direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C22⋊C4, C23⋊3C28, C24.2C14, C2.1(D4×C14), C22⋊2(C2×C28), (C22×C14)⋊3C4, (C22×C4)⋊1C14, (C22×C28)⋊3C2, (C2×C14).50D4, C14.64(C2×D4), (C2×C28)⋊11C22, C23.5(C2×C14), C2.1(C22×C28), (C23×C14).1C2, C22.12(C7×D4), C14.29(C22×C4), (C2×C14).70C23, C22.4(C22×C14), (C22×C14).24C22, (C2×C4)⋊3(C2×C14), (C2×C14)⋊7(C2×C4), SmallGroup(224,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C22⋊C4
G = < a,b,c,d | a14=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 188 in 132 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C7×C22⋊C4, C22×C28, C23×C14, C14×C22⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C2×C28, C7×D4, C22×C14, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4
►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 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 43)(13 44)(14 45)(15 67)(16 68)(17 69)(18 70)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 15)(12 16)(13 17)(14 18)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 68)(44 69)(45 70)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 99)(92 100)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)
(1 96 46 83)(2 97 47 84)(3 98 48 71)(4 85 49 72)(5 86 50 73)(6 87 51 74)(7 88 52 75)(8 89 53 76)(9 90 54 77)(10 91 55 78)(11 92 56 79)(12 93 43 80)(13 94 44 81)(14 95 45 82)(15 100 67 30)(16 101 68 31)(17 102 69 32)(18 103 70 33)(19 104 57 34)(20 105 58 35)(21 106 59 36)(22 107 60 37)(23 108 61 38)(24 109 62 39)(25 110 63 40)(26 111 64 41)(27 112 65 42)(28 99 66 29)
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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,43)(13,44)(14,45)(15,67)(16,68)(17,69)(18,70)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,15)(12,16)(13,17)(14,18)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,68)(44,69)(45,70)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106), (1,96,46,83)(2,97,47,84)(3,98,48,71)(4,85,49,72)(5,86,50,73)(6,87,51,74)(7,88,52,75)(8,89,53,76)(9,90,54,77)(10,91,55,78)(11,92,56,79)(12,93,43,80)(13,94,44,81)(14,95,45,82)(15,100,67,30)(16,101,68,31)(17,102,69,32)(18,103,70,33)(19,104,57,34)(20,105,58,35)(21,106,59,36)(22,107,60,37)(23,108,61,38)(24,109,62,39)(25,110,63,40)(26,111,64,41)(27,112,65,42)(28,99,66,29)>;
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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,43)(13,44)(14,45)(15,67)(16,68)(17,69)(18,70)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,15)(12,16)(13,17)(14,18)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,68)(44,69)(45,70)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106), (1,96,46,83)(2,97,47,84)(3,98,48,71)(4,85,49,72)(5,86,50,73)(6,87,51,74)(7,88,52,75)(8,89,53,76)(9,90,54,77)(10,91,55,78)(11,92,56,79)(12,93,43,80)(13,94,44,81)(14,95,45,82)(15,100,67,30)(16,101,68,31)(17,102,69,32)(18,103,70,33)(19,104,57,34)(20,105,58,35)(21,106,59,36)(22,107,60,37)(23,108,61,38)(24,109,62,39)(25,110,63,40)(26,111,64,41)(27,112,65,42)(28,99,66,29) );
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)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,43),(13,44),(14,45),(15,67),(16,68),(17,69),(18,70),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,15),(12,16),(13,17),(14,18),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,68),(44,69),(45,70),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,99),(92,100),(93,101),(94,102),(95,103),(96,104),(97,105),(98,106)], [(1,96,46,83),(2,97,47,84),(3,98,48,71),(4,85,49,72),(5,86,50,73),(6,87,51,74),(7,88,52,75),(8,89,53,76),(9,90,54,77),(10,91,55,78),(11,92,56,79),(12,93,43,80),(13,94,44,81),(14,95,45,82),(15,100,67,30),(16,101,68,31),(17,102,69,32),(18,103,70,33),(19,104,57,34),(20,105,58,35),(21,106,59,36),(22,107,60,37),(23,108,61,38),(24,109,62,39),(25,110,63,40),(26,111,64,41),(27,112,65,42),(28,99,66,29)]])
C14×C22⋊C4 is a maximal subgroup of
C24.Dic7 C24.D14 C24.2D14 C24.44D14 C23.42D28 C24.3D14 C24.4D14 C24.46D14 C23⋊Dic14 C24.6D14 C24.7D14 C24.47D14 C24.8D14 C24.9D14 C24.10D14 C23.44D28 C24.12D14 C24.13D14 C23.45D28 C24.14D14 C23⋊2D28 C23.16D28 C23⋊2Dic14 C24.24D14 C24.27D14 C23⋊3D28 C24.30D14 C24.31D14 D4×C2×C28
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BN | 28A | ··· | 28AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D4 | C7×D4 |
| kernel | C14×C22⋊C4 | C7×C22⋊C4 | C22×C28 | C23×C14 | C22×C14 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C2×C14 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 6 | 24 | 12 | 6 | 48 | 4 | 24 |
Matrix representation of C14×C22⋊C4 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,23,0,0,0,0,23],[28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,17,0,0,0,0,0,2,0,0,14,0] >;
C14×C22⋊C4 in GAP, Magma, Sage, TeX
C_{14}\times C_2^2\rtimes C_4 % in TeX
G:=Group("C14xC2^2:C4"); // GroupNames label
G:=SmallGroup(224,150);
// by ID
G=gap.SmallGroup(224,150);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations