direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8×C4⋊C4, C23.232C24, C22.492- 1+4, C22.682+ 1+4, C2.1Q82, C4⋊3(C4×Q8), C2.3(D4×Q8), (C4×Q8)⋊19C4, (C2×Q8).26Q8, (C2×Q8).260D4, C42.186(C2×C4), C2.1(Q8⋊3Q8), C2.2(Q8⋊6D4), C42⋊9C4.23C2, C22.40(C22×Q8), C22.123(C23×C4), (C2×C42).431C22, C22.107(C22×D4), (C22×C4).1247C23, (C22×Q8).509C22, C2.C42.521C22, C23.65C23.33C2, C2.28(C23.33C23), C4.18(C2×C4⋊C4), C2.16(C2×C4×Q8), (C4×C4⋊C4).38C2, (C2×C4×Q8).23C2, C4⋊C4.208(C2×C4), C2.16(C22×C4⋊C4), (C2×C4).301(C2×Q8), (C2×C4).1071(C2×D4), (C2×Q8).219(C2×C4), (C2×C4).889(C4○D4), (C2×C4⋊C4).821C22, (C2×C4).567(C22×C4), C22.117(C2×C4○D4), SmallGroup(128,1082)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8×C4⋊C4
G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 396 in 284 conjugacy classes, 196 normal (21 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, C4×C4⋊C4, C42⋊9C4, C23.65C23, C2×C4×Q8, C2×C4×Q8, Q8×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C4×Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C4⋊C4, C2×C4×Q8, C23.33C23, D4×Q8, Q8⋊6D4, Q8⋊3Q8, Q82, Q8×C4⋊C4
►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 54 3 56)(2 53 4 55)(5 106 7 108)(6 105 8 107)(9 100 11 98)(10 99 12 97)(13 104 15 102)(14 103 16 101)(17 47 19 45)(18 46 20 48)(21 59 23 57)(22 58 24 60)(25 49 27 51)(26 52 28 50)(29 61 31 63)(30 64 32 62)(33 65 35 67)(34 68 36 66)(37 69 39 71)(38 72 40 70)(41 73 43 75)(42 76 44 74)(77 111 79 109)(78 110 80 112)(81 115 83 113)(82 114 84 116)(85 119 87 117)(86 118 88 120)(89 123 91 121)(90 122 92 124)(93 127 95 125)(94 126 96 128)
(1 40 20 44)(2 37 17 41)(3 38 18 42)(4 39 19 43)(5 110 15 120)(6 111 16 117)(7 112 13 118)(8 109 14 119)(9 114 128 122)(10 115 125 123)(11 116 126 124)(12 113 127 121)(21 29 27 35)(22 30 28 36)(23 31 25 33)(24 32 26 34)(45 75 55 71)(46 76 56 72)(47 73 53 69)(48 74 54 70)(49 65 57 63)(50 66 58 64)(51 67 59 61)(52 68 60 62)(77 103 87 107)(78 104 88 108)(79 101 85 105)(80 102 86 106)(81 95 89 97)(82 96 90 98)(83 93 91 99)(84 94 92 100)
(1 79 22 81)(2 80 23 82)(3 77 24 83)(4 78 21 84)(5 61 11 71)(6 62 12 72)(7 63 9 69)(8 64 10 70)(13 65 128 73)(14 66 125 74)(15 67 126 75)(16 68 127 76)(17 86 25 90)(18 87 26 91)(19 88 27 92)(20 85 28 89)(29 100 39 108)(30 97 40 105)(31 98 37 106)(32 99 38 107)(33 96 41 102)(34 93 42 103)(35 94 43 104)(36 95 44 101)(45 120 51 124)(46 117 52 121)(47 118 49 122)(48 119 50 123)(53 112 57 114)(54 109 58 115)(55 110 59 116)(56 111 60 113)
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), (1,54,3,56)(2,53,4,55)(5,106,7,108)(6,105,8,107)(9,100,11,98)(10,99,12,97)(13,104,15,102)(14,103,16,101)(17,47,19,45)(18,46,20,48)(21,59,23,57)(22,58,24,60)(25,49,27,51)(26,52,28,50)(29,61,31,63)(30,64,32,62)(33,65,35,67)(34,68,36,66)(37,69,39,71)(38,72,40,70)(41,73,43,75)(42,76,44,74)(77,111,79,109)(78,110,80,112)(81,115,83,113)(82,114,84,116)(85,119,87,117)(86,118,88,120)(89,123,91,121)(90,122,92,124)(93,127,95,125)(94,126,96,128), (1,40,20,44)(2,37,17,41)(3,38,18,42)(4,39,19,43)(5,110,15,120)(6,111,16,117)(7,112,13,118)(8,109,14,119)(9,114,128,122)(10,115,125,123)(11,116,126,124)(12,113,127,121)(21,29,27,35)(22,30,28,36)(23,31,25,33)(24,32,26,34)(45,75,55,71)(46,76,56,72)(47,73,53,69)(48,74,54,70)(49,65,57,63)(50,66,58,64)(51,67,59,61)(52,68,60,62)(77,103,87,107)(78,104,88,108)(79,101,85,105)(80,102,86,106)(81,95,89,97)(82,96,90,98)(83,93,91,99)(84,94,92,100), (1,79,22,81)(2,80,23,82)(3,77,24,83)(4,78,21,84)(5,61,11,71)(6,62,12,72)(7,63,9,69)(8,64,10,70)(13,65,128,73)(14,66,125,74)(15,67,126,75)(16,68,127,76)(17,86,25,90)(18,87,26,91)(19,88,27,92)(20,85,28,89)(29,100,39,108)(30,97,40,105)(31,98,37,106)(32,99,38,107)(33,96,41,102)(34,93,42,103)(35,94,43,104)(36,95,44,101)(45,120,51,124)(46,117,52,121)(47,118,49,122)(48,119,50,123)(53,112,57,114)(54,109,58,115)(55,110,59,116)(56,111,60,113)>;
