direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8×C42, C43.11C2, C23.154C24, C42○4(C4⋊C4), C4.9(C2×C42), C42.276(C2×C4), C2.7(C22×C42), C22.26(C23×C4), C22.16(C22×Q8), C42○3(C2.C42), (C2×C42).1084C22, (C22×C4).1645C23, (C22×Q8).505C22, C2.C42.566C22, C4○4(C4×C4⋊C4), C2.2(C2×C4×Q8), C42○3(C4×C4⋊C4), C42○3(C2×C4⋊C4), C2.3(C4×C4○D4), (C4×C4⋊C4).81C2, (C2×C4×Q8).58C2, C4⋊C4.236(C2×C4), (C2×C4).350(C2×Q8), (C2×Q8).216(C2×C4), C22.52(C2×C4○D4), (C2×C4).949(C4○D4), (C2×C4⋊C4).969C22, (C2×C4).448(C22×C4), SmallGroup(128,1004)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8×C42
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 396 in 342 conjugacy classes, 288 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, C43, C4×C4⋊C4, C2×C4×Q8, Q8×C42
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C42, C22×C4, C2×Q8, C4○D4, C24, C2×C42, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C22×C42, C2×C4×Q8, C4×C4○D4, Q8×C42
►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 31 15)(2 40 32 16)(3 37 29 13)(4 38 30 14)(5 20 121 106)(6 17 122 107)(7 18 123 108)(8 19 124 105)(9 49 33 28)(10 50 34 25)(11 51 35 26)(12 52 36 27)(21 127 111 117)(22 128 112 118)(23 125 109 119)(24 126 110 120)(41 81 65 60)(42 82 66 57)(43 83 67 58)(44 84 68 59)(45 55 69 61)(46 56 70 62)(47 53 71 63)(48 54 72 64)(73 113 97 92)(74 114 98 89)(75 115 99 90)(76 116 100 91)(77 87 101 93)(78 88 102 94)(79 85 103 95)(80 86 104 96)
(1 43 10 54)(2 44 11 55)(3 41 12 56)(4 42 9 53)(5 102 21 113)(6 103 22 114)(7 104 23 115)(8 101 24 116)(13 60 27 46)(14 57 28 47)(15 58 25 48)(16 59 26 45)(17 95 128 98)(18 96 125 99)(19 93 126 100)(20 94 127 97)(29 65 36 62)(30 66 33 63)(31 67 34 64)(32 68 35 61)(37 81 52 70)(38 82 49 71)(39 83 50 72)(40 84 51 69)(73 106 88 117)(74 107 85 118)(75 108 86 119)(76 105 87 120)(77 110 91 124)(78 111 92 121)(79 112 89 122)(80 109 90 123)
(1 114 10 103)(2 115 11 104)(3 116 12 101)(4 113 9 102)(5 42 21 53)(6 43 22 54)(7 44 23 55)(8 41 24 56)(13 76 27 87)(14 73 28 88)(15 74 25 85)(16 75 26 86)(17 83 128 72)(18 84 125 69)(19 81 126 70)(20 82 127 71)(29 91 36 77)(30 92 33 78)(31 89 34 79)(32 90 35 80)(37 100 52 93)(38 97 49 94)(39 98 50 95)(40 99 51 96)(45 108 59 119)(46 105 60 120)(47 106 57 117)(48 107 58 118)(61 123 68 109)(62 124 65 110)(63 121 66 111)(64 122 67 112)
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), (1,39,31,15)(2,40,32,16)(3,37,29,13)(4,38,30,14)(5,20,121,106)(6,17,122,107)(7,18,123,108)(8,19,124,105)(9,49,33,28)(10,50,34,25)(11,51,35,26)(12,52,36,27)(21,127,111,117)(22,128,112,118)(23,125,109,119)(24,126,110,120)(41,81,65,60)(42,82,66,57)(43,83,67,58)(44,84,68,59)(45,55,69,61)(46,56,70,62)(47,53,71,63)(48,54,72,64)(73,113,97,92)(74,114,98,89)(75,115,99,90)(76,116,100,91)(77,87,101,93)(78,88,102,94)(79,85,103,95)(80,86,104,96), (1,43,10,54)(2,44,11,55)(3,41,12,56)(4,42,9,53)(5,102,21,113)(6,103,22,114)(7,104,23,115)(8,101,24,116)(13,60,27,46)(14,57,28,47)(15,58,25,48)(16,59,26,45)(17,95,128,98)(18,96,125,99)(19,93,126,100)(20,94,127,97)(29,65,36,62)(30,66,33,63)(31,67,34,64)(32,68,35,61)(37,81,52,70)(38,82,49,71)(39,83,50,72)(40,84,51,69)(73,106,88,117)(74,107,85,118)(75,108,86,119)(76,105,87,120)(77,110,91,124)(78,111,92,121)(79,112,89,122)(80,109,90,123), (1,114,10,103)(2,115,11,104)(3,116,12,101)(4,113,9,102)(5,42,21,53)(6,43,22,54)(7,44,23,55)(8,41,24,56)(13,76,27,87)(14,73,28,88)(15,74,25,85)(16,75,26,86)(17,83,128,72)(18,84,125,69)(19,81,126,70)(20,82,127,71)(29,91,36,77)(30,92,33,78)(31,89,34,79)(32,90,35,80)(37,100,52,93)(38,97,49,94)(39,98,50,95)(40,99,51,96)(45,108,59,119)(46,105,60,120)(47,106,57,117)(48,107,58,118)(61,123,68,109)(62,124,65,110)(63,121,66,111)(64,122,67,112)>;
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), (1,39,31,15)(2,40,32,16)(3,37,29,13)(4,38,30,14)(5,20,121,106)(6,17,122,107)(7,18,123,108)(8,19,124,105)(9,49,33,28)(10,50,34,25)(11,51,35,26)(12,52,36,27)(21,127,111,117)(22,128,112,118)(23,125,109,119)(24,126,110,120)(41,81,65,60)(42,82,66,57)(43,83,67,58)(44,84,68,59)(45,55,69,61)(46,56,70,62)(47,53,71,63)(48,54,72,64)(73,113,97,92)(74,114,98,89)(75,115,99,90)(76,116,100,91)(77,87,101,93)(78,88,102,94)(79,85,103,95)(80,86,104,96), (1,43,10,54)(2,44,11,55)(3,41,12,56)(4,42,9,53)(5,102,21,113)(6,103,22,114)(7,104,23,115)(8,101,24,116)(13,60,27,46)(14,57,28,47)(15,58,25,48)(16,59,26,45)(17,95,128,98)(18,96,125,99)(19,93,126,100)(20,94,127,97)(29,65,36,62)(30,66,33,63)(31,67,34,64)(32,68,35,61)(37,81,52,70)(38,82,49,71)(39,83,50,72)(40,84,51,69)(73,106,88,117)(74,107,85,118)(75,108,86,119)(76,105,87,120)(77,110,91,124)(78,111,92,121)(79,112,89,122)(80,109,90,123), (1,114,10,103)(2,115,11,104)(3,116,12,101)(4,113,9,102)(5,42,21,53)(6,43,22,54)(7,44,23,55)(8,41,24,56)(13,76,27,87)(14,73,28,88)(15,74,25,85)(16,75,26,86)(17,83,128,72)(18,84,125,69)(19,81,126,70)(20,82,127,71)(29,91,36,77)(30,92,33,78)(31,89,34,79)(32,90,35,80)(37,100,52,93)(38,97,49,94)(39,98,50,95)(40,99,51,96)(45,108,59,119)(46,105,60,120)(47,106,57,117)(48,107,58,118)(61,123,68,109)(62,124,65,110)(63,121,66,111)(64,122,67,112) );
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)], [(1,39,31,15),(2,40,32,16),(3,37,29,13),(4,38,30,14),(5,20,121,106),(6,17,122,107),(7,18,123,108),(8,19,124,105),(9,49,33,28),(10,50,34,25),(11,51,35,26),(12,52,36,27),(21,127,111,117),(22,128,112,118),(23,125,109,119),(24,126,110,120),(41,81,65,60),(42,82,66,57),(43,83,67,58),(44,84,68,59),(45,55,69,61),(46,56,70,62),(47,53,71,63),(48,54,72,64),(73,113,97,92),(74,114,98,89),(75,115,99,90),(76,116,100,91),(77,87,101,93),(78,88,102,94),(79,85,103,95),(80,86,104,96)], [(1,43,10,54),(2,44,11,55),(3,41,12,56),(4,42,9,53),(5,102,21,113),(6,103,22,114),(7,104,23,115),(8,101,24,116),(13,60,27,46),(14,57,28,47),(15,58,25,48),(16,59,26,45),(17,95,128,98),(18,96,125,99),(19,93,126,100),(20,94,127,97),(29,65,36,62),(30,66,33,63),(31,67,34,64),(32,68,35,61),(37,81,52,70),(38,82,49,71),(39,83,50,72),(40,84,51,69),(73,106,88,117),(74,107,85,118),(75,108,86,119),(76,105,87,120),(77,110,91,124),(78,111,92,121),(79,112,89,122),(80,109,90,123)], [(1,114,10,103),(2,115,11,104),(3,116,12,101),(4,113,9,102),(5,42,21,53),(6,43,22,54),(7,44,23,55),(8,41,24,56),(13,76,27,87),(14,73,28,88),(15,74,25,85),(16,75,26,86),(17,83,128,72),(18,84,125,69),(19,81,126,70),(20,82,127,71),(29,91,36,77),(30,92,33,78),(31,89,34,79),(32,90,35,80),(37,100,52,93),(38,97,49,94),(39,98,50,95),(40,99,51,96),(45,108,59,119),(46,105,60,120),(47,106,57,117),(48,107,58,118),(61,123,68,109),(62,124,65,110),(63,121,66,111),(64,122,67,112)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 4Y | ··· | 4BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | C4○D4 |
| kernel | Q8×C42 | C43 | C4×C4⋊C4 | C2×C4×Q8 | C4×Q8 | C42 | C2×C4 |
| # reps | 1 | 3 | 9 | 3 | 48 | 4 | 12 |
Matrix representation of Q8×C42 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,3,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0],[4,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0] >;
Q8×C42 in GAP, Magma, Sage, TeX
Q_8\times C_4^2
% in TeX
G:=Group("Q8xC4^2"); // GroupNames label
G:=SmallGroup(128,1004);
// by ID
G=gap.SmallGroup(128,1004);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,436,192]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations