direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C4○D4, C28.55C23, C14.18C24, D4⋊3(C2×C14), (C2×D4)⋊7C14, (C2×Q8)⋊6C14, Q8⋊3(C2×C14), (D4×C14)⋊16C2, (C22×C4)⋊6C14, (Q8×C14)⋊13C2, (C22×C28)⋊13C2, (C2×C28)⋊16C22, (C7×D4)⋊12C22, (C2×C14).6C23, C2.3(C23×C14), C4.8(C22×C14), (C7×Q8)⋊11C22, C23.11(C2×C14), C22.1(C22×C14), (C22×C14).30C22, (C2×C28)○(C7×D4), (C2×C28)○(C7×Q8), (C2×C4)⋊5(C2×C14), (C2×C28)○(Q8×C14), SmallGroup(224,192)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C4○D4
G = < a,b,c,d | a14=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C2×C14, C2×C14, C2×C4○D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C4○D4
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, C22×C14, C7×C4○D4, C23×C14, C14×C4○D4
►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 59 109 45)(2 60 110 46)(3 61 111 47)(4 62 112 48)(5 63 99 49)(6 64 100 50)(7 65 101 51)(8 66 102 52)(9 67 103 53)(10 68 104 54)(11 69 105 55)(12 70 106 56)(13 57 107 43)(14 58 108 44)(15 72 42 97)(16 73 29 98)(17 74 30 85)(18 75 31 86)(19 76 32 87)(20 77 33 88)(21 78 34 89)(22 79 35 90)(23 80 36 91)(24 81 37 92)(25 82 38 93)(26 83 39 94)(27 84 40 95)(28 71 41 96)
(1 32 109 19)(2 33 110 20)(3 34 111 21)(4 35 112 22)(5 36 99 23)(6 37 100 24)(7 38 101 25)(8 39 102 26)(9 40 103 27)(10 41 104 28)(11 42 105 15)(12 29 106 16)(13 30 107 17)(14 31 108 18)(43 74 57 85)(44 75 58 86)(45 76 59 87)(46 77 60 88)(47 78 61 89)(48 79 62 90)(49 80 63 91)(50 81 64 92)(51 82 65 93)(52 83 66 94)(53 84 67 95)(54 71 68 96)(55 72 69 97)(56 73 70 98)
(1 39)(2 40)(3 41)(4 42)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(43 81)(44 82)(45 83)(46 84)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)
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), (1,59,109,45)(2,60,110,46)(3,61,111,47)(4,62,112,48)(5,63,99,49)(6,64,100,50)(7,65,101,51)(8,66,102,52)(9,67,103,53)(10,68,104,54)(11,69,105,55)(12,70,106,56)(13,57,107,43)(14,58,108,44)(15,72,42,97)(16,73,29,98)(17,74,30,85)(18,75,31,86)(19,76,32,87)(20,77,33,88)(21,78,34,89)(22,79,35,90)(23,80,36,91)(24,81,37,92)(25,82,38,93)(26,83,39,94)(27,84,40,95)(28,71,41,96), (1,32,109,19)(2,33,110,20)(3,34,111,21)(4,35,112,22)(5,36,99,23)(6,37,100,24)(7,38,101,25)(8,39,102,26)(9,40,103,27)(10,41,104,28)(11,42,105,15)(12,29,106,16)(13,30,107,17)(14,31,108,18)(43,74,57,85)(44,75,58,86)(45,76,59,87)(46,77,60,88)(47,78,61,89)(48,79,62,90)(49,80,63,91)(50,81,64,92)(51,82,65,93)(52,83,66,94)(53,84,67,95)(54,71,68,96)(55,72,69,97)(56,73,70,98), (1,39)(2,40)(3,41)(4,42)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(43,81)(44,82)(45,83)(46,84)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)>;
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), (1,59,109,45)(2,60,110,46)(3,61,111,47)(4,62,112,48)(5,63,99,49)(6,64,100,50)(7,65,101,51)(8,66,102,52)(9,67,103,53)(10,68,104,54)(11,69,105,55)(12,70,106,56)(13,57,107,43)(14,58,108,44)(15,72,42,97)(16,73,29,98)(17,74,30,85)(18,75,31,86)(19,76,32,87)(20,77,33,88)(21,78,34,89)(22,79,35,90)(23,80,36,91)(24,81,37,92)(25,82,38,93)(26,83,39,94)(27,84,40,95)(28,71,41,96), (1,32,109,19)(2,33,110,20)(3,34,111,21)(4,35,112,22)(5,36,99,23)(6,37,100,24)(7,38,101,25)(8,39,102,26)(9,40,103,27)(10,41,104,28)(11,42,105,15)(12,29,106,16)(13,30,107,17)(14,31,108,18)(43,74,57,85)(44,75,58,86)(45,76,59,87)(46,77,60,88)(47,78,61,89)(48,79,62,90)(49,80,63,91)(50,81,64,92)(51,82,65,93)(52,83,66,94)(53,84,67,95)(54,71,68,96)(55,72,69,97)(56,73,70,98), (1,39)(2,40)(3,41)(4,42)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(43,81)(44,82)(45,83)(46,84)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91) );
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)], [(1,59,109,45),(2,60,110,46),(3,61,111,47),(4,62,112,48),(5,63,99,49),(6,64,100,50),(7,65,101,51),(8,66,102,52),(9,67,103,53),(10,68,104,54),(11,69,105,55),(12,70,106,56),(13,57,107,43),(14,58,108,44),(15,72,42,97),(16,73,29,98),(17,74,30,85),(18,75,31,86),(19,76,32,87),(20,77,33,88),(21,78,34,89),(22,79,35,90),(23,80,36,91),(24,81,37,92),(25,82,38,93),(26,83,39,94),(27,84,40,95),(28,71,41,96)], [(1,32,109,19),(2,33,110,20),(3,34,111,21),(4,35,112,22),(5,36,99,23),(6,37,100,24),(7,38,101,25),(8,39,102,26),(9,40,103,27),(10,41,104,28),(11,42,105,15),(12,29,106,16),(13,30,107,17),(14,31,108,18),(43,74,57,85),(44,75,58,86),(45,76,59,87),(46,77,60,88),(47,78,61,89),(48,79,62,90),(49,80,63,91),(50,81,64,92),(51,82,65,93),(52,83,66,94),(53,84,67,95),(54,71,68,96),(55,72,69,97),(56,73,70,98)], [(1,39),(2,40),(3,41),(4,42),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,112),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(43,81),(44,82),(45,83),(46,84),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91)]])
C14×C4○D4 is a maximal subgroup of
C4○D4⋊Dic7 C28.(C2×D4) (D4×C14).11C4 (D4×C14)⋊9C4 (D4×C14).16C4 (C7×D4)⋊14D4 (C7×D4).32D4 (D4×C14)⋊10C4 C28.76C24 C28.C24 C14.1042- 1+4 C14.1052- 1+4 C14.1062- 1+4 (C2×C28)⋊15D4 C14.1452+ 1+4 C14.1462+ 1+4 C14.1072- 1+4 (C2×C28)⋊17D4 C14.1082- 1+4 C14.1482+ 1+4 C14.C25
C14×C4○D4 is a maximal quotient of
D4×C2×C28 Q8×C2×C28
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 28A | ··· | 28X | 28Y | ··· | 28BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C4○D4 | C7×C4○D4 |
| kernel | C14×C4○D4 | C22×C28 | D4×C14 | Q8×C14 | C7×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C14 | C2 |
| # reps | 1 | 3 | 3 | 1 | 8 | 6 | 18 | 18 | 6 | 48 | 4 | 24 |
Matrix representation of C14×C4○D4 ►in GL3(𝔽29) generated by
| 28 | 0 | 0 |
| 0 | 20 | 0 |
| 0 | 0 | 20 |
| 1 | 0 | 0 |
| 0 | 17 | 0 |
| 0 | 0 | 17 |
| 28 | 0 | 0 |
| 0 | 27 | 1 |
| 0 | 24 | 2 |
| 28 | 0 | 0 |
| 0 | 2 | 28 |
| 0 | 3 | 27 |
G:=sub<GL(3,GF(29))| [28,0,0,0,20,0,0,0,20],[1,0,0,0,17,0,0,0,17],[28,0,0,0,27,24,0,1,2],[28,0,0,0,2,3,0,28,27] >;
C14×C4○D4 in GAP, Magma, Sage, TeX
C_{14}\times C_4\circ D_4 % in TeX
G:=Group("C14xC4oD4"); // GroupNames label
G:=SmallGroup(224,192);
// by ID
G=gap.SmallGroup(224,192);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-7,-2,1369,518]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations