direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4.4D4, C42⋊5C14, C28.39D4, C4.4(C7×D4), (C4×C28)⋊12C2, (Q8×C14)⋊9C2, (C2×Q8)⋊2C14, C2.8(D4×C14), C22⋊C4⋊5C14, (C2×D4).5C14, C14.71(C2×D4), (D4×C14).12C2, C23.2(C2×C14), C14.44(C4○D4), (C2×C28).66C22, (C2×C14).79C23, (C22×C14).2C22, C22.14(C22×C14), C2.7(C7×C4○D4), (C2×C4).6(C2×C14), (C7×C22⋊C4)⋊13C2, SmallGroup(224,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.4D4
G = < a,b,c,d | a7=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C4.4D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C7×C4.4D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4.4D4, C7×D4, C22×C14, D4×C14, C7×C4○D4, C7×C4.4D4
►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 24 47 19)(2 25 48 20)(3 26 49 21)(4 27 43 15)(5 28 44 16)(6 22 45 17)(7 23 46 18)(8 34 112 38)(9 35 106 39)(10 29 107 40)(11 30 108 41)(12 31 109 42)(13 32 110 36)(14 33 111 37)(50 103 63 94)(51 104 57 95)(52 105 58 96)(53 99 59 97)(54 100 60 98)(55 101 61 92)(56 102 62 93)(64 89 73 83)(65 90 74 84)(66 91 75 78)(67 85 76 79)(68 86 77 80)(69 87 71 81)(70 88 72 82)
(1 94 35 91)(2 95 29 85)(3 96 30 86)(4 97 31 87)(5 98 32 88)(6 92 33 89)(7 93 34 90)(8 65 18 62)(9 66 19 63)(10 67 20 57)(11 68 21 58)(12 69 15 59)(13 70 16 60)(14 64 17 61)(22 55 111 73)(23 56 112 74)(24 50 106 75)(25 51 107 76)(26 52 108 77)(27 53 109 71)(28 54 110 72)(36 82 44 100)(37 83 45 101)(38 84 46 102)(39 78 47 103)(40 79 48 104)(41 80 49 105)(42 81 43 99)
(1 75 47 66)(2 76 48 67)(3 77 49 68)(4 71 43 69)(5 72 44 70)(6 73 45 64)(7 74 46 65)(8 102 112 93)(9 103 106 94)(10 104 107 95)(11 105 108 96)(12 99 109 97)(13 100 110 98)(14 101 111 92)(15 81 27 87)(16 82 28 88)(17 83 22 89)(18 84 23 90)(19 78 24 91)(20 79 25 85)(21 80 26 86)(29 51 40 57)(30 52 41 58)(31 53 42 59)(32 54 36 60)(33 55 37 61)(34 56 38 62)(35 50 39 63)
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), (1,24,47,19)(2,25,48,20)(3,26,49,21)(4,27,43,15)(5,28,44,16)(6,22,45,17)(7,23,46,18)(8,34,112,38)(9,35,106,39)(10,29,107,40)(11,30,108,41)(12,31,109,42)(13,32,110,36)(14,33,111,37)(50,103,63,94)(51,104,57,95)(52,105,58,96)(53,99,59,97)(54,100,60,98)(55,101,61,92)(56,102,62,93)(64,89,73,83)(65,90,74,84)(66,91,75,78)(67,85,76,79)(68,86,77,80)(69,87,71,81)(70,88,72,82), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,65,18,62)(9,66,19,63)(10,67,20,57)(11,68,21,58)(12,69,15,59)(13,70,16,60)(14,64,17,61)(22,55,111,73)(23,56,112,74)(24,50,106,75)(25,51,107,76)(26,52,108,77)(27,53,109,71)(28,54,110,72)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99), (1,75,47,66)(2,76,48,67)(3,77,49,68)(4,71,43,69)(5,72,44,70)(6,73,45,64)(7,74,46,65)(8,102,112,93)(9,103,106,94)(10,104,107,95)(11,105,108,96)(12,99,109,97)(13,100,110,98)(14,101,111,92)(15,81,27,87)(16,82,28,88)(17,83,22,89)(18,84,23,90)(19,78,24,91)(20,79,25,85)(21,80,26,86)(29,51,40,57)(30,52,41,58)(31,53,42,59)(32,54,36,60)(33,55,37,61)(34,56,38,62)(35,50,39,63)>;
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), (1,24,47,19)(2,25,48,20)(3,26,49,21)(4,27,43,15)(5,28,44,16)(6,22,45,17)(7,23,46,18)(8,34,112,38)(9,35,106,39)(10,29,107,40)(11,30,108,41)(12,31,109,42)(13,32,110,36)(14,33,111,37)(50,103,63,94)(51,104,57,95)(52,105,58,96)(53,99,59,97)(54,100,60,98)(55,101,61,92)(56,102,62,93)(64,89,73,83)(65,90,74,84)(66,91,75,78)(67,85,76,79)(68,86,77,80)(69,87,71,81)(70,88,72,82), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,65,18,62)(9,66,19,63)(10,67,20,57)(11,68,21,58)(12,69,15,59)(13,70,16,60)(14,64,17,61)(22,55,111,73)(23,56,112,74)(24,50,106,75)(25,51,107,76)(26,52,108,77)(27,53,109,71)(28,54,110,72)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99), (1,75,47,66)(2,76,48,67)(3,77,49,68)(4,71,43,69)(5,72,44,70)(6,73,45,64)(7,74,46,65)(8,102,112,93)(9,103,106,94)(10,104,107,95)(11,105,108,96)(12,99,109,97)(13,100,110,98)(14,101,111,92)(15,81,27,87)(16,82,28,88)(17,83,22,89)(18,84,23,90)(19,78,24,91)(20,79,25,85)(21,80,26,86)(29,51,40,57)(30,52,41,58)(31,53,42,59)(32,54,36,60)(33,55,37,61)(34,56,38,62)(35,50,39,63) );
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)], [(1,24,47,19),(2,25,48,20),(3,26,49,21),(4,27,43,15),(5,28,44,16),(6,22,45,17),(7,23,46,18),(8,34,112,38),(9,35,106,39),(10,29,107,40),(11,30,108,41),(12,31,109,42),(13,32,110,36),(14,33,111,37),(50,103,63,94),(51,104,57,95),(52,105,58,96),(53,99,59,97),(54,100,60,98),(55,101,61,92),(56,102,62,93),(64,89,73,83),(65,90,74,84),(66,91,75,78),(67,85,76,79),(68,86,77,80),(69,87,71,81),(70,88,72,82)], [(1,94,35,91),(2,95,29,85),(3,96,30,86),(4,97,31,87),(5,98,32,88),(6,92,33,89),(7,93,34,90),(8,65,18,62),(9,66,19,63),(10,67,20,57),(11,68,21,58),(12,69,15,59),(13,70,16,60),(14,64,17,61),(22,55,111,73),(23,56,112,74),(24,50,106,75),(25,51,107,76),(26,52,108,77),(27,53,109,71),(28,54,110,72),(36,82,44,100),(37,83,45,101),(38,84,46,102),(39,78,47,103),(40,79,48,104),(41,80,49,105),(42,81,43,99)], [(1,75,47,66),(2,76,48,67),(3,77,49,68),(4,71,43,69),(5,72,44,70),(6,73,45,64),(7,74,46,65),(8,102,112,93),(9,103,106,94),(10,104,107,95),(11,105,108,96),(12,99,109,97),(13,100,110,98),(14,101,111,92),(15,81,27,87),(16,82,28,88),(17,83,22,89),(18,84,23,90),(19,78,24,91),(20,79,25,85),(21,80,26,86),(29,51,40,57),(30,52,41,58),(31,53,42,59),(32,54,36,60),(33,55,37,61),(34,56,38,62),(35,50,39,63)]])
C7×C4.4D4 is a maximal subgroup of
C42.7D14 C42⋊2Dic7 C42.Dic7 C42.61D14 C42.62D14 C42.213D14 D28.23D4 C42.64D14 C42.214D14 C42.65D14 C42⋊5D14 D28.14D4 C42.233D14 C42.137D14 C42.138D14 C42.139D14 C42.140D14 C42⋊18D14 C42.141D14 D28⋊10D4 Dic14⋊10D4 C42⋊20D14 C42⋊21D14 C42.234D14 C42.143D14 C42.144D14 C42⋊22D14 C42.145D14
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28AJ | 28AK | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C4.4D4 | C4×C28 | C7×C22⋊C4 | D4×C14 | Q8×C14 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C28 | C14 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 6 | 6 | 24 | 6 | 6 | 2 | 4 | 12 | 24 |
Matrix representation of C7×C4.4D4 ►in GL4(𝔽29) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 17 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 12 |
| 28 | 0 | 0 | 0 |
| 27 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 17 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 1 | 28 |
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,16,0,0,0,0,16],[17,5,0,0,0,12,0,0,0,0,17,0,0,0,24,12],[28,27,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,17,17,0,0,0,0,1,1,0,0,27,28] >;
C7×C4.4D4 in GAP, Magma, Sage, TeX
C_7\times C_4._4D_4
% in TeX
G:=Group("C7xC4.4D4"); // GroupNames label
G:=SmallGroup(224,159);
// by ID
G=gap.SmallGroup(224,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,679,2090,266]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations