direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C42⋊3C4, C42⋊3C28, (C4×C28)⋊6C4, (C2×Q8)⋊2C28, (Q8×C14)⋊4C4, C23.4(C7×D4), C23⋊C4.2C14, (C22×C14).4D4, C4.4D4.2C14, C14.35(C23⋊C4), (D4×C14).177C22, (C2×C4).2(C2×C28), C2.9(C7×C23⋊C4), (C2×C28).13(C2×C4), (C2×D4).4(C2×C14), (C7×C23⋊C4).4C2, (C7×C4.4D4).11C2, C22.13(C7×C22⋊C4), (C2×C14).76(C22⋊C4), SmallGroup(448,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42⋊3C4
G = < a,b,c,d | a7=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >
Subgroups: 178 in 70 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, C28, C2×C14, C2×C14, C23⋊C4, C4.4D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C42⋊3C4, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C7×C23⋊C4, C7×C4.4D4, 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 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)
(8 43 31 99)(9 44 32 100)(10 45 33 101)(11 46 34 102)(12 47 35 103)(13 48 29 104)(14 49 30 105)(15 26 38 112)(16 27 39 106)(17 28 40 107)(18 22 41 108)(19 23 42 109)(20 24 36 110)(21 25 37 111)(71 95)(72 96)(73 97)(74 98)(75 92)(76 93)(77 94)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)
(1 50 69 59)(2 51 70 60)(3 52 64 61)(4 53 65 62)(5 54 66 63)(6 55 67 57)(7 56 68 58)(8 43 31 99)(9 44 32 100)(10 45 33 101)(11 46 34 102)(12 47 35 103)(13 48 29 104)(14 49 30 105)(15 112 38 26)(16 106 39 27)(17 107 40 28)(18 108 41 22)(19 109 42 23)(20 110 36 24)(21 111 37 25)(71 83 95 85)(72 84 96 86)(73 78 97 87)(74 79 98 88)(75 80 92 89)(76 81 93 90)(77 82 94 91)
(1 13 85 16)(2 14 86 17)(3 8 87 18)(4 9 88 19)(5 10 89 20)(6 11 90 21)(7 12 91 15)(22 61 43 73)(23 62 44 74)(24 63 45 75)(25 57 46 76)(26 58 47 77)(27 59 48 71)(28 60 49 72)(29 83 39 69)(30 84 40 70)(31 78 41 64)(32 79 42 65)(33 80 36 66)(34 81 37 67)(35 82 38 68)(50 104 95 106)(51 105 96 107)(52 99 97 108)(53 100 98 109)(54 101 92 110)(55 102 93 111)(56 103 94 112)
G:=sub<Sym(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), (8,43,31,99)(9,44,32,100)(10,45,33,101)(11,46,34,102)(12,47,35,103)(13,48,29,104)(14,49,30,105)(15,26,38,112)(16,27,39,106)(17,28,40,107)(18,22,41,108)(19,23,42,109)(20,24,36,110)(21,25,37,111)(71,95)(72,96)(73,97)(74,98)(75,92)(76,93)(77,94)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86), (1,50,69,59)(2,51,70,60)(3,52,64,61)(4,53,65,62)(5,54,66,63)(6,55,67,57)(7,56,68,58)(8,43,31,99)(9,44,32,100)(10,45,33,101)(11,46,34,102)(12,47,35,103)(13,48,29,104)(14,49,30,105)(15,112,38,26)(16,106,39,27)(17,107,40,28)(18,108,41,22)(19,109,42,23)(20,110,36,24)(21,111,37,25)(71,83,95,85)(72,84,96,86)(73,78,97,87)(74,79,98,88)(75,80,92,89)(76,81,93,90)(77,82,94,91), (1,13,85,16)(2,14,86,17)(3,8,87,18)(4,9,88,19)(5,10,89,20)(6,11,90,21)(7,12,91,15)(22,61,43,73)(23,62,44,74)(24,63,45,75)(25,57,46,76)(26,58,47,77)(27,59,48,71)(28,60,49,72)(29,83,39,69)(30,84,40,70)(31,78,41,64)(32,79,42,65)(33,80,36,66)(34,81,37,67)(35,82,38,68)(50,104,95,106)(51,105,96,107)(52,99,97,108)(53,100,98,109)(54,101,92,110)(55,102,93,111)(56,103,94,112)>;
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), (8,43,31,99)(9,44,32,100)(10,45,33,101)(11,46,34,102)(12,47,35,103)(13,48,29,104)(14,49,30,105)(15,26,38,112)(16,27,39,106)(17,28,40,107)(18,22,41,108)(19,23,42,109)(20,24,36,110)(21,25,37,111)(71,95)(72,96)(73,97)(74,98)(75,92)(76,93)(77,94)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86), (1,50,69,59)(2,51,70,60)(3,52,64,61)(4,53,65,62)(5,54,66,63)(6,55,67,57)(7,56,68,58)(8,43,31,99)(9,44,32,100)(10,45,33,101)(11,46,34,102)(12,47,35,103)(13,48,29,104)(14,49,30,105)(15,112,38,26)(16,106,39,27)(17,107,40,28)(18,108,41,22)(19,109,42,23)(20,110,36,24)(21,111,37,25)(71,83,95,85)(72,84,96,86)(73,78,97,87)(74,79,98,88)(75,80,92,89)(76,81,93,90)(77,82,94,91), (1,13,85,16)(2,14,86,17)(3,8,87,18)(4,9,88,19)(5,10,89,20)(6,11,90,21)(7,12,91,15)(22,61,43,73)(23,62,44,74)(24,63,45,75)(25,57,46,76)(26,58,47,77)(27,59,48,71)(28,60,49,72)(29,83,39,69)(30,84,40,70)(31,78,41,64)(32,79,42,65)(33,80,36,66)(34,81,37,67)(35,82,38,68)(50,104,95,106)(51,105,96,107)(52,99,97,108)(53,100,98,109)(54,101,92,110)(55,102,93,111)(56,103,94,112) );
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)], [(8,43,31,99),(9,44,32,100),(10,45,33,101),(11,46,34,102),(12,47,35,103),(13,48,29,104),(14,49,30,105),(15,26,38,112),(16,27,39,106),(17,28,40,107),(18,22,41,108),(19,23,42,109),(20,24,36,110),(21,25,37,111),(71,95),(72,96),(73,97),(74,98),(75,92),(76,93),(77,94),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86)], [(1,50,69,59),(2,51,70,60),(3,52,64,61),(4,53,65,62),(5,54,66,63),(6,55,67,57),(7,56,68,58),(8,43,31,99),(9,44,32,100),(10,45,33,101),(11,46,34,102),(12,47,35,103),(13,48,29,104),(14,49,30,105),(15,112,38,26),(16,106,39,27),(17,107,40,28),(18,108,41,22),(19,109,42,23),(20,110,36,24),(21,111,37,25),(71,83,95,85),(72,84,96,86),(73,78,97,87),(74,79,98,88),(75,80,92,89),(76,81,93,90),(77,82,94,91)], [(1,13,85,16),(2,14,86,17),(3,8,87,18),(4,9,88,19),(5,10,89,20),(6,11,90,21),(7,12,91,15),(22,61,43,73),(23,62,44,74),(24,63,45,75),(25,57,46,76),(26,58,47,77),(27,59,48,71),(28,60,49,72),(29,83,39,69),(30,84,40,70),(31,78,41,64),(32,79,42,65),(33,80,36,66),(34,81,37,67),(35,82,38,68),(50,104,95,106),(51,105,96,107),(52,99,97,108),(53,100,98,109),(54,101,92,110),(55,102,93,111),(56,103,94,112)]])
91 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | ··· | 4H | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 28A | ··· | 28R | 28S | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 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⋊3C4 | C7×C23⋊C4 | C7×C42⋊3C4 |
| kernel | C7×C42⋊3C4 | C7×C23⋊C4 | C7×C4.4D4 | C4×C28 | Q8×C14 | C42⋊3C4 | C23⋊C4 | C4.4D4 | C42 | C2×Q8 | C22×C14 | C23 | 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(𝔽29) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 12 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,28,0,0,0,0,17,0,0,0,0,17],[12,0,0,0,0,17,0,0,0,0,12,0,0,0,0,17],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C7×C42⋊3C4 in GAP, Magma, Sage, TeX
C_7\times C_4^2\rtimes_3C_4
% in TeX
G:=Group("C7xC4^2:3C4"); // GroupNames label
G:=SmallGroup(448,158);
// by ID
G=gap.SmallGroup(448,158);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,1576,3923,3538,248,6871,375,14117]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^4=1,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