direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C4⋊Q8, C42⋊17Q8, C43.14C2, C42.344D4, C23.189C24, C4⋊1(C4×Q8), C4.36(C4×D4), C4○3(C42⋊9C4), C42.279(C2×C4), C42⋊9C4.43C2, C22.80(C23×C4), C22.83(C22×D4), C22.29(C22×Q8), (C22×C4).752C23, (C2×C42).404C22, (C22×Q8).394C22, C4○5(C23.67C23), C4○7(C23.65C23), C2.7(C22.26C24), C23.67C23.68C2, C2.C42.519C22, C23.65C23.95C2, C2.6(C23.37C23), C2.19(C2×C4×D4), C2.4(C2×C4⋊Q8), C2.10(C2×C4×Q8), (C4×C4⋊C4).33C2, (C2×C4×Q8).20C2, (C2×C4⋊Q8).58C2, C4⋊C4.155(C2×C4), (C2×C4).826(C2×D4), (C2×C4).227(C2×Q8), (C2×C4).23(C22×C4), (C2×Q8).147(C2×C4), C22.81(C2×C4○D4), (C2×C4)○3(C42⋊9C4), (C2×C4).642(C4○D4), (C2×C4⋊C4).803C22, (C2×C4)○3(C23.67C23), SmallGroup(128,1039)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C4⋊Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 412 in 294 conjugacy classes, 184 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C4⋊Q8, C22×Q8, C43, C4×C4⋊C4, C42⋊9C4, C23.65C23, C23.67C23, C2×C4×Q8, C2×C4⋊Q8, C4×C4⋊Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C4×D4, C4×Q8, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4×D4, C2×C4×Q8, C2×C4⋊Q8, C22.26C24, C23.37C23, C4×C4⋊Q8
►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 70 28 68)(2 71 25 65)(3 72 26 66)(4 69 27 67)(5 88 128 82)(6 85 125 83)(7 86 126 84)(8 87 127 81)(9 77 15 91)(10 78 16 92)(11 79 13 89)(12 80 14 90)(17 63 23 73)(18 64 24 74)(19 61 21 75)(20 62 22 76)(29 59 43 45)(30 60 44 46)(31 57 41 47)(32 58 42 48)(33 55 39 49)(34 56 40 50)(35 53 37 51)(36 54 38 52)(93 109 107 123)(94 110 108 124)(95 111 105 121)(96 112 106 122)(97 113 103 119)(98 114 104 120)(99 115 101 117)(100 116 102 118)
(1 58 22 56)(2 59 23 53)(3 60 24 54)(4 57 21 55)(5 108 14 102)(6 105 15 103)(7 106 16 104)(8 107 13 101)(9 97 125 95)(10 98 126 96)(11 99 127 93)(12 100 128 94)(17 51 25 45)(18 52 26 46)(19 49 27 47)(20 50 28 48)(29 63 35 65)(30 64 36 66)(31 61 33 67)(32 62 34 68)(37 71 43 73)(38 72 44 74)(39 69 41 75)(40 70 42 76)(77 113 83 111)(78 114 84 112)(79 115 81 109)(80 116 82 110)(85 121 91 119)(86 122 92 120)(87 123 89 117)(88 124 90 118)
(1 77 22 83)(2 78 23 84)(3 79 24 81)(4 80 21 82)(5 67 14 61)(6 68 15 62)(7 65 16 63)(8 66 13 64)(9 76 125 70)(10 73 126 71)(11 74 127 72)(12 75 128 69)(17 86 25 92)(18 87 26 89)(19 88 27 90)(20 85 28 91)(29 106 35 104)(30 107 36 101)(31 108 33 102)(32 105 34 103)(37 98 43 96)(38 99 44 93)(39 100 41 94)(40 97 42 95)(45 122 51 120)(46 123 52 117)(47 124 49 118)(48 121 50 119)(53 114 59 112)(54 115 60 109)(55 116 57 110)(56 113 58 111)
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), (1,70,28,68)(2,71,25,65)(3,72,26,66)(4,69,27,67)(5,88,128,82)(6,85,125,83)(7,86,126,84)(8,87,127,81)(9,77,15,91)(10,78,16,92)(11,79,13,89)(12,80,14,90)(17,63,23,73)(18,64,24,74)(19,61,21,75)(20,62,22,76)(29,59,43,45)(30,60,44,46)(31,57,41,47)(32,58,42,48)(33,55,39,49)(34,56,40,50)(35,53,37,51)(36,54,38,52)(93,109,107,123)(94,110,108,124)(95,111,105,121)(96,112,106,122)(97,113,103,119)(98,114,104,120)(99,115,101,117)(100,116,102,118), (1,58,22,56)(2,59,23,53)(3,60,24,54)(4,57,21,55)(5,108,14,102)(6,105,15,103)(7,106,16,104)(8,107,13,101)(9,97,125,95)(10,98,126,96)(11,99,127,93)(12,100,128,94)(17,51,25,45)(18,52,26,46)(19,49,27,47)(20,50,28,48)(29,63,35,65)(30,64,36,66)(31,61,33,67)(32,62,34,68)(37,71,43,73)(38,72,44,74)(39,69,41,75)(40,70,42,76)(77,113,83,111)(78,114,84,112)(79,115,81,109)(80,116,82,110)(85,121,91,119)(86,122,92,120)(87,123,89,117)(88,124,90,118), (1,77,22,83)(2,78,23,84)(3,79,24,81)(4,80,21,82)(5,67,14,61)(6,68,15,62)(7,65,16,63)(8,66,13,64)(9,76,125,70)(10,73,126,71)(11,74,127,72)(12,75,128,69)(17,86,25,92)(18,87,26,89)(19,88,27,90)(20,85,28,91)(29,106,35,104)(30,107,36,101)(31,108,33,102)(32,105,34,103)(37,98,43,96)(38,99,44,93)(39,100,41,94)(40,97,42,95)(45,122,51,120)(46,123,52,117)(47,124,49,118)(48,121,50,119)(53,114,59,112)(54,115,60,109)(55,116,57,110)(56,113,58,111)>;
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), (1,70,28,68)(2,71,25,65)(3,72,26,66)(4,69,27,67)(5,88,128,82)(6,85,125,83)(7,86,126,84)(8,87,127,81)(9,77,15,91)(10,78,16,92)(11,79,13,89)(12,80,14,90)(17,63,23,73)(18,64,24,74)(19,61,21,75)(20,62,22,76)(29,59,43,45)(30,60,44,46)(31,57,41,47)(32,58,42,48)(33,55,39,49)(34,56,40,50)(35,53,37,51)(36,54,38,52)(93,109,107,123)(94,110,108,124)(95,111,105,121)(96,112,106,122)(97,113,103,119)(98,114,104,120)(99,115,101,117)(100,116,102,118), (1,58,22,56)(2,59,23,53)(3,60,24,54)(4,57,21,55)(5,108,14,102)(6,105,15,103)(7,106,16,104)(8,107,13,101)(9,97,125,95)(10,98,126,96)(11,99,127,93)(12,100,128,94)(17,51,25,45)(18,52,26,46)(19,49,27,47)(20,50,28,48)(29,63,35,65)(30,64,36,66)(31,61,33,67)(32,62,34,68)(37,71,43,73)(38,72,44,74)(39,69,41,75)(40,70,42,76)(77,113,83,111)(78,114,84,112)(79,115,81,109)(80,116,82,110)(85,121,91,119)(86,122,92,120)(87,123,89,117)(88,124,90,118), (1,77,22,83)(2,78,23,84)(3,79,24,81)(4,80,21,82)(5,67,14,61)(6,68,15,62)(7,65,16,63)(8,66,13,64)(9,76,125,70)(10,73,126,71)(11,74,127,72)(12,75,128,69)(17,86,25,92)(18,87,26,89)(19,88,27,90)(20,85,28,91)(29,106,35,104)(30,107,36,101)(31,108,33,102)(32,105,34,103)(37,98,43,96)(38,99,44,93)(39,100,41,94)(40,97,42,95)(45,122,51,120)(46,123,52,117)(47,124,49,118)(48,121,50,119)(53,114,59,112)(54,115,60,109)(55,116,57,110)(56,113,58,111) );
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)], [(1,70,28,68),(2,71,25,65),(3,72,26,66),(4,69,27,67),(5,88,128,82),(6,85,125,83),(7,86,126,84),(8,87,127,81),(9,77,15,91),(10,78,16,92),(11,79,13,89),(12,80,14,90),(17,63,23,73),(18,64,24,74),(19,61,21,75),(20,62,22,76),(29,59,43,45),(30,60,44,46),(31,57,41,47),(32,58,42,48),(33,55,39,49),(34,56,40,50),(35,53,37,51),(36,54,38,52),(93,109,107,123),(94,110,108,124),(95,111,105,121),(96,112,106,122),(97,113,103,119),(98,114,104,120),(99,115,101,117),(100,116,102,118)], [(1,58,22,56),(2,59,23,53),(3,60,24,54),(4,57,21,55),(5,108,14,102),(6,105,15,103),(7,106,16,104),(8,107,13,101),(9,97,125,95),(10,98,126,96),(11,99,127,93),(12,100,128,94),(17,51,25,45),(18,52,26,46),(19,49,27,47),(20,50,28,48),(29,63,35,65),(30,64,36,66),(31,61,33,67),(32,62,34,68),(37,71,43,73),(38,72,44,74),(39,69,41,75),(40,70,42,76),(77,113,83,111),(78,114,84,112),(79,115,81,109),(80,116,82,110),(85,121,91,119),(86,122,92,120),(87,123,89,117),(88,124,90,118)], [(1,77,22,83),(2,78,23,84),(3,79,24,81),(4,80,21,82),(5,67,14,61),(6,68,15,62),(7,65,16,63),(8,66,13,64),(9,76,125,70),(10,73,126,71),(11,74,127,72),(12,75,128,69),(17,86,25,92),(18,87,26,89),(19,88,27,90),(20,85,28,91),(29,106,35,104),(30,107,36,101),(31,108,33,102),(32,105,34,103),(37,98,43,96),(38,99,44,93),(39,100,41,94),(40,97,42,95),(45,122,51,120),(46,123,52,117),(47,124,49,118),(48,121,50,119),(53,114,59,112),(54,115,60,109),(55,116,57,110),(56,113,58,111)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 4AG | ··· | 4AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | C4○D4 |
| kernel | C4×C4⋊Q8 | C43 | C4×C4⋊C4 | C42⋊9C4 | C23.65C23 | C23.67C23 | C2×C4×Q8 | C2×C4⋊Q8 | C4⋊Q8 | C42 | C42 | C2×C4 |
| # reps | 1 | 1 | 4 | 1 | 4 | 2 | 2 | 1 | 16 | 4 | 8 | 12 |
Matrix representation of C4×C4⋊Q8 ►in GL5(𝔽5)
| 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,2,1,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,3,4,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3],[1,0,0,0,0,0,2,0,0,0,0,2,3,0,0,0,0,0,0,4,0,0,0,1,0] >;
C4×C4⋊Q8 in GAP, Magma, Sage, TeX
C_4\times C_4\rtimes Q_8
% in TeX
G:=Group("C4xC4:Q8"); // GroupNames label
G:=SmallGroup(128,1039);
// by ID
G=gap.SmallGroup(128,1039);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,184,304]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations