direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.10D4, C28.59D4, M4(2).1C14, (C2×C4).C28, (C2×C28).2C4, C4.10(C7×D4), (C2×Q8).1C14, (Q8×C14).6C2, C22.4(C2×C28), (C2×C28).60C22, (C7×M4(2)).3C2, C14.23(C22⋊C4), (C2×C4).2(C2×C14), C2.5(C7×C22⋊C4), (C2×C14).21(C2×C4), SmallGroup(224,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.10D4
G = < a,b,c,d | a7=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C7×C4.10D4
►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 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(74 80 78 76)(81 83 85 87)(82 88 86 84)(89 91 93 95)(90 96 94 92)(97 99 101 103)(98 104 102 100)(105 107 109 111)(106 112 110 108)
(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)
(1 100 7 102 5 104 3 98)(2 101 4 99 6 97 8 103)(9 47 11 45 13 43 15 41)(10 44 16 46 14 48 12 42)(17 50 23 52 21 54 19 56)(18 51 20 49 22 55 24 53)(25 62 31 64 29 58 27 60)(26 63 28 61 30 59 32 57)(33 82 39 84 37 86 35 88)(34 83 36 81 38 87 40 85)(65 94 71 96 69 90 67 92)(66 95 68 93 70 91 72 89)(73 106 79 108 77 110 75 112)(74 107 76 105 78 111 80 109)
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), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76)(81,83,85,87)(82,88,86,84)(89,91,93,95)(90,96,94,92)(97,99,101,103)(98,104,102,100)(105,107,109,111)(106,112,110,108), (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), (1,100,7,102,5,104,3,98)(2,101,4,99,6,97,8,103)(9,47,11,45,13,43,15,41)(10,44,16,46,14,48,12,42)(17,50,23,52,21,54,19,56)(18,51,20,49,22,55,24,53)(25,62,31,64,29,58,27,60)(26,63,28,61,30,59,32,57)(33,82,39,84,37,86,35,88)(34,83,36,81,38,87,40,85)(65,94,71,96,69,90,67,92)(66,95,68,93,70,91,72,89)(73,106,79,108,77,110,75,112)(74,107,76,105,78,111,80,109)>;
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), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76)(81,83,85,87)(82,88,86,84)(89,91,93,95)(90,96,94,92)(97,99,101,103)(98,104,102,100)(105,107,109,111)(106,112,110,108), (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), (1,100,7,102,5,104,3,98)(2,101,4,99,6,97,8,103)(9,47,11,45,13,43,15,41)(10,44,16,46,14,48,12,42)(17,50,23,52,21,54,19,56)(18,51,20,49,22,55,24,53)(25,62,31,64,29,58,27,60)(26,63,28,61,30,59,32,57)(33,82,39,84,37,86,35,88)(34,83,36,81,38,87,40,85)(65,94,71,96,69,90,67,92)(66,95,68,93,70,91,72,89)(73,106,79,108,77,110,75,112)(74,107,76,105,78,111,80,109) );
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)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20),(25,27,29,31),(26,32,30,28),(33,35,37,39),(34,40,38,36),(41,43,45,47),(42,48,46,44),(49,51,53,55),(50,56,54,52),(57,59,61,63),(58,64,62,60),(65,67,69,71),(66,72,70,68),(73,75,77,79),(74,80,78,76),(81,83,85,87),(82,88,86,84),(89,91,93,95),(90,96,94,92),(97,99,101,103),(98,104,102,100),(105,107,109,111),(106,112,110,108)], [(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)], [(1,100,7,102,5,104,3,98),(2,101,4,99,6,97,8,103),(9,47,11,45,13,43,15,41),(10,44,16,46,14,48,12,42),(17,50,23,52,21,54,19,56),(18,51,20,49,22,55,24,53),(25,62,31,64,29,58,27,60),(26,63,28,61,30,59,32,57),(33,82,39,84,37,86,35,88),(34,83,36,81,38,87,40,85),(65,94,71,96,69,90,67,92),(66,95,68,93,70,91,72,89),(73,106,79,108,77,110,75,112),(74,107,76,105,78,111,80,109)]])
C7×C4.10D4 is a maximal subgroup of
(C2×C4).D28 (C2×Q8).D14 M4(2).21D14 D28.4D4 D28.5D4 D28.6D4 D28.7D4
77 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14L | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | C7×D4 | C4.10D4 | C7×C4.10D4 |
| kernel | C7×C4.10D4 | C7×M4(2) | Q8×C14 | C2×C28 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | C28 | C4 | C7 | C1 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 2 | 12 | 1 | 6 |
Matrix representation of C7×C4.10D4 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 108 | 103 | 0 | 112 |
| 10 | 108 | 1 | 0 |
| 108 | 10 | 111 | 0 |
| 0 | 98 | 0 | 111 |
| 69 | 69 | 5 | 103 |
| 57 | 56 | 0 | 15 |
| 55 | 6 | 22 | 23 |
| 1 | 59 | 23 | 91 |
| 3 | 3 | 60 | 2 |
| 111 | 100 | 110 | 52 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[0,112,108,10,1,0,103,108,0,0,0,1,0,0,112,0],[108,0,69,57,10,98,69,56,111,0,5,0,0,111,103,15],[55,1,3,111,6,59,3,100,22,23,60,110,23,91,2,52] >;
C7×C4.10D4 in GAP, Magma, Sage, TeX
C_7\times C_4._{10}D_4 % in TeX
G:=Group("C7xC4.10D4"); // GroupNames label
G:=SmallGroup(224,50);
// by ID
G=gap.SmallGroup(224,50);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,679,3363,2530,88]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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