direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C42.3C4, C42.3C28, (C4×C28).6C4, C4⋊Q8.3C14, (C2×C28).20D4, (C2×Q8).3C28, (Q8×C14).6C4, C4.10D4.C14, C14.37(C23⋊C4), (Q8×C14).156C22, (C2×C4).4(C7×D4), (C2×C4).4(C2×C28), (C7×C4⋊Q8).18C2, (C2×C28).15(C2×C4), C2.11(C7×C23⋊C4), (C2×Q8).2(C2×C14), (C7×C4.10D4).2C2, C22.15(C7×C22⋊C4), (C2×C14).78(C22⋊C4), SmallGroup(448,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42.3C4
G = < a,b,c,d | a7=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >
Subgroups: 114 in 60 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, M4(2), C2×Q8, C28, C2×C14, C4.10D4, C4⋊Q8, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C42.3C4, C4×C28, C7×C4⋊C4, C7×M4(2), Q8×C14, C7×C4.10D4, C7×C4⋊Q8, C7×C42.3C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, C28, C2×C14, C23⋊C4, C2×C28, C7×D4, C42.3C4, C7×C22⋊C4, C7×C23⋊C4, C7×C42.3C4
►On 112 points
Generators in S112
(1 14 71 87 31 79 23)(2 15 72 88 32 80 24)(3 16 65 81 25 73 17)(4 9 66 82 26 74 18)(5 10 67 83 27 75 19)(6 11 68 84 28 76 20)(7 12 69 85 29 77 21)(8 13 70 86 30 78 22)(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 36 6 40)(3 7)(4 34 8 38)(9 62 13 58)(10 14)(11 60 15 64)(12 16)(17 21)(18 90 22 94)(19 23)(20 96 24 92)(25 29)(26 98 30 102)(27 31)(28 104 32 100)(33 37)(35 39)(41 45)(42 78 46 74)(43 47)(44 76 48 80)(49 53)(50 86 54 82)(51 55)(52 84 56 88)(57 61)(59 63)(65 69)(66 106 70 110)(67 71)(68 112 72 108)(73 77)(75 79)(81 85)(83 87)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)
(1 39 5 35)(2 40 6 36)(3 37 7 33)(4 38 8 34)(9 58 13 62)(10 63 14 59)(11 64 15 60)(12 61 16 57)(17 93 21 89)(18 94 22 90)(19 91 23 95)(20 92 24 96)(25 101 29 97)(26 102 30 98)(27 99 31 103)(28 100 32 104)(41 73 45 77)(42 74 46 78)(43 79 47 75)(44 80 48 76)(49 81 53 85)(50 82 54 86)(51 87 55 83)(52 88 56 84)(65 109 69 105)(66 110 70 106)(67 107 71 111)(68 108 72 112)
(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)| (1,14,71,87,31,79,23)(2,15,72,88,32,80,24)(3,16,65,81,25,73,17)(4,9,66,82,26,74,18)(5,10,67,83,27,75,19)(6,11,68,84,28,76,20)(7,12,69,85,29,77,21)(8,13,70,86,30,78,22)(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)(2,36,6,40)(3,7)(4,34,8,38)(9,62,13,58)(10,14)(11,60,15,64)(12,16)(17,21)(18,90,22,94)(19,23)(20,96,24,92)(25,29)(26,98,30,102)(27,31)(28,104,32,100)(33,37)(35,39)(41,45)(42,78,46,74)(43,47)(44,76,48,80)(49,53)(50,86,54,82)(51,55)(52,84,56,88)(57,61)(59,63)(65,69)(66,106,70,110)(67,71)(68,112,72,108)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,39,5,35)(2,40,6,36)(3,37,7,33)(4,38,8,34)(9,58,13,62)(10,63,14,59)(11,64,15,60)(12,61,16,57)(17,93,21,89)(18,94,22,90)(19,91,23,95)(20,92,24,96)(25,101,29,97)(26,102,30,98)(27,99,31,103)(28,100,32,104)(41,73,45,77)(42,74,46,78)(43,79,47,75)(44,80,48,76)(49,81,53,85)(50,82,54,86)(51,87,55,83)(52,88,56,84)(65,109,69,105)(66,110,70,106)(67,107,71,111)(68,108,72,112), (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( (1,14,71,87,31,79,23)(2,15,72,88,32,80,24)(3,16,65,81,25,73,17)(4,9,66,82,26,74,18)(5,10,67,83,27,75,19)(6,11,68,84,28,76,20)(7,12,69,85,29,77,21)(8,13,70,86,30,78,22)(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)(2,36,6,40)(3,7)(4,34,8,38)(9,62,13,58)(10,14)(11,60,15,64)(12,16)(17,21)(18,90,22,94)(19,23)(20,96,24,92)(25,29)(26,98,30,102)(27,31)(28,104,32,100)(33,37)(35,39)(41,45)(42,78,46,74)(43,47)(44,76,48,80)(49,53)(50,86,54,82)(51,55)(52,84,56,88)(57,61)(59,63)(65,69)(66,106,70,110)(67,71)(68,112,72,108)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,39,5,35)(2,40,6,36)(3,37,7,33)(4,38,8,34)(9,58,13,62)(10,63,14,59)(11,64,15,60)(12,61,16,57)(17,93,21,89)(18,94,22,90)(19,91,23,95)(20,92,24,96)(25,101,29,97)(26,102,30,98)(27,99,31,103)(28,100,32,104)(41,73,45,77)(42,74,46,78)(43,79,47,75)(44,80,48,76)(49,81,53,85)(50,82,54,86)(51,87,55,83)(52,88,56,84)(65,109,69,105)(66,110,70,106)(67,107,71,111)(68,108,72,112), (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([[(1,14,71,87,31,79,23),(2,15,72,88,32,80,24),(3,16,65,81,25,73,17),(4,9,66,82,26,74,18),(5,10,67,83,27,75,19),(6,11,68,84,28,76,20),(7,12,69,85,29,77,21),(8,13,70,86,30,78,22),(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),(2,36,6,40),(3,7),(4,34,8,38),(9,62,13,58),(10,14),(11,60,15,64),(12,16),(17,21),(18,90,22,94),(19,23),(20,96,24,92),(25,29),(26,98,30,102),(27,31),(28,104,32,100),(33,37),(35,39),(41,45),(42,78,46,74),(43,47),(44,76,48,80),(49,53),(50,86,54,82),(51,55),(52,84,56,88),(57,61),(59,63),(65,69),(66,106,70,110),(67,71),(68,112,72,108),(73,77),(75,79),(81,85),(83,87),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111)], [(1,39,5,35),(2,40,6,36),(3,37,7,33),(4,38,8,34),(9,58,13,62),(10,63,14,59),(11,64,15,60),(12,61,16,57),(17,93,21,89),(18,94,22,90),(19,91,23,95),(20,92,24,96),(25,101,29,97),(26,102,30,98),(27,99,31,103),(28,100,32,104),(41,73,45,77),(42,74,46,78),(43,79,47,75),(44,80,48,76),(49,81,53,85),(50,82,54,86),(51,87,55,83),(52,88,56,84),(65,109,69,105),(66,110,70,106),(67,107,71,111),(68,108,72,112)], [(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 | 4A | ··· | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14L | 28A | ··· | 28AD | 28AE | ··· | 28AJ | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 4 | ··· | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | ··· | 4 | 8 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 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 | C42.3C4 | C7×C23⋊C4 | C7×C42.3C4 |
| kernel | C7×C42.3C4 | C7×C4.10D4 | C7×C4⋊Q8 | C4×C28 | Q8×C14 | C42.3C4 | C4.10D4 | C4⋊Q8 | C42 | C2×Q8 | 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×C42.3C4 ►in GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 97 | 60 |
| 0 | 0 | 88 | 16 |
| 97 | 60 | 0 | 0 |
| 88 | 16 | 0 | 0 |
| 0 | 0 | 16 | 53 |
| 0 | 0 | 25 | 97 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 112 | 0 | 0 |
| 2 | 112 | 0 | 0 |
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[112,0,0,0,0,112,0,0,0,0,97,88,0,0,60,16],[97,88,0,0,60,16,0,0,0,0,16,25,0,0,53,97],[0,0,1,2,0,0,112,112,1,0,0,0,0,1,0,0] >;
C7×C42.3C4 in GAP, Magma, Sage, TeX
C_7\times C_4^2._3C_4
% in TeX
G:=Group("C7xC4^2.3C4"); // GroupNames label
G:=SmallGroup(448,160);
// by ID
G=gap.SmallGroup(448,160);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,1576,3923,3538,248,6871,375,172,14117]);
// 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,d*c*d^-1=b^2*c>;
// generators/relations