direct product, non-abelian, soluble
Aliases: C7×C4.A4, Q8.C42, C28.2A4, SL2(𝔽3)⋊2C14, C4○D4⋊C21, C4.(C7×A4), C2.3(A4×C14), (C7×Q8).5C6, C14.12(C2×A4), (C7×SL2(𝔽3))⋊5C2, (C7×C4○D4)⋊1C3, SmallGroup(336,170)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C7×Q8 — C7×SL2(𝔽3) — C7×C4.A4 |
| Q8 — C7×C4.A4 |
Generators and relations for C7×C4.A4
G = < a,b,c,d,e | a7=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >
Smallest permutation representation of C7×C4.A4
►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 39 78 32)(2 40 79 33)(3 41 80 34)(4 42 81 35)(5 36 82 29)(6 37 83 30)(7 38 84 31)(8 71 25 64)(9 72 26 65)(10 73 27 66)(11 74 28 67)(12 75 22 68)(13 76 23 69)(14 77 24 70)(15 62 109 53)(16 63 110 54)(17 57 111 55)(18 58 112 56)(19 59 106 50)(20 60 107 51)(21 61 108 52)(43 89 99 97)(44 90 100 98)(45 91 101 92)(46 85 102 93)(47 86 103 94)(48 87 104 95)(49 88 105 96)
(1 87 78 95)(2 88 79 96)(3 89 80 97)(4 90 81 98)(5 91 82 92)(6 85 83 93)(7 86 84 94)(8 109 25 15)(9 110 26 16)(10 111 27 17)(11 112 28 18)(12 106 22 19)(13 107 23 20)(14 108 24 21)(29 45 36 101)(30 46 37 102)(31 47 38 103)(32 48 39 104)(33 49 40 105)(34 43 41 99)(35 44 42 100)(50 68 59 75)(51 69 60 76)(52 70 61 77)(53 64 62 71)(54 65 63 72)(55 66 57 73)(56 67 58 74)
(1 75 78 68)(2 76 79 69)(3 77 80 70)(4 71 81 64)(5 72 82 65)(6 73 83 66)(7 74 84 67)(8 42 25 35)(9 36 26 29)(10 37 27 30)(11 38 28 31)(12 39 22 32)(13 40 23 33)(14 41 24 34)(15 100 109 44)(16 101 110 45)(17 102 111 46)(18 103 112 47)(19 104 106 48)(20 105 107 49)(21 99 108 43)(50 87 59 95)(51 88 60 96)(52 89 61 97)(53 90 62 98)(54 91 63 92)(55 85 57 93)(56 86 58 94)
(8 109 44)(9 110 45)(10 111 46)(11 112 47)(12 106 48)(13 107 49)(14 108 43)(15 100 25)(16 101 26)(17 102 27)(18 103 28)(19 104 22)(20 105 23)(21 99 24)(50 87 75)(51 88 76)(52 89 77)(53 90 71)(54 91 72)(55 85 73)(56 86 74)(57 93 66)(58 94 67)(59 95 68)(60 96 69)(61 97 70)(62 98 64)(63 92 65)
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), (1,39,78,32)(2,40,79,33)(3,41,80,34)(4,42,81,35)(5,36,82,29)(6,37,83,30)(7,38,84,31)(8,71,25,64)(9,72,26,65)(10,73,27,66)(11,74,28,67)(12,75,22,68)(13,76,23,69)(14,77,24,70)(15,62,109,53)(16,63,110,54)(17,57,111,55)(18,58,112,56)(19,59,106,50)(20,60,107,51)(21,61,108,52)(43,89,99,97)(44,90,100,98)(45,91,101,92)(46,85,102,93)(47,86,103,94)(48,87,104,95)(49,88,105,96), (1,87,78,95)(2,88,79,96)(3,89,80,97)(4,90,81,98)(5,91,82,92)(6,85,83,93)(7,86,84,94)(8,109,25,15)(9,110,26,16)(10,111,27,17)(11,112,28,18)(12,106,22,19)(13,107,23,20)(14,108,24,21)(29,45,36,101)(30,46,37,102)(31,47,38,103)(32,48,39,104)(33,49,40,105)(34,43,41,99)(35,44,42,100)(50,68,59,75)(51,69,60,76)(52,70,61,77)(53,64,62,71)(54,65,63,72)(55,66,57,73)(56,67,58,74), (1,75,78,68)(2,76,79,69)(3,77,80,70)(4,71,81,64)(5,72,82,65)(6,73,83,66)(7,74,84,67)(8,42,25,35)(9,36,26,29)(10,37,27,30)(11,38,28,31)(12,39,22,32)(13,40,23,33)(14,41,24,34)(15,100,109,44)(16,101,110,45)(17,102,111,46)(18,103,112,47)(19,104,106,48)(20,105,107,49)(21,99,108,43)(50,87,59,95)(51,88,60,96)(52,89,61,97)(53,90,62,98)(54,91,63,92)(55,85,57,93)(56,86,58,94), (8,109,44)(9,110,45)(10,111,46)(11,112,47)(12,106,48)(13,107,49)(14,108,43)(15,100,25)(16,101,26)(17,102,27)(18,103,28)(19,104,22)(20,105,23)(21,99,24)(50,87,75)(51,88,76)(52,89,77)(53,90,71)(54,91,72)(55,85,73)(56,86,74)(57,93,66)(58,94,67)(59,95,68)(60,96,69)(61,97,70)(62,98,64)(63,92,65)>;
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), (1,39,78,32)(2,40,79,33)(3,41,80,34)(4,42,81,35)(5,36,82,29)(6,37,83,30)(7,38,84,31)(8,71,25,64)(9,72,26,65)(10,73,27,66)(11,74,28,67)(12,75,22,68)(13,76,23,69)(14,77,24,70)(15,62,109,53)(16,63,110,54)(17,57,111,55)(18,58,112,56)(19,59,106,50)(20,60,107,51)(21,61,108,52)(43,89,99,97)(44,90,100,98)(45,91,101,92)(46,85,102,93)(47,86,103,94)(48,87,104,95)(49,88,105,96), (1,87,78,95)(2,88,79,96)(3,89,80,97)(4,90,81,98)(5,91,82,92)(6,85,83,93)(7,86,84,94)(8,109,25,15)(9,110,26,16)(10,111,27,17)(11,112,28,18)(12,106,22,19)(13,107,23,20)(14,108,24,21)(29,45,36,101)(30,46,37,102)(31,47,38,103)(32,48,39,104)(33,49,40,105)(34,43,41,99)(35,44,42,100)(50,68,59,75)(51,69,60,76)(52,70,61,77)(53,64,62,71)(54,65,63,72)(55,66,57,73)(56,67,58,74), (1,75,78,68)(2,76,79,69)(3,77,80,70)(4,71,81,64)(5,72,82,65)(6,73,83,66)(7,74,84,67)(8,42,25,35)(9,36,26,29)(10,37,27,30)(11,38,28,31)(12,39,22,32)(13,40,23,33)(14,41,24,34)(15,100,109,44)(16,101,110,45)(17,102,111,46)(18,103,112,47)(19,104,106,48)(20,105,107,49)(21,99,108,43)(50,87,59,95)(51,88,60,96)(52,89,61,97)(53,90,62,98)(54,91,63,92)(55,85,57,93)(56,86,58,94), (8,109,44)(9,110,45)(10,111,46)(11,112,47)(12,106,48)(13,107,49)(14,108,43)(15,100,25)(16,101,26)(17,102,27)(18,103,28)(19,104,22)(20,105,23)(21,99,24)(50,87,75)(51,88,76)(52,89,77)(53,90,71)(54,91,72)(55,85,73)(56,86,74)(57,93,66)(58,94,67)(59,95,68)(60,96,69)(61,97,70)(62,98,64)(63,92,65) );
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)], [(1,39,78,32),(2,40,79,33),(3,41,80,34),(4,42,81,35),(5,36,82,29),(6,37,83,30),(7,38,84,31),(8,71,25,64),(9,72,26,65),(10,73,27,66),(11,74,28,67),(12,75,22,68),(13,76,23,69),(14,77,24,70),(15,62,109,53),(16,63,110,54),(17,57,111,55),(18,58,112,56),(19,59,106,50),(20,60,107,51),(21,61,108,52),(43,89,99,97),(44,90,100,98),(45,91,101,92),(46,85,102,93),(47,86,103,94),(48,87,104,95),(49,88,105,96)], [(1,87,78,95),(2,88,79,96),(3,89,80,97),(4,90,81,98),(5,91,82,92),(6,85,83,93),(7,86,84,94),(8,109,25,15),(9,110,26,16),(10,111,27,17),(11,112,28,18),(12,106,22,19),(13,107,23,20),(14,108,24,21),(29,45,36,101),(30,46,37,102),(31,47,38,103),(32,48,39,104),(33,49,40,105),(34,43,41,99),(35,44,42,100),(50,68,59,75),(51,69,60,76),(52,70,61,77),(53,64,62,71),(54,65,63,72),(55,66,57,73),(56,67,58,74)], [(1,75,78,68),(2,76,79,69),(3,77,80,70),(4,71,81,64),(5,72,82,65),(6,73,83,66),(7,74,84,67),(8,42,25,35),(9,36,26,29),(10,37,27,30),(11,38,28,31),(12,39,22,32),(13,40,23,33),(14,41,24,34),(15,100,109,44),(16,101,110,45),(17,102,111,46),(18,103,112,47),(19,104,106,48),(20,105,107,49),(21,99,108,43),(50,87,59,95),(51,88,60,96),(52,89,61,97),(53,90,62,98),(54,91,63,92),(55,85,57,93),(56,86,58,94)], [(8,109,44),(9,110,45),(10,111,46),(11,112,47),(12,106,48),(13,107,49),(14,108,43),(15,100,25),(16,101,26),(17,102,27),(18,103,28),(19,104,22),(20,105,23),(21,99,24),(50,87,75),(51,88,76),(52,89,77),(53,90,71),(54,91,72),(55,85,73),(56,86,74),(57,93,66),(58,94,67),(59,95,68),(60,96,69),(61,97,70),(62,98,64),(63,92,65)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 14A | ··· | 14F | 14G | ··· | 14L | 21A | ··· | 21L | 28A | ··· | 28L | 28M | ··· | 28R | 42A | ··· | 42L | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 6 | 4 | 4 | 1 | 1 | 6 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | C4.A4 | C7×C4.A4 | A4 | C2×A4 | C7×A4 | A4×C14 |
| kernel | C7×C4.A4 | C7×SL2(𝔽3) | C7×C4○D4 | C7×Q8 | C4.A4 | SL2(𝔽3) | C4○D4 | Q8 | C7 | C1 | C28 | C14 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 6 | 36 | 1 | 1 | 6 | 6 |
Matrix representation of C7×C4.A4 ►in GL2(𝔽29) generated by
| 20 | 0 |
| 0 | 20 |
| 12 | 0 |
| 0 | 12 |
| 17 | 0 |
| 16 | 12 |
| 12 | 18 |
| 0 | 17 |
| 28 | 12 |
| 12 | 0 |
G:=sub<GL(2,GF(29))| [20,0,0,20],[12,0,0,12],[17,16,0,12],[12,0,18,17],[28,12,12,0] >;
C7×C4.A4 in GAP, Magma, Sage, TeX
C_7\times C_4.A_4
% in TeX
G:=Group("C7xC4.A4"); // GroupNames label
G:=SmallGroup(336,170);
// by ID
G=gap.SmallGroup(336,170);
# by ID
G:=PCGroup([6,-2,-3,-7,-2,2,-2,1008,1017,117,1900,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=e^3=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations
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