 64)(30 67 40 61)(31 68 37 62)(32 65 38 63)(33 76 41 72)(34 73 42 69)(35 74 43 70)(36 75 44 71)(77 114 81 110)(78 115 82 111)(79 116 83 112)(80 113 84 109)(85 124 91 118)(86 121 92 119)(87 122 89 120)(88 123 90 117)
(1 87 19 80)(2 86 20 79)(3 85 17 78)(4 88 18 77)(5 70 14 63)(6 69 15 62)(7 72 16 61)(8 71 13 64)(9 74 127 65)(10 73 128 68)(11 76 125 67)(12 75 126 66)(21 90 28 81)(22 89 25 84)(23 92 26 83)(24 91 27 82)(29 106 36 97)(30 105 33 100)(31 108 34 99)(32 107 35 98)(37 102 42 95)(38 101 43 94)(39 104 44 93)(40 103 41 96)(45 122 52 113)(46 121 49 116)(47 124 50 115)(48 123 51 114)(53 118 58 111)(54 117 59 110)(55 120 60 109)(56 119 57 112)
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), (1,101,3,103)(2,104,4,102)(5,45,7,47)(6,48,8,46)(9,55,11,53)(10,54,12,56)(13,49,15,51)(14,52,16,50)(17,96,19,94)(18,95,20,93)(21,108,23,106)(22,107,24,105)(25,98,27,100)(26,97,28,99)(29,81,31,83)(30,84,32,82)(33,89,35,91)(34,92,36,90)(37,79,39,77)(38,78,40,80)(41,87,43,85)(42,86,44,88)(57,128,59,126)(58,127,60,125)(61,115,63,113)(62,114,64,116)(65,109,67,111)(66,112,68,110)(69,123,71,121)(70,122,72,124)(73,117,75,119)(74,120,76,118), (1,45,22,55)(2,46,23,56)(3,47,24,53)(4,48,21,54)(5,107,9,101)(6,108,10,102)(7,105,11,103)(8,106,12,104)(13,97,126,93)(14,98,127,94)(15,99,128,95)(16,100,125,96)(17,50,27,58)(18,51,28,59)(19,52,25,60)(20,49,26,57)(29,66,39,64)(30,67,40,61)(31,68,37,62)(32,65,38,63)(33,76,41,72)(34,73,42,69)(35,74,43,70)(36,75,44,71)(77,114,81,110)(78,115,82,111)(79,116,83,112)(80,113,84,109)(85,124,91,118)(86,121,92,119)(87,122,89,120)(88,123,90,117), (1,87,19,80)(2,86,20,79)(3,85,17,78)(4,88,18,77)(5,70,14,63)(6,69,15,62)(7,72,16,61)(8,71,13,64)(9,74,127,65)(10,73,128,68)(11,76,125,67)(12,75,126,66)(21,90,28,81)(22,89,25,84)(23,92,26,83)(24,91,27,82)(29,106,36,97)(30,105,33,100)(31,108,34,99)(32,107,35,98)(37,102,42,95)(38,101,43,94)(39,104,44,93)(40,103,41,96)(45,122,52,113)(46,121,49,116)(47,124,50,115)(48,123,51,114)(53,118,58,111)(54,117,59,110)(55,120,60,109)(56,119,57,112)>;
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), (1,101,3,103)(2,104,4,102)(5,45,7,47)(6,48,8,46)(9,55,11,53)(10,54,12,56)(13,49,15,51)(14,52,16,50)(17,96,19,94)(18,95,20,93)(21,108,23,106)(22,107,24,105)(25,98,27,100)(26,97,28,99)(29,81,31,83)(30,84,32,82)(33,89,35,91)(34,92,36,90)(37,79,39,77)(38,78,40,80)(41,87,43,85)(42,86,44,88)(57,128,59,126)(58,127,60,125)(61,115,63,113)(62,114,64,116)(65,109,67,111)(66,112,68,110)(69,123,71,121)(70,122,72,124)(73,117,75,119)(74,120,76,118), (1,45,22,55)(2,46,23,56)(3,47,24,53)(4,48,21,54)(5,107,9,101)(6,108,10,102)(7,105,11,103)(8,106,12,104)(13,97,126,93)(14,98,127,94)(15,99,128,95)(16,100,125,96)(17,50,27,58)(18,51,28,59)(19,52,25,60)(20,49,26,57)(29,66,39,64)(30,67,40,61)(31,68,37,62)(32,65,38,63)(33,76,41,72)(34,73,42,69)(35,74,43,70)(36,75,44,71)(77,114,81,110)(78,115,82,111)(79,116,83,112)(80,113,84,109)(85,124,91,118)(86,121,92,119)(87,122,89,120)(88,123,90,117), (1,87,19,80)(2,86,20,79)(3,85,17,78)(4,88,18,77)(5,70,14,63)(6,69,15,62)(7,72,16,61)(8,71,13,64)(9,74,127,65)(10,73,128,68)(11,76,125,67)(12,75,126,66)(21,90,28,81)(22,89,25,84)(23,92,26,83)(24,91,27,82)(29,106,36,97)(30,105,33,100)(31,108,34,99)(32,107,35,98)(37,102,42,95)(38,101,43,94)(39,104,44,93)(40,103,41,96)(45,122,52,113)(46,121,49,116)(47,124,50,115)(48,123,51,114)(53,118,58,111)(54,117,59,110)(55,120,60,109)(56,119,57,112) );
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)], [(1,101,3,103),(2,104,4,102),(5,45,7,47),(6,48,8,46),(9,55,11,53),(10,54,12,56),(13,49,15,51),(14,52,16,50),(17,96,19,94),(18,95,20,93),(21,108,23,106),(22,107,24,105),(25,98,27,100),(26,97,28,99),(29,81,31,83),(30,84,32,82),(33,89,35,91),(34,92,36,90),(37,79,39,77),(38,78,40,80),(41,87,43,85),(42,86,44,88),(57,128,59,126),(58,127,60,125),(61,115,63,113),(62,114,64,116),(65,109,67,111),(66,112,68,110),(69,123,71,121),(70,122,72,124),(73,117,75,119),(74,120,76,118)], [(1,45,22,55),(2,46,23,56),(3,47,24,53),(4,48,21,54),(5,107,9,101),(6,108,10,102),(7,105,11,103),(8,106,12,104),(13,97,126,93),(14,98,127,94),(15,99,128,95),(16,100,125,96),(17,50,27,58),(18,51,28,59),(19,52,25,60),(20,49,26,57),(29,66,39,64),(30,67,40,61),(31,68,37,62),(32,65,38,63),(33,76,41,72),(34,73,42,69),(35,74,43,70),(36,75,44,71),(77,114,81,110),(78,115,82,111),(79,116,83,112),(80,113,84,109),(85,124,91,118),(86,121,92,119),(87,122,89,120),(88,123,90,117)], [(1,87,19,80),(2,86,20,79),(3,85,17,78),(4,88,18,77),(5,70,14,63),(6,69,15,62),(7,72,16,61),(8,71,13,64),(9,74,127,65),(10,73,128,68),(11,76,125,67),(12,75,126,66),(21,90,28,81),(22,89,25,84),(23,92,26,83),(24,91,27,82),(29,106,36,97),(30,105,33,100),(31,108,34,99),(32,107,35,98),(37,102,42,95),(38,101,43,94),(39,104,44,93),(40,103,41,96),(45,122,52,113),(46,121,49,116),(47,124,50,115),(48,123,51,114),(53,118,58,111),(54,117,59,110),(55,120,60,109),(56,119,57,112)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4BH |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | 2+ 1+4 | 2- 1+4 |
| kernel | Q8⋊4C42 | C42⋊4C4 | C4×C4⋊C4 | C2×C4×Q8 | C4×Q8 | C22 | C22 |
| # reps | 1 | 3 | 9 | 3 | 48 | 1 | 3 |
Matrix representation of Q8⋊4C42 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 4 | 3 | 2 | 0 |
| 0 | 0 | 3 | 2 | 4 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 4 | 1 | 2 |
| 0 | 0 | 1 | 2 | 0 | 2 |
| 0 | 0 | 2 | 3 | 3 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 2 | 4 |
| 0 | 0 | 1 | 4 | 1 | 1 |
| 0 | 0 | 4 | 2 | 0 | 4 |
| 0 | 0 | 4 | 0 | 4 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 4 | 2 | 1 | 2 |
| 0 | 0 | 0 | 2 | 0 | 0 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,4,3,0,0,3,4,3,2,0,0,0,0,2,4,0,0,0,0,0,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,1,2,0,0,1,4,2,3,0,0,1,1,0,3,0,0,1,2,2,0],[3,0,0,0,0,0,0,4,0,0,0,0,0,0,3,1,4,4,0,0,3,4,2,0,0,0,2,1,0,4,0,0,4,1,4,3],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,4,0,0,0,1,0,2,2,0,0,2,0,1,0,0,0,1,2,2,0] >;
Q8⋊4C42 in GAP, Magma, Sage, TeX
Q_8\rtimes_4C_4^2
% in TeX
G:=Group("Q8:4C4^2"); // GroupNames label
G:=SmallGroup(128,1008);
// by ID
G=gap.SmallGroup(128,1008);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,219,268,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=d^4=1,b^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,c*d=d*c>;
// generators/relations