direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C42⋊2C2, C42⋊2C14, C4⋊C4⋊5C14, (C4×C28)⋊3C2, C22⋊C4.2C14, C23.3(C2×C14), C14.46(C4○D4), (C2×C28).68C22, (C2×C14).81C23, (C22×C14).3C22, C22.16(C22×C14), (C7×C4⋊C4)⋊14C2, C2.9(C7×C4○D4), (C2×C4).8(C2×C14), (C7×C22⋊C4).5C2, SmallGroup(224,161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42⋊2C2
G = < a,b,c,d | a7=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >
Subgroups: 84 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2×C14, C42⋊2C2, C2×C28, C22×C14, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C42⋊2C2
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C2×C14, C42⋊2C2, C22×C14, C7×C4○D4, C7×C42⋊2C2
►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)
(1 99 43 94)(2 100 44 95)(3 101 45 96)(4 102 46 97)(5 103 47 98)(6 104 48 92)(7 105 49 93)(8 62 112 56)(9 63 106 50)(10 57 107 51)(11 58 108 52)(12 59 109 53)(13 60 110 54)(14 61 111 55)(15 66 24 71)(16 67 25 72)(17 68 26 73)(18 69 27 74)(19 70 28 75)(20 64 22 76)(21 65 23 77)(29 79 40 85)(30 80 41 86)(31 81 42 87)(32 82 36 88)(33 83 37 89)(34 84 38 90)(35 78 39 91)
(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 90 23 105)(9 91 24 99)(10 85 25 100)(11 86 26 101)(12 87 27 102)(13 88 28 103)(14 89 22 104)(15 94 106 78)(16 95 107 79)(17 96 108 80)(18 97 109 81)(19 98 110 82)(20 92 111 83)(21 93 112 84)(29 72 44 57)(30 73 45 58)(31 74 46 59)(32 75 47 60)(33 76 48 61)(34 77 49 62)(35 71 43 63)
(8 112)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
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), (1,99,43,94)(2,100,44,95)(3,101,45,96)(4,102,46,97)(5,103,47,98)(6,104,48,92)(7,105,49,93)(8,62,112,56)(9,63,106,50)(10,57,107,51)(11,58,108,52)(12,59,109,53)(13,60,110,54)(14,61,111,55)(15,66,24,71)(16,67,25,72)(17,68,26,73)(18,69,27,74)(19,70,28,75)(20,64,22,76)(21,65,23,77)(29,79,40,85)(30,80,41,86)(31,81,42,87)(32,82,36,88)(33,83,37,89)(34,84,38,90)(35,78,39,91), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,90,23,105)(9,91,24,99)(10,85,25,100)(11,86,26,101)(12,87,27,102)(13,88,28,103)(14,89,22,104)(15,94,106,78)(16,95,107,79)(17,96,108,80)(18,97,109,81)(19,98,110,82)(20,92,111,83)(21,93,112,84)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)>;
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), (1,99,43,94)(2,100,44,95)(3,101,45,96)(4,102,46,97)(5,103,47,98)(6,104,48,92)(7,105,49,93)(8,62,112,56)(9,63,106,50)(10,57,107,51)(11,58,108,52)(12,59,109,53)(13,60,110,54)(14,61,111,55)(15,66,24,71)(16,67,25,72)(17,68,26,73)(18,69,27,74)(19,70,28,75)(20,64,22,76)(21,65,23,77)(29,79,40,85)(30,80,41,86)(31,81,42,87)(32,82,36,88)(33,83,37,89)(34,84,38,90)(35,78,39,91), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,90,23,105)(9,91,24,99)(10,85,25,100)(11,86,26,101)(12,87,27,102)(13,88,28,103)(14,89,22,104)(15,94,106,78)(16,95,107,79)(17,96,108,80)(18,97,109,81)(19,98,110,82)(20,92,111,83)(21,93,112,84)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99) );
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)], [(1,99,43,94),(2,100,44,95),(3,101,45,96),(4,102,46,97),(5,103,47,98),(6,104,48,92),(7,105,49,93),(8,62,112,56),(9,63,106,50),(10,57,107,51),(11,58,108,52),(12,59,109,53),(13,60,110,54),(14,61,111,55),(15,66,24,71),(16,67,25,72),(17,68,26,73),(18,69,27,74),(19,70,28,75),(20,64,22,76),(21,65,23,77),(29,79,40,85),(30,80,41,86),(31,81,42,87),(32,82,36,88),(33,83,37,89),(34,84,38,90),(35,78,39,91)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,90,23,105),(9,91,24,99),(10,85,25,100),(11,86,26,101),(12,87,27,102),(13,88,28,103),(14,89,22,104),(15,94,106,78),(16,95,107,79),(17,96,108,80),(18,97,109,81),(19,98,110,82),(20,92,111,83),(21,93,112,84),(29,72,44,57),(30,73,45,58),(31,74,46,59),(32,75,47,60),(33,76,48,61),(34,77,49,62),(35,71,43,63)], [(8,112),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,24),(16,25),(17,26),(18,27),(19,28),(20,22),(21,23),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)]])
C7×C42⋊2C2 is a maximal subgroup of
C42.159D14 C42.160D14 C42⋊23D14 C42⋊24D14 C42.189D14 C42.161D14 C42.162D14 C42.163D14 C42.164D14 C42⋊25D14 C42.165D14
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14X | 28A | ··· | 28AJ | 28AK | ··· | 28BB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C4○D4 | C7×C4○D4 |
| kernel | C7×C42⋊2C2 | C4×C28 | C7×C22⋊C4 | C7×C4⋊C4 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C14 | C2 |
| # reps | 1 | 1 | 3 | 3 | 6 | 6 | 18 | 18 | 6 | 36 |
Matrix representation of C7×C42⋊2C2 ►in GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 0 | 12 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,23,0,0,0,0,23],[0,17,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[0,28,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28] >;
C7×C42⋊2C2 in GAP, Magma, Sage, TeX
C_7\times C_4^2\rtimes_2C_2
% in TeX
G:=Group("C7xC4^2:2C2"); // GroupNames label
G:=SmallGroup(224,161);
// by ID
G=gap.SmallGroup(224,161);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,1015,2090,266]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,d*c*d=b^2*c^-1>;
// generators/relations