direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4.10C42, C28.33C42, (C2×C8).1C28, (C2×C56).4C4, C4.10(C4×C28), (C2×C28).278D4, C23.1(C7×Q8), (C22×C14).1Q8, (C2×M4(2)).5C14, C28.102(C22⋊C4), (C14×M4(2)).17C2, (C22×C28).387C22, C14.22(C2.C42), (C2×C4).9(C7×D4), C22.2(C7×C4⋊C4), (C2×C4).64(C2×C28), C4.18(C7×C22⋊C4), (C2×C14).19(C4⋊C4), (C2×C28).325(C2×C4), (C22×C4).17(C2×C14), C2.3(C7×C2.C42), SmallGroup(448,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.10C42
G = < a,b,c,d | a7=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 122 in 86 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C23, C14, C14, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×M4(2), C56, C2×C28, C22×C14, C4.10C42, C2×C56, C7×M4(2), C22×C28, C14×M4(2), C7×C4.10C42
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C2×C28, C7×D4, C7×Q8, C4.10C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×C4.10C42
►On 112 points
Generators in S112
(1 31 47 79 95 23 39)(2 32 48 80 96 24 40)(3 25 41 73 89 17 33)(4 26 42 74 90 18 34)(5 27 43 75 91 19 35)(6 28 44 76 92 20 36)(7 29 45 77 93 21 37)(8 30 46 78 94 22 38)(9 106 66 50 82 98 62)(10 107 67 51 83 99 63)(11 108 68 52 84 100 64)(12 109 69 53 85 101 57)(13 110 70 54 86 102 58)(14 111 71 55 87 103 59)(15 112 72 56 88 104 60)(16 105 65 49 81 97 61)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 79 77 75)(74 80 78 76)(81 83 85 87)(82 84 86 88)(89 95 93 91)(90 96 94 92)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)
(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 97 3 103 5 101 7 99)(2 104 8 98 6 100 4 102)(9 44 11 42 13 48 15 46)(10 47 16 41 14 43 12 45)(17 55 19 53 21 51 23 49)(18 54 24 56 22 50 20 52)(25 59 27 57 29 63 31 61)(26 58 32 60 30 62 28 64)(33 87 35 85 37 83 39 81)(34 86 40 88 38 82 36 84)(65 89 71 91 69 93 67 95)(66 92 68 90 70 96 72 94)(73 111 75 109 77 107 79 105)(74 110 80 112 78 106 76 108)
G:=sub<Sym(112)| (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,106,66,50,82,98,62)(10,107,67,51,83,99,63)(11,108,68,52,84,100,64)(12,109,69,53,85,101,57)(13,110,70,54,86,102,58)(14,111,71,55,87,103,59)(15,112,72,56,88,104,60)(16,105,65,49,81,97,61), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,79,77,75)(74,80,78,76)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112), (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), (1,97,3,103,5,101,7,99)(2,104,8,98,6,100,4,102)(9,44,11,42,13,48,15,46)(10,47,16,41,14,43,12,45)(17,55,19,53,21,51,23,49)(18,54,24,56,22,50,20,52)(25,59,27,57,29,63,31,61)(26,58,32,60,30,62,28,64)(33,87,35,85,37,83,39,81)(34,86,40,88,38,82,36,84)(65,89,71,91,69,93,67,95)(66,92,68,90,70,96,72,94)(73,111,75,109,77,107,79,105)(74,110,80,112,78,106,76,108)>;
G:=Group( (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,106,66,50,82,98,62)(10,107,67,51,83,99,63)(11,108,68,52,84,100,64)(12,109,69,53,85,101,57)(13,110,70,54,86,102,58)(14,111,71,55,87,103,59)(15,112,72,56,88,104,60)(16,105,65,49,81,97,61), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,79,77,75)(74,80,78,76)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112), (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), (1,97,3,103,5,101,7,99)(2,104,8,98,6,100,4,102)(9,44,11,42,13,48,15,46)(10,47,16,41,14,43,12,45)(17,55,19,53,21,51,23,49)(18,54,24,56,22,50,20,52)(25,59,27,57,29,63,31,61)(26,58,32,60,30,62,28,64)(33,87,35,85,37,83,39,81)(34,86,40,88,38,82,36,84)(65,89,71,91,69,93,67,95)(66,92,68,90,70,96,72,94)(73,111,75,109,77,107,79,105)(74,110,80,112,78,106,76,108) );
G=PermutationGroup([[(1,31,47,79,95,23,39),(2,32,48,80,96,24,40),(3,25,41,73,89,17,33),(4,26,42,74,90,18,34),(5,27,43,75,91,19,35),(6,28,44,76,92,20,36),(7,29,45,77,93,21,37),(8,30,46,78,94,22,38),(9,106,66,50,82,98,62),(10,107,67,51,83,99,63),(11,108,68,52,84,100,64),(12,109,69,53,85,101,57),(13,110,70,54,86,102,58),(14,111,71,55,87,103,59),(15,112,72,56,88,104,60),(16,105,65,49,81,97,61)], [(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,79,77,75),(74,80,78,76),(81,83,85,87),(82,84,86,88),(89,95,93,91),(90,96,94,92),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112)], [(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)], [(1,97,3,103,5,101,7,99),(2,104,8,98,6,100,4,102),(9,44,11,42,13,48,15,46),(10,47,16,41,14,43,12,45),(17,55,19,53,21,51,23,49),(18,54,24,56,22,50,20,52),(25,59,27,57,29,63,31,61),(26,58,32,60,30,62,28,64),(33,87,35,85,37,83,39,81),(34,86,40,88,38,82,36,84),(65,89,71,91,69,93,67,95),(66,92,68,90,70,96,72,94),(73,111,75,109,77,107,79,105),(74,110,80,112,78,106,76,108)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | ··· | 8L | 14A | ··· | 14F | 14G | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28AD | 56A | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | D4 | Q8 | C7×D4 | C7×Q8 | C4.10C42 | C7×C4.10C42 |
| kernel | C7×C4.10C42 | C14×M4(2) | C2×C56 | C4.10C42 | C2×M4(2) | C2×C8 | C2×C28 | C22×C14 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 3 | 12 | 6 | 18 | 72 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C7×C4.10C42 ►in GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 112 | 0 | 111 | 0 |
| 0 | 0 | 1 | 1 |
| 8 | 0 | 1 | 0 |
| 7 | 15 | 112 | 0 |
| 112 | 111 | 0 | 0 |
| 8 | 1 | 0 | 0 |
| 106 | 1 | 0 | 15 |
| 0 | 112 | 1 | 0 |
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[112,0,8,7,0,0,0,15,111,1,1,112,0,1,0,0],[112,8,106,0,111,1,1,112,0,0,0,1,0,0,15,0] >;
C7×C4.10C42 in GAP, Magma, Sage, TeX
C_7\times C_4._{10}C_4^2 % in TeX
G:=Group("C7xC4.10C4^2"); // GroupNames label
G:=SmallGroup(448,142);
// by ID
G=gap.SmallGroup(448,142);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,248,4911,172,14117,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations