direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4⋊C4, C4⋊C28, C28⋊3C4, C14.3Q8, C14.13D4, C2.(C7×Q8), C2.2(C7×D4), (C2×C4).1C14, C2.2(C2×C28), (C2×C28).2C2, C14.11(C2×C4), C22.3(C2×C14), (C2×C14).14C22, SmallGroup(112,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4⋊C4
G = < a,b,c | a7=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C4⋊C4
►Regular action 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 67 47 63)(2 68 48 57)(3 69 49 58)(4 70 43 59)(5 64 44 60)(6 65 45 61)(7 66 46 62)(8 92 110 88)(9 93 111 89)(10 94 112 90)(11 95 106 91)(12 96 107 85)(13 97 108 86)(14 98 109 87)(15 83 22 100)(16 84 23 101)(17 78 24 102)(18 79 25 103)(19 80 26 104)(20 81 27 105)(21 82 28 99)(29 51 40 75)(30 52 41 76)(31 53 42 77)(32 54 36 71)(33 55 37 72)(34 56 38 73)(35 50 39 74)
(1 91 35 102)(2 85 29 103)(3 86 30 104)(4 87 31 105)(5 88 32 99)(6 89 33 100)(7 90 34 101)(8 71 21 60)(9 72 15 61)(10 73 16 62)(11 74 17 63)(12 75 18 57)(13 76 19 58)(14 77 20 59)(22 65 111 55)(23 66 112 56)(24 67 106 50)(25 68 107 51)(26 69 108 52)(27 70 109 53)(28 64 110 54)(36 82 44 92)(37 83 45 93)(38 84 46 94)(39 78 47 95)(40 79 48 96)(41 80 49 97)(42 81 43 98)
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), (1,67,47,63)(2,68,48,57)(3,69,49,58)(4,70,43,59)(5,64,44,60)(6,65,45,61)(7,66,46,62)(8,92,110,88)(9,93,111,89)(10,94,112,90)(11,95,106,91)(12,96,107,85)(13,97,108,86)(14,98,109,87)(15,83,22,100)(16,84,23,101)(17,78,24,102)(18,79,25,103)(19,80,26,104)(20,81,27,105)(21,82,28,99)(29,51,40,75)(30,52,41,76)(31,53,42,77)(32,54,36,71)(33,55,37,72)(34,56,38,73)(35,50,39,74), (1,91,35,102)(2,85,29,103)(3,86,30,104)(4,87,31,105)(5,88,32,99)(6,89,33,100)(7,90,34,101)(8,71,21,60)(9,72,15,61)(10,73,16,62)(11,74,17,63)(12,75,18,57)(13,76,19,58)(14,77,20,59)(22,65,111,55)(23,66,112,56)(24,67,106,50)(25,68,107,51)(26,69,108,52)(27,70,109,53)(28,64,110,54)(36,82,44,92)(37,83,45,93)(38,84,46,94)(39,78,47,95)(40,79,48,96)(41,80,49,97)(42,81,43,98)>;
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), (1,67,47,63)(2,68,48,57)(3,69,49,58)(4,70,43,59)(5,64,44,60)(6,65,45,61)(7,66,46,62)(8,92,110,88)(9,93,111,89)(10,94,112,90)(11,95,106,91)(12,96,107,85)(13,97,108,86)(14,98,109,87)(15,83,22,100)(16,84,23,101)(17,78,24,102)(18,79,25,103)(19,80,26,104)(20,81,27,105)(21,82,28,99)(29,51,40,75)(30,52,41,76)(31,53,42,77)(32,54,36,71)(33,55,37,72)(34,56,38,73)(35,50,39,74), (1,91,35,102)(2,85,29,103)(3,86,30,104)(4,87,31,105)(5,88,32,99)(6,89,33,100)(7,90,34,101)(8,71,21,60)(9,72,15,61)(10,73,16,62)(11,74,17,63)(12,75,18,57)(13,76,19,58)(14,77,20,59)(22,65,111,55)(23,66,112,56)(24,67,106,50)(25,68,107,51)(26,69,108,52)(27,70,109,53)(28,64,110,54)(36,82,44,92)(37,83,45,93)(38,84,46,94)(39,78,47,95)(40,79,48,96)(41,80,49,97)(42,81,43,98) );
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)], [(1,67,47,63),(2,68,48,57),(3,69,49,58),(4,70,43,59),(5,64,44,60),(6,65,45,61),(7,66,46,62),(8,92,110,88),(9,93,111,89),(10,94,112,90),(11,95,106,91),(12,96,107,85),(13,97,108,86),(14,98,109,87),(15,83,22,100),(16,84,23,101),(17,78,24,102),(18,79,25,103),(19,80,26,104),(20,81,27,105),(21,82,28,99),(29,51,40,75),(30,52,41,76),(31,53,42,77),(32,54,36,71),(33,55,37,72),(34,56,38,73),(35,50,39,74)], [(1,91,35,102),(2,85,29,103),(3,86,30,104),(4,87,31,105),(5,88,32,99),(6,89,33,100),(7,90,34,101),(8,71,21,60),(9,72,15,61),(10,73,16,62),(11,74,17,63),(12,75,18,57),(13,76,19,58),(14,77,20,59),(22,65,111,55),(23,66,112,56),(24,67,106,50),(25,68,107,51),(26,69,108,52),(27,70,109,53),(28,64,110,54),(36,82,44,92),(37,83,45,93),(38,84,46,94),(39,78,47,95),(40,79,48,96),(41,80,49,97),(42,81,43,98)]])
C7×C4⋊C4 is a maximal subgroup of
C28.Q8 C4.Dic14 C14.D8 C14.Q16 Dic7⋊3Q8 C28⋊Q8 Dic7.Q8 C28.3Q8 C4⋊C4⋊7D7 D28⋊C4 D14.5D4 C4⋊D28 D14⋊Q8 D14⋊2Q8 C4⋊C4⋊D7 D4×C28 Q8×C28
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 7A | ··· | 7F | 14A | ··· | 14R | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | D4 | Q8 | C7×D4 | C7×Q8 |
| kernel | C7×C4⋊C4 | C2×C28 | C28 | C4⋊C4 | C2×C4 | C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 3 | 4 | 6 | 18 | 24 | 1 | 1 | 6 | 6 |
Matrix representation of C7×C4⋊C4 ►in GL3(𝔽29) generated by
| 1 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 24 |
| 28 | 0 | 0 |
| 0 | 0 | 28 |
| 0 | 1 | 0 |
| 17 | 0 | 0 |
| 0 | 27 | 16 |
| 0 | 16 | 2 |
G:=sub<GL(3,GF(29))| [1,0,0,0,24,0,0,0,24],[28,0,0,0,0,1,0,28,0],[17,0,0,0,27,16,0,16,2] >;
C7×C4⋊C4 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes C_4
% in TeX
G:=Group("C7xC4:C4"); // GroupNames label
G:=SmallGroup(112,21);
// by ID
G=gap.SmallGroup(112,21);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-2,280,301,146]);
// Polycyclic
G:=Group<a,b,c|a^7=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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