metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊5D4, C23⋊1D14, C7⋊2C22≀C2, (C2×D4)⋊3D7, (C2×C4)⋊2D14, (C2×C14)⋊2D4, (D4×C14)⋊8C2, C2.25(D4×D7), D14⋊C4⋊14C2, (C2×C28)⋊7C22, C14.49(C2×D4), (C23×D7)⋊2C2, C22⋊2(C7⋊D4), C23.D7⋊10C2, (C2×C14).52C23, (C22×C14)⋊3C22, (C2×Dic7)⋊2C22, C22.59(C22×D7), (C22×D7).25C22, (C2×C7⋊D4)⋊4C2, C2.13(C2×C7⋊D4), SmallGroup(224,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊D14
G = < a,b,c,d,e | a2=b2=c2=d14=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 638 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22≀C2, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, D14⋊C4, C23.D7, C2×C7⋊D4, D4×C14, C23×D7, C23⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4×D7, C2×C7⋊D4, C23⋊D14
►On 56 points
(1 56)(2 27)(3 44)(4 15)(5 46)(6 17)(7 48)(8 19)(9 50)(10 21)(11 52)(12 23)(13 54)(14 25)(16 40)(18 42)(20 30)(22 32)(24 34)(26 36)(28 38)(29 49)(31 51)(33 53)(35 55)(37 43)(39 45)(41 47)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 43)(28 44)
(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)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 45)(16 44)(17 43)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)
G:=sub<Sym(56)| (1,56)(2,27)(3,44)(4,15)(5,46)(6,17)(7,48)(8,19)(9,50)(10,21)(11,52)(12,23)(13,54)(14,25)(16,40)(18,42)(20,30)(22,32)(24,34)(26,36)(28,38)(29,49)(31,51)(33,53)(35,55)(37,43)(39,45)(41,47), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,45)(16,44)(17,43)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)>;
G:=Group( (1,56)(2,27)(3,44)(4,15)(5,46)(6,17)(7,48)(8,19)(9,50)(10,21)(11,52)(12,23)(13,54)(14,25)(16,40)(18,42)(20,30)(22,32)(24,34)(26,36)(28,38)(29,49)(31,51)(33,53)(35,55)(37,43)(39,45)(41,47), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,45)(16,44)(17,43)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46) );
G=PermutationGroup([[(1,56),(2,27),(3,44),(4,15),(5,46),(6,17),(7,48),(8,19),(9,50),(10,21),(11,52),(12,23),(13,54),(14,25),(16,40),(18,42),(20,30),(22,32),(24,34),(26,36),(28,38),(29,49),(31,51),(33,53),(35,55),(37,43),(39,45),(41,47)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,43),(28,44)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,45),(16,44),(17,43),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,49),(26,48),(27,47),(28,46)]])
C23⋊D14 is a maximal subgroup of
C7⋊C2≀C4 C23.3D28 C23⋊D28 2+ 1+4⋊2D7 C42⋊12D14 D28⋊23D4 C42⋊16D14 C42⋊17D14 D7×C22≀C2 C24⋊2D14 C24⋊3D14 C24.33D14 C24.34D14 C24⋊4D14 C14.372+ 1+4 C4⋊C4⋊21D14 C14.382+ 1+4 D28⋊19D4 C14.402+ 1+4 D28⋊20D4 C14.422+ 1+4 C14.462+ 1+4 C14.482+ 1+4 C14.1202+ 1+4 C4⋊C4⋊28D14 C14.612+ 1+4 C14.1222+ 1+4 C14.622+ 1+4 C14.682+ 1+4 C42⋊20D14 C42⋊21D14 C42⋊22D14 C42⋊26D14 D28⋊11D4 C42⋊28D14 D4×C7⋊D4 C24⋊7D14 (C2×C28)⋊15D4 C14.1452+ 1+4 C14.1462+ 1+4
C23⋊D14 is a maximal quotient of
C24.46D14 C23⋊Dic14 C23.44D28 C24.12D14 C24.14D14 C23⋊2D28 (C2×C4)⋊Dic14 D14⋊C4⋊6C4 (C2×C4)⋊3D28 C24⋊D14 D28⋊16D4 D28⋊17D4 Dic14⋊17D4 D28.36D4 D28.37D4 Dic14.37D4 C22⋊C4⋊D14 C42⋊5D14 D28.14D4 D28⋊5D4 D28.15D4 D28⋊D4 Dic14⋊D4 D14⋊6SD16 Dic14⋊7D4 D28⋊7D4 Dic14.16D4 D14⋊5Q16 D28.17D4 D28⋊18D4 D28.38D4 D28.39D4 D28.40D4 C24.18D14 C24.21D14
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | D4×D7 |
| kernel | C23⋊D14 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | D4×C14 | C23×D7 | D14 | C2×C14 | C2×D4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 12 | 6 |
Matrix representation of C23⋊D14 ►in GL4(𝔽29) generated by
| 9 | 16 | 0 | 0 |
| 24 | 20 | 0 | 0 |
| 0 | 0 | 28 | 28 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 10 | 8 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 2 | 1 |
| 22 | 26 | 0 | 0 |
| 16 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 28 |
G:=sub<GL(4,GF(29))| [9,24,0,0,16,20,0,0,0,0,28,0,0,0,28,1],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[10,12,0,0,8,1,0,0,0,0,28,2,0,0,0,1],[22,16,0,0,26,7,0,0,0,0,1,27,0,0,0,28] >;
C23⋊D14 in GAP, Magma, Sage, TeX
C_2^3\rtimes D_{14} % in TeX
G:=Group("C2^3:D14"); // GroupNames label
G:=SmallGroup(224,132);
// by ID
G=gap.SmallGroup(224,132);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^14=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations