direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C42.C4, C42.2C28, (C4×C28).5C4, (D4×C14).6C4, (C2×D4).3C28, (C2×C28).19D4, C4.10D4⋊5C14, C4.4D4.3C14, C14.36(C23⋊C4), (Q8×C14).155C22, (C2×C4).3(C7×D4), (C2×C4).3(C2×C28), (C2×C28).14(C2×C4), C2.10(C7×C23⋊C4), (C2×Q8).1(C2×C14), (C7×C4.10D4)⋊12C2, (C7×C4.4D4).12C2, C22.14(C7×C22⋊C4), (C2×C14).77(C22⋊C4), SmallGroup(448,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42.C4
G = < a,b,c,d | a7=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >
Subgroups: 146 in 64 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C28, C2×C14, C2×C14, C4.10D4, C4.4D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C42.C4, C4×C28, C7×C22⋊C4, C7×M4(2), D4×C14, Q8×C14, C7×C4.10D4, C7×C4.4D4, C7×C42.C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, C28, C2×C14, C23⋊C4, C2×C28, C7×D4, C42.C4, C7×C22⋊C4, C7×C23⋊C4, C7×C42.C4
►On 112 points
Generators in S112
(1 16 71 81 25 73 17)(2 9 72 82 26 74 18)(3 10 65 83 27 75 19)(4 11 66 84 28 76 20)(5 12 67 85 29 77 21)(6 13 68 86 30 78 22)(7 14 69 87 31 79 23)(8 15 70 88 32 80 24)(33 61 105 49 97 41 89)(34 62 106 50 98 42 90)(35 63 107 51 99 43 91)(36 64 108 52 100 44 92)(37 57 109 53 101 45 93)(38 58 110 54 102 46 94)(39 59 111 55 103 47 95)(40 60 112 56 104 48 96)
(1 5)(2 40 6 36)(4 34 8 38)(9 60 13 64)(11 62 15 58)(12 16)(17 21)(18 96 22 92)(20 90 24 94)(25 29)(26 104 30 100)(28 98 32 102)(35 39)(42 80 46 76)(43 47)(44 74 48 78)(50 88 54 84)(51 55)(52 82 56 86)(59 63)(66 106 70 110)(67 71)(68 108 72 112)(73 77)(81 85)(91 95)(99 103)(107 111)
(1 39 5 35)(2 36 6 40)(3 37 7 33)(4 34 8 38)(9 64 13 60)(10 57 14 61)(11 62 15 58)(12 63 16 59)(17 95 21 91)(18 92 22 96)(19 93 23 89)(20 90 24 94)(25 103 29 99)(26 100 30 104)(27 101 31 97)(28 98 32 102)(41 75 45 79)(42 80 46 76)(43 73 47 77)(44 78 48 74)(49 83 53 87)(50 88 54 84)(51 81 55 85)(52 86 56 82)(65 109 69 105)(66 106 70 110)(67 107 71 111)(68 112 72 108)
(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)
G:=sub<Sym(112)| (1,16,71,81,25,73,17)(2,9,72,82,26,74,18)(3,10,65,83,27,75,19)(4,11,66,84,28,76,20)(5,12,67,85,29,77,21)(6,13,68,86,30,78,22)(7,14,69,87,31,79,23)(8,15,70,88,32,80,24)(33,61,105,49,97,41,89)(34,62,106,50,98,42,90)(35,63,107,51,99,43,91)(36,64,108,52,100,44,92)(37,57,109,53,101,45,93)(38,58,110,54,102,46,94)(39,59,111,55,103,47,95)(40,60,112,56,104,48,96), (1,5)(2,40,6,36)(4,34,8,38)(9,60,13,64)(11,62,15,58)(12,16)(17,21)(18,96,22,92)(20,90,24,94)(25,29)(26,104,30,100)(28,98,32,102)(35,39)(42,80,46,76)(43,47)(44,74,48,78)(50,88,54,84)(51,55)(52,82,56,86)(59,63)(66,106,70,110)(67,71)(68,108,72,112)(73,77)(81,85)(91,95)(99,103)(107,111), (1,39,5,35)(2,36,6,40)(3,37,7,33)(4,34,8,38)(9,64,13,60)(10,57,14,61)(11,62,15,58)(12,63,16,59)(17,95,21,91)(18,92,22,96)(19,93,23,89)(20,90,24,94)(25,103,29,99)(26,100,30,104)(27,101,31,97)(28,98,32,102)(41,75,45,79)(42,80,46,76)(43,73,47,77)(44,78,48,74)(49,83,53,87)(50,88,54,84)(51,81,55,85)(52,86,56,82)(65,109,69,105)(66,106,70,110)(67,107,71,111)(68,112,72,108), (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)>;
G:=Group( (1,16,71,81,25,73,17)(2,9,72,82,26,74,18)(3,10,65,83,27,75,19)(4,11,66,84,28,76,20)(5,12,67,85,29,77,21)(6,13,68,86,30,78,22)(7,14,69,87,31,79,23)(8,15,70,88,32,80,24)(33,61,105,49,97,41,89)(34,62,106,50,98,42,90)(35,63,107,51,99,43,91)(36,64,108,52,100,44,92)(37,57,109,53,101,45,93)(38,58,110,54,102,46,94)(39,59,111,55,103,47,95)(40,60,112,56,104,48,96), (1,5)(2,40,6,36)(4,34,8,38)(9,60,13,64)(11,62,15,58)(12,16)(17,21)(18,96,22,92)(20,90,24,94)(25,29)(26,104,30,100)(28,98,32,102)(35,39)(42,80,46,76)(43,47)(44,74,48,78)(50,88,54,84)(51,55)(52,82,56,86)(59,63)(66,106,70,110)(67,71)(68,108,72,112)(73,77)(81,85)(91,95)(99,103)(107,111), (1,39,5,35)(2,36,6,40)(3,37,7,33)(4,34,8,38)(9,64,13,60)(10,57,14,61)(11,62,15,58)(12,63,16,59)(17,95,21,91)(18,92,22,96)(19,93,23,89)(20,90,24,94)(25,103,29,99)(26,100,30,104)(27,101,31,97)(28,98,32,102)(41,75,45,79)(42,80,46,76)(43,73,47,77)(44,78,48,74)(49,83,53,87)(50,88,54,84)(51,81,55,85)(52,86,56,82)(65,109,69,105)(66,106,70,110)(67,107,71,111)(68,112,72,108), (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) );
G=PermutationGroup([[(1,16,71,81,25,73,17),(2,9,72,82,26,74,18),(3,10,65,83,27,75,19),(4,11,66,84,28,76,20),(5,12,67,85,29,77,21),(6,13,68,86,30,78,22),(7,14,69,87,31,79,23),(8,15,70,88,32,80,24),(33,61,105,49,97,41,89),(34,62,106,50,98,42,90),(35,63,107,51,99,43,91),(36,64,108,52,100,44,92),(37,57,109,53,101,45,93),(38,58,110,54,102,46,94),(39,59,111,55,103,47,95),(40,60,112,56,104,48,96)], [(1,5),(2,40,6,36),(4,34,8,38),(9,60,13,64),(11,62,15,58),(12,16),(17,21),(18,96,22,92),(20,90,24,94),(25,29),(26,104,30,100),(28,98,32,102),(35,39),(42,80,46,76),(43,47),(44,74,48,78),(50,88,54,84),(51,55),(52,82,56,86),(59,63),(66,106,70,110),(67,71),(68,108,72,112),(73,77),(81,85),(91,95),(99,103),(107,111)], [(1,39,5,35),(2,36,6,40),(3,37,7,33),(4,34,8,38),(9,64,13,60),(10,57,14,61),(11,62,15,58),(12,63,16,59),(17,95,21,91),(18,92,22,96),(19,93,23,89),(20,90,24,94),(25,103,29,99),(26,100,30,104),(27,101,31,97),(28,98,32,102),(41,75,45,79),(42,80,46,76),(43,73,47,77),(44,78,48,74),(49,83,53,87),(50,88,54,84),(51,81,55,85),(52,86,56,82),(65,109,69,105),(66,106,70,110),(67,107,71,111),(68,112,72,108)], [(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)]])
91 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 28A | ··· | 28AD | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 4 | ··· | 4 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
91 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | D4 | C7×D4 | C23⋊C4 | C42.C4 | C7×C23⋊C4 | C7×C42.C4 |
| kernel | C7×C42.C4 | C7×C4.10D4 | C7×C4.4D4 | C4×C28 | D4×C14 | C42.C4 | C4.10D4 | C4.4D4 | C42 | C2×D4 | C2×C28 | C2×C4 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 12 | 6 | 12 | 12 | 2 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of C7×C42.C4 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 112 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 7 | 0 | 15 | 0 |
| 106 | 0 | 0 | 15 |
| 15 | 0 | 0 | 0 |
| 98 | 98 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 112 | 0 | 0 | 98 |
| 98 | 0 | 111 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 15 | 0 |
| 1 | 0 | 98 | 0 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[112,1,7,106,0,1,0,0,0,0,15,0,0,0,0,15],[15,98,0,112,0,98,0,0,0,0,15,0,0,0,0,98],[98,0,0,1,0,0,1,0,111,1,15,98,0,1,0,0] >;
C7×C42.C4 in GAP, Magma, Sage, TeX
C_7\times C_4^2.C_4
% in TeX
G:=Group("C7xC4^2.C4"); // GroupNames label
G:=SmallGroup(448,159);
// by ID
G=gap.SmallGroup(448,159);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,3538,248,6871,375,172,14117]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=1,d^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations