metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊1D14, C7⋊2C2≀C22, (C2×C28)⋊2D4, C22≀C2⋊1D7, C22⋊C4⋊3D14, (C2×Dic7)⋊2D4, (C22×D7)⋊2D4, (C22×C14)⋊3D4, D4⋊6D14⋊3C2, (C2×D4).33D14, C24⋊D7⋊1C2, C23⋊1(C7⋊D4), C23⋊Dic7⋊5C2, C22.33(D4×D7), C14.43C22≀C2, (C23×C14)⋊7C22, C23.D7⋊4C22, (D4×C14).49C22, C23.1D14⋊5C2, C2.11(C23⋊D14), C23.74(C22×D7), (C22×C14).113C23, (C2×C4)⋊1(C7⋊D4), (C7×C22≀C2)⋊1C2, (C2×C14).30(C2×D4), (C2×C7⋊D4).5C22, C22.29(C2×C7⋊D4), (C7×C22⋊C4)⋊34C22, SmallGroup(448,566)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=abcd, bc=cb, ebe-1=fbf=bd=db, fcf=cd=dc, ce=ec, de=ed, df=fd, fef=e-1 >
Subgroups: 1164 in 198 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C23⋊C4, C22≀C2, C22≀C2, 2+ 1+4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C2≀C22, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, C2×C7⋊D4, D4×C14, D4×C14, C23×C14, C23.1D14, C23⋊Dic7, C24⋊D7, C7×C22≀C2, D4⋊6D14, C24⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, C2≀C22, D4×D7, C2×C7⋊D4, C23⋊D14, C24⋊D14
►On 56 points
(1 45)(2 53)(3 47)(4 55)(5 49)(6 43)(7 51)(22 48)(23 56)(24 50)(25 44)(26 52)(27 46)(28 54)
(1 45)(2 53)(3 47)(4 55)(5 49)(6 43)(7 51)(8 37)(9 31)(10 39)(11 33)(12 41)(13 35)(14 29)(15 38)(16 32)(17 40)(18 34)(19 42)(20 36)(21 30)(22 48)(23 56)(24 50)(25 44)(26 52)(27 46)(28 54)
(1 26)(2 27)(3 28)(4 22)(5 23)(6 24)(7 25)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(1 26)(2 27)(3 28)(4 22)(5 23)(6 24)(7 25)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 26)(9 25)(10 24)(11 23)(12 22)(13 28)(14 27)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 56)(41 55)(42 54)
G:=sub<Sym(56)| (1,45)(2,53)(3,47)(4,55)(5,49)(6,43)(7,51)(22,48)(23,56)(24,50)(25,44)(26,52)(27,46)(28,54), (1,45)(2,53)(3,47)(4,55)(5,49)(6,43)(7,51)(8,37)(9,31)(10,39)(11,33)(12,41)(13,35)(14,29)(15,38)(16,32)(17,40)(18,34)(19,42)(20,36)(21,30)(22,48)(23,56)(24,50)(25,44)(26,52)(27,46)(28,54), (1,26)(2,27)(3,28)(4,22)(5,23)(6,24)(7,25)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (1,26)(2,27)(3,28)(4,22)(5,23)(6,24)(7,25)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,26)(9,25)(10,24)(11,23)(12,22)(13,28)(14,27)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,56)(41,55)(42,54)>;
G:=Group( (1,45)(2,53)(3,47)(4,55)(5,49)(6,43)(7,51)(22,48)(23,56)(24,50)(25,44)(26,52)(27,46)(28,54), (1,45)(2,53)(3,47)(4,55)(5,49)(6,43)(7,51)(8,37)(9,31)(10,39)(11,33)(12,41)(13,35)(14,29)(15,38)(16,32)(17,40)(18,34)(19,42)(20,36)(21,30)(22,48)(23,56)(24,50)(25,44)(26,52)(27,46)(28,54), (1,26)(2,27)(3,28)(4,22)(5,23)(6,24)(7,25)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (1,26)(2,27)(3,28)(4,22)(5,23)(6,24)(7,25)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,26)(9,25)(10,24)(11,23)(12,22)(13,28)(14,27)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,56)(41,55)(42,54) );
G=PermutationGroup([[(1,45),(2,53),(3,47),(4,55),(5,49),(6,43),(7,51),(22,48),(23,56),(24,50),(25,44),(26,52),(27,46),(28,54)], [(1,45),(2,53),(3,47),(4,55),(5,49),(6,43),(7,51),(8,37),(9,31),(10,39),(11,33),(12,41),(13,35),(14,29),(15,38),(16,32),(17,40),(18,34),(19,42),(20,36),(21,30),(22,48),(23,56),(24,50),(25,44),(26,52),(27,46),(28,54)], [(1,26),(2,27),(3,28),(4,22),(5,23),(6,24),(7,25),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(1,26),(2,27),(3,28),(4,22),(5,23),(6,24),(7,25),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,26),(9,25),(10,24),(11,23),(12,22),(13,28),(14,27),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,56),(41,55),(42,54)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 4 | 8 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C2≀C22 | D4×D7 | C24⋊D14 |
| kernel | C24⋊D14 | C23.1D14 | C23⋊Dic7 | C24⋊D7 | C7×C22≀C2 | D4⋊6D14 | C2×Dic7 | C2×C28 | C22×D7 | C22×C14 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C4 | C23 | C7 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 6 | 12 |
Matrix representation of C24⋊D14 ►in GL4(𝔽29) generated by
| 1 | 0 | 23 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 3 | 1 |
| 28 | 0 | 23 | 25 |
| 25 | 1 | 0 | 21 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 3 | 1 |
| 1 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 7 | 11 | 0 | 6 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 25 | 7 |
| 0 | 0 | 0 | 4 |
| 0 | 4 | 25 | 7 |
| 0 | 0 | 0 | 14 |
| 7 | 11 | 0 | 14 |
| 0 | 27 | 0 | 0 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,23,0,28,3,0,0,0,1],[28,25,0,0,0,1,0,0,23,0,28,3,25,21,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,4,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[7,0,0,0,11,22,0,0,0,0,25,0,6,0,7,4],[0,0,7,0,4,0,11,27,25,0,0,0,7,14,14,0] >;
C24⋊D14 in GAP, Magma, Sage, TeX
C_2^4\rtimes D_{14} % in TeX
G:=Group("C2^4:D14"); // GroupNames label
G:=SmallGroup(448,566);
// by ID
G=gap.SmallGroup(448,566);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,570,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*b*c*d,b*c=c*b,e*b*e^-1=f*b*f=b*d=d*b,f*c*f=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations