direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C42.C2, C43.13C2, C42.63Q8, C23.187C24, C4.18(C4×Q8), C4○4(C42⋊8C4), C42.278(C2×C4), C42⋊8C4.56C2, C22.78(C23×C4), C22.28(C22×Q8), (C22×C4).751C23, (C2×C42).1010C22, C4○7(C23.63C23), C4○6(C23.65C23), C23.65C23.94C2, C23.63C23.67C2, C2.C42.469C22, C2.5(C23.37C23), C2.11(C23.36C23), C2.9(C2×C4×Q8), (C4×C4⋊C4).32C2, C2.16(C4×C4○D4), C4⋊C4.154(C2×C4), (C2×C4).226(C2×Q8), C2.4(C2×C42.C2), (C2×C4).22(C22×C4), C22.79(C2×C4○D4), (C2×C4)○3(C42⋊8C4), (C2×C4).640(C4○D4), (C2×C4⋊C4).801C22, (C2×C42.C2).39C2, (C2×C4)○3(C23.63C23), SmallGroup(128,1037)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C42.C2
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 332 in 242 conjugacy classes, 160 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C42.C2, C43, C4×C4⋊C4, C42⋊8C4, C23.63C23, C23.65C23, C2×C42.C2, C4×C42.C2
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C42.C2, C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, C4×C4○D4, C2×C42.C2, C23.36C23, C23.37C23, C4×C42.C2
►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 48 25 56)(2 45 26 53)(3 46 27 54)(4 47 28 55)(5 108 10 98)(6 105 11 99)(7 106 12 100)(8 107 9 97)(13 101 127 93)(14 102 128 94)(15 103 125 95)(16 104 126 96)(17 51 24 57)(18 52 21 58)(19 49 22 59)(20 50 23 60)(29 68 37 74)(30 65 38 75)(31 66 39 76)(32 67 40 73)(33 62 43 70)(34 63 44 71)(35 64 41 72)(36 61 42 69)(77 119 85 111)(78 120 86 112)(79 117 87 109)(80 118 88 110)(81 121 91 115)(82 122 92 116)(83 123 89 113)(84 124 90 114)
(1 33 22 40)(2 34 23 37)(3 35 24 38)(4 36 21 39)(5 117 127 114)(6 118 128 115)(7 119 125 116)(8 120 126 113)(9 112 16 123)(10 109 13 124)(11 110 14 121)(12 111 15 122)(17 30 27 41)(18 31 28 42)(19 32 25 43)(20 29 26 44)(45 63 60 74)(46 64 57 75)(47 61 58 76)(48 62 59 73)(49 67 56 70)(50 68 53 71)(51 65 54 72)(52 66 55 69)(77 103 92 100)(78 104 89 97)(79 101 90 98)(80 102 91 99)(81 105 88 94)(82 106 85 95)(83 107 86 96)(84 108 87 93)
(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), (1,48,25,56)(2,45,26,53)(3,46,27,54)(4,47,28,55)(5,108,10,98)(6,105,11,99)(7,106,12,100)(8,107,9,97)(13,101,127,93)(14,102,128,94)(15,103,125,95)(16,104,126,96)(17,51,24,57)(18,52,21,58)(19,49,22,59)(20,50,23,60)(29,68,37,74)(30,65,38,75)(31,66,39,76)(32,67,40,73)(33,62,43,70)(34,63,44,71)(35,64,41,72)(36,61,42,69)(77,119,85,111)(78,120,86,112)(79,117,87,109)(80,118,88,110)(81,121,91,115)(82,122,92,116)(83,123,89,113)(84,124,90,114), (1,33,22,40)(2,34,23,37)(3,35,24,38)(4,36,21,39)(5,117,127,114)(6,118,128,115)(7,119,125,116)(8,120,126,113)(9,112,16,123)(10,109,13,124)(11,110,14,121)(12,111,15,122)(17,30,27,41)(18,31,28,42)(19,32,25,43)(20,29,26,44)(45,63,60,74)(46,64,57,75)(47,61,58,76)(48,62,59,73)(49,67,56,70)(50,68,53,71)(51,65,54,72)(52,66,55,69)(77,103,92,100)(78,104,89,97)(79,101,90,98)(80,102,91,99)(81,105,88,94)(82,106,85,95)(83,107,86,96)(84,108,87,93), (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), (1,48,25,56)(2,45,26,53)(3,46,27,54)(4,47,28,55)(5,108,10,98)(6,105,11,99)(7,106,12,100)(8,107,9,97)(13,101,127,93)(14,102,128,94)(15,103,125,95)(16,104,126,96)(17,51,24,57)(18,52,21,58)(19,49,22,59)(20,50,23,60)(29,68,37,74)(30,65,38,75)(31,66,39,76)(32,67,40,73)(33,62,43,70)(34,63,44,71)(35,64,41,72)(36,61,42,69)(77,119,85,111)(78,120,86,112)(79,117,87,109)(80,118,88,110)(81,121,91,115)(82,122,92,116)(83,123,89,113)(84,124,90,114), (1,33,22,40)(2,34,23,37)(3,35,24,38)(4,36,21,39)(5,117,127,114)(6,118,128,115)(7,119,125,116)(8,120,126,113)(9,112,16,123)(10,109,13,124)(11,110,14,121)(12,111,15,122)(17,30,27,41)(18,31,28,42)(19,32,25,43)(20,29,26,44)(45,63,60,74)(46,64,57,75)(47,61,58,76)(48,62,59,73)(49,67,56,70)(50,68,53,71)(51,65,54,72)(52,66,55,69)(77,103,92,100)(78,104,89,97)(79,101,90,98)(80,102,91,99)(81,105,88,94)(82,106,85,95)(83,107,86,96)(84,108,87,93), (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)], [(1,48,25,56),(2,45,26,53),(3,46,27,54),(4,47,28,55),(5,108,10,98),(6,105,11,99),(7,106,12,100),(8,107,9,97),(13,101,127,93),(14,102,128,94),(15,103,125,95),(16,104,126,96),(17,51,24,57),(18,52,21,58),(19,49,22,59),(20,50,23,60),(29,68,37,74),(30,65,38,75),(31,66,39,76),(32,67,40,73),(33,62,43,70),(34,63,44,71),(35,64,41,72),(36,61,42,69),(77,119,85,111),(78,120,86,112),(79,117,87,109),(80,118,88,110),(81,121,91,115),(82,122,92,116),(83,123,89,113),(84,124,90,114)], [(1,33,22,40),(2,34,23,37),(3,35,24,38),(4,36,21,39),(5,117,127,114),(6,118,128,115),(7,119,125,116),(8,120,126,113),(9,112,16,123),(10,109,13,124),(11,110,14,121),(12,111,15,122),(17,30,27,41),(18,31,28,42),(19,32,25,43),(20,29,26,44),(45,63,60,74),(46,64,57,75),(47,61,58,76),(48,62,59,73),(49,67,56,70),(50,68,53,71),(51,65,54,72),(52,66,55,69),(77,103,92,100),(78,104,89,97),(79,101,90,98),(80,102,91,99),(81,105,88,94),(82,106,85,95),(83,107,86,96),(84,108,87,93)], [(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)]])
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 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | C4○D4 |
| kernel | C4×C42.C2 | C43 | C4×C4⋊C4 | C42⋊8C4 | C23.63C23 | C23.65C23 | C2×C42.C2 | C42.C2 | C42 | C2×C4 |
| # reps | 1 | 1 | 6 | 1 | 4 | 2 | 1 | 16 | 4 | 20 |
Matrix representation of C4×C42.C2 ►in GL5(𝔽5)
| 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[4,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,4] >;
C4×C42.C2 in GAP, Magma, Sage, TeX
C_4\times C_4^2.C_2
% in TeX
G:=Group("C4xC4^2.C2"); // GroupNames label
G:=SmallGroup(128,1037);
// by ID
G=gap.SmallGroup(128,1037);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100,304]);
// 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*c^2,d*c*d^-1=b^2*c>;
// generators/relations