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○7(C23.65C23), C4○5(C23.67C23), C2.7(C22.26C24), C23.65C23.95C2, C23.67C23.68C2, C2.C42.519C22, 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
Subgroups: 412 in 294 conjugacy classes, 184 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×16], C4 [×18], C22 [×3], C22 [×4], C2×C4 [×38], C2×C4 [×26], Q8 [×16], C23, C42 [×16], C42 [×14], C4⋊C4 [×16], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×8], C2×Q8 [×8], C2.C42 [×8], C2×C42 [×3], C2×C42 [×8], C2×C4⋊C4 [×14], C4×Q8 [×8], C4⋊Q8 [×8], C22×Q8 [×2], C43, C4×C4⋊C4 [×4], C42⋊9C4, C23.65C23 [×4], C23.67C23 [×2], C2×C4×Q8 [×2], C2×C4⋊Q8, C4×C4⋊Q8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×8], C23 [×15], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×6], C24, C4×D4 [×4], C4×Q8 [×8], C4⋊Q8 [×4], C23×C4, C22×D4, C22×Q8 [×2], C2×C4○D4 [×3], C2×C4×D4, C2×C4×Q8 [×2], C2×C4⋊Q8, C22.26C24, C23.37C23 [×2], C4×C4⋊Q8
Generators and relations
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 >
(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 19 73)(2 71 20 74)(3 72 17 75)(4 69 18 76)(5 79 13 84)(6 80 14 81)(7 77 15 82)(8 78 16 83)(9 86 126 89)(10 87 127 90)(11 88 128 91)(12 85 125 92)(21 66 28 61)(22 67 25 62)(23 68 26 63)(24 65 27 64)(29 50 34 45)(30 51 35 46)(31 52 36 47)(32 49 33 48)(37 60 44 53)(38 57 41 54)(39 58 42 55)(40 59 43 56)(93 109 98 114)(94 110 99 115)(95 111 100 116)(96 112 97 113)(101 117 108 124)(102 118 105 121)(103 119 106 122)(104 120 107 123)
(1 49 22 56)(2 50 23 53)(3 51 24 54)(4 52 21 55)(5 108 127 93)(6 105 128 94)(7 106 125 95)(8 107 126 96)(9 97 16 104)(10 98 13 101)(11 99 14 102)(12 100 15 103)(17 46 27 57)(18 47 28 58)(19 48 25 59)(20 45 26 60)(29 63 44 74)(30 64 41 75)(31 61 42 76)(32 62 43 73)(33 67 40 70)(34 68 37 71)(35 65 38 72)(36 66 39 69)(77 122 92 111)(78 123 89 112)(79 124 90 109)(80 121 91 110)(81 118 88 115)(82 119 85 116)(83 120 86 113)(84 117 87 114)
(1 77 22 92)(2 78 23 89)(3 79 24 90)(4 80 21 91)(5 65 127 72)(6 66 128 69)(7 67 125 70)(8 68 126 71)(9 74 16 63)(10 75 13 64)(11 76 14 61)(12 73 15 62)(17 84 27 87)(18 81 28 88)(19 82 25 85)(20 83 26 86)(29 97 44 104)(30 98 41 101)(31 99 42 102)(32 100 43 103)(33 95 40 106)(34 96 37 107)(35 93 38 108)(36 94 39 105)(45 113 60 120)(46 114 57 117)(47 115 58 118)(48 116 59 119)(49 111 56 122)(50 112 53 123)(51 109 54 124)(52 110 55 121)
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), (1,70,19,73)(2,71,20,74)(3,72,17,75)(4,69,18,76)(5,79,13,84)(6,80,14,81)(7,77,15,82)(8,78,16,83)(9,86,126,89)(10,87,127,90)(11,88,128,91)(12,85,125,92)(21,66,28,61)(22,67,25,62)(23,68,26,63)(24,65,27,64)(29,50,34,45)(30,51,35,46)(31,52,36,47)(32,49,33,48)(37,60,44,53)(38,57,41,54)(39,58,42,55)(40,59,43,56)(93,109,98,114)(94,110,99,115)(95,111,100,116)(96,112,97,113)(101,117,108,124)(102,118,105,121)(103,119,106,122)(104,120,107,123), (1,49,22,56)(2,50,23,53)(3,51,24,54)(4,52,21,55)(5,108,127,93)(6,105,128,94)(7,106,125,95)(8,107,126,96)(9,97,16,104)(10,98,13,101)(11,99,14,102)(12,100,15,103)(17,46,27,57)(18,47,28,58)(19,48,25,59)(20,45,26,60)(29,63,44,74)(30,64,41,75)(31,61,42,76)(32,62,43,73)(33,67,40,70)(34,68,37,71)(35,65,38,72)(36,66,39,69)(77,122,92,111)(78,123,89,112)(79,124,90,109)(80,121,91,110)(81,118,88,115)(82,119,85,116)(83,120,86,113)(84,117,87,114), (1,77,22,92)(2,78,23,89)(3,79,24,90)(4,80,21,91)(5,65,127,72)(6,66,128,69)(7,67,125,70)(8,68,126,71)(9,74,16,63)(10,75,13,64)(11,76,14,61)(12,73,15,62)(17,84,27,87)(18,81,28,88)(19,82,25,85)(20,83,26,86)(29,97,44,104)(30,98,41,101)(31,99,42,102)(32,100,43,103)(33,95,40,106)(34,96,37,107)(35,93,38,108)(36,94,39,105)(45,113,60,120)(46,114,57,117)(47,115,58,118)(48,116,59,119)(49,111,56,122)(50,112,53,123)(51,109,54,124)(52,110,55,121)>;
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), (1,70,19,73)(2,71,20,74)(3,72,17,75)(4,69,18,76)(5,79,13,84)(6,80,14,81)(7,77,15,82)(8,78,16,83)(9,86,126,89)(10,87,127,90)(11,88,128,91)(12,85,125,92)(21,66,28,61)(22,67,25,62)(23,68,26,63)(24,65,27,64)(29,50,34,45)(30,51,35,46)(31,52,36,47)(32,49,33,48)(37,60,44,53)(38,57,41,54)(39,58,42,55)(40,59,43,56)(93,109,98,114)(94,110,99,115)(95,111,100,116)(96,112,97,113)(101,117,108,124)(102,118,105,121)(103,119,106,122)(104,120,107,123), (1,49,22,56)(2,50,23,53)(3,51,24,54)(4,52,21,55)(5,108,127,93)(6,105,128,94)(7,106,125,95)(8,107,126,96)(9,97,16,104)(10,98,13,101)(11,99,14,102)(12,100,15,103)(17,46,27,57)(18,47,28,58)(19,48,25,59)(20,45,26,60)(29,63,44,74)(30,64,41,75)(31,61,42,76)(32,62,43,73)(33,67,40,70)(34,68,37,71)(35,65,38,72)(36,66,39,69)(77,122,92,111)(78,123,89,112)(79,124,90,109)(80,121,91,110)(81,118,88,115)(82,119,85,116)(83,120,86,113)(84,117,87,114), (1,77,22,92)(2,78,23,89)(3,79,24,90)(4,80,21,91)(5,65,127,72)(6,66,128,69)(7,67,125,70)(8,68,126,71)(9,74,16,63)(10,75,13,64)(11,76,14,61)(12,73,15,62)(17,84,27,87)(18,81,28,88)(19,82,25,85)(20,83,26,86)(29,97,44,104)(30,98,41,101)(31,99,42,102)(32,100,43,103)(33,95,40,106)(34,96,37,107)(35,93,38,108)(36,94,39,105)(45,113,60,120)(46,114,57,117)(47,115,58,118)(48,116,59,119)(49,111,56,122)(50,112,53,123)(51,109,54,124)(52,110,55,121) );
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)], [(1,70,19,73),(2,71,20,74),(3,72,17,75),(4,69,18,76),(5,79,13,84),(6,80,14,81),(7,77,15,82),(8,78,16,83),(9,86,126,89),(10,87,127,90),(11,88,128,91),(12,85,125,92),(21,66,28,61),(22,67,25,62),(23,68,26,63),(24,65,27,64),(29,50,34,45),(30,51,35,46),(31,52,36,47),(32,49,33,48),(37,60,44,53),(38,57,41,54),(39,58,42,55),(40,59,43,56),(93,109,98,114),(94,110,99,115),(95,111,100,116),(96,112,97,113),(101,117,108,124),(102,118,105,121),(103,119,106,122),(104,120,107,123)], [(1,49,22,56),(2,50,23,53),(3,51,24,54),(4,52,21,55),(5,108,127,93),(6,105,128,94),(7,106,125,95),(8,107,126,96),(9,97,16,104),(10,98,13,101),(11,99,14,102),(12,100,15,103),(17,46,27,57),(18,47,28,58),(19,48,25,59),(20,45,26,60),(29,63,44,74),(30,64,41,75),(31,61,42,76),(32,62,43,73),(33,67,40,70),(34,68,37,71),(35,65,38,72),(36,66,39,69),(77,122,92,111),(78,123,89,112),(79,124,90,109),(80,121,91,110),(81,118,88,115),(82,119,85,116),(83,120,86,113),(84,117,87,114)], [(1,77,22,92),(2,78,23,89),(3,79,24,90),(4,80,21,91),(5,65,127,72),(6,66,128,69),(7,67,125,70),(8,68,126,71),(9,74,16,63),(10,75,13,64),(11,76,14,61),(12,73,15,62),(17,84,27,87),(18,81,28,88),(19,82,25,85),(20,83,26,86),(29,97,44,104),(30,98,41,101),(31,99,42,102),(32,100,43,103),(33,95,40,106),(34,96,37,107),(35,93,38,108),(36,94,39,105),(45,113,60,120),(46,114,57,117),(47,115,58,118),(48,116,59,119),(49,111,56,122),(50,112,53,123),(51,109,54,124),(52,110,55,121)])
Matrix representation ►G ⊆ 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] >;
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 |
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