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), (1,54,3,56)(2,53,4,55)(5,106,7,108)(6,105,8,107)(9,100,11,98)(10,99,12,97)(13,104,15,102)(14,103,16,101)(17,47,19,45)(18,46,20,48)(21,59,23,57)(22,58,24,60)(25,49,27,51)(26,52,28,50)(29,61,31,63)(30,64,32,62)(33,65,35,67)(34,68,36,66)(37,69,39,71)(38,72,40,70)(41,73,43,75)(42,76,44,74)(77,111,79,109)(78,110,80,112)(81,115,83,113)(82,114,84,116)(85,119,87,117)(86,118,88,120)(89,123,91,121)(90,122,92,124)(93,127,95,125)(94,126,96,128), (1,40,20,44)(2,37,17,41)(3,38,18,42)(4,39,19,43)(5,110,15,120)(6,111,16,117)(7,112,13,118)(8,109,14,119)(9,114,128,122)(10,115,125,123)(11,116,126,124)(12,113,127,121)(21,29,27,35)(22,30,28,36)(23,31,25,33)(24,32,26,34)(45,75,55,71)(46,76,56,72)(47,73,53,69)(48,74,54,70)(49,65,57,63)(50,66,58,64)(51,67,59,61)(52,68,60,62)(77,103,87,107)(78,104,88,108)(79,101,85,105)(80,102,86,106)(81,95,89,97)(82,96,90,98)(83,93,91,99)(84,94,92,100), (1,79,22,81)(2,80,23,82)(3,77,24,83)(4,78,21,84)(5,61,11,71)(6,62,12,72)(7,63,9,69)(8,64,10,70)(13,65,128,73)(14,66,125,74)(15,67,126,75)(16,68,127,76)(17,86,25,90)(18,87,26,91)(19,88,27,92)(20,85,28,89)(29,100,39,108)(30,97,40,105)(31,98,37,106)(32,99,38,107)(33,96,41,102)(34,93,42,103)(35,94,43,104)(36,95,44,101)(45,120,51,124)(46,117,52,121)(47,118,49,122)(48,119,50,123)(53,112,57,114)(54,109,58,115)(55,110,59,116)(56,111,60,113) );
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)], [(1,54,3,56),(2,53,4,55),(5,106,7,108),(6,105,8,107),(9,100,11,98),(10,99,12,97),(13,104,15,102),(14,103,16,101),(17,47,19,45),(18,46,20,48),(21,59,23,57),(22,58,24,60),(25,49,27,51),(26,52,28,50),(29,61,31,63),(30,64,32,62),(33,65,35,67),(34,68,36,66),(37,69,39,71),(38,72,40,70),(41,73,43,75),(42,76,44,74),(77,111,79,109),(78,110,80,112),(81,115,83,113),(82,114,84,116),(85,119,87,117),(86,118,88,120),(89,123,91,121),(90,122,92,124),(93,127,95,125),(94,126,96,128)], [(1,40,20,44),(2,37,17,41),(3,38,18,42),(4,39,19,43),(5,110,15,120),(6,111,16,117),(7,112,13,118),(8,109,14,119),(9,114,128,122),(10,115,125,123),(11,116,126,124),(12,113,127,121),(21,29,27,35),(22,30,28,36),(23,31,25,33),(24,32,26,34),(45,75,55,71),(46,76,56,72),(47,73,53,69),(48,74,54,70),(49,65,57,63),(50,66,58,64),(51,67,59,61),(52,68,60,62),(77,103,87,107),(78,104,88,108),(79,101,85,105),(80,102,86,106),(81,95,89,97),(82,96,90,98),(83,93,91,99),(84,94,92,100)], [(1,79,22,81),(2,80,23,82),(3,77,24,83),(4,78,21,84),(5,61,11,71),(6,62,12,72),(7,63,9,69),(8,64,10,70),(13,65,128,73),(14,66,125,74),(15,67,126,75),(16,68,127,76),(17,86,25,90),(18,87,26,91),(19,88,27,92),(20,85,28,89),(29,100,39,108),(30,97,40,105),(31,98,37,106),(32,99,38,107),(33,96,41,102),(34,93,42,103),(35,94,43,104),(36,95,44,101),(45,120,51,124),(46,117,52,121),(47,118,49,122),(48,119,50,123),(53,112,57,114),(54,109,58,115),(55,110,59,116),(56,111,60,113)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 4Y | ··· | 4AP |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | Q8×C4⋊C4 | C4×C4⋊C4 | C42⋊9C4 | C23.65C23 | C2×C4×Q8 | C4×Q8 | C4⋊C4 | C2×Q8 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 3 | 3 | 6 | 3 | 16 | 4 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of Q8×C4⋊C4 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,3,0,0,0,0,3],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,4,0,0,0,2,4],[1,0,0,0,0,0,2,1,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,3,0,0,0,0,3,2,0,0,0,0,0,1,0,0,0,0,0,1] >;
Q8×C4⋊C4 in GAP, Magma, Sage, TeX
Q_8\times C_4\rtimes C_4
% in TeX
G:=Group("Q8xC4:C4"); // GroupNames label
G:=SmallGroup(128,1082);
// by ID
G=gap.SmallGroup(128,1082);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,184,346,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=d^4=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations