direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22≀C2, C24⋊2C14, (C2×C14)⋊7D4, (C2×D4)⋊1C14, C2.4(D4×C14), C22⋊2(C7×D4), (D4×C14)⋊10C2, C22⋊C4⋊2C14, (C23×C14)⋊1C2, (C2×C28)⋊8C22, C23⋊2(C2×C14), C14.67(C2×D4), (C2×C14).75C23, (C22×C14)⋊1C22, C22.10(C22×C14), (C2×C4)⋊1(C2×C14), (C7×C22⋊C4)⋊10C2, SmallGroup(224,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22≀C2
G = < a,b,c,d,e,f | a7=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, C23, C23, C23, C14, C14, C22⋊C4, C2×D4, C24, C28, C2×C14, C2×C14, C2×C14, C22≀C2, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C7×C22⋊C4, D4×C14, C23×C14, C7×C22≀C2
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, D4×C14, C7×C22≀C2
►On 56 points
Generators in S56
(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 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 28)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 55)(16 56)(17 50)(18 51)(19 52)(20 53)(21 54)(29 41)(30 42)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 54)(9 55)(10 56)(11 50)(12 51)(13 52)(14 53)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 54)(9 55)(10 56)(11 50)(12 51)(13 52)(14 53)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(29 41)(30 42)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 11)(2 12)(3 13)(4 14)(5 8)(6 9)(7 10)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 43)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 36)(29 52)(30 53)(31 54)(32 55)(33 56)(34 50)(35 51)
G:=sub<Sym(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), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,55)(16,56)(17,50)(18,51)(19,52)(20,53)(21,54)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,54)(9,55)(10,56)(11,50)(12,51)(13,52)(14,53)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,54)(9,55)(10,56)(11,50)(12,51)(13,52)(14,53)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,43)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,36)(29,52)(30,53)(31,54)(32,55)(33,56)(34,50)(35,51)>;
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), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,55)(16,56)(17,50)(18,51)(19,52)(20,53)(21,54)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,54)(9,55)(10,56)(11,50)(12,51)(13,52)(14,53)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,54)(9,55)(10,56)(11,50)(12,51)(13,52)(14,53)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,43)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,36)(29,52)(30,53)(31,54)(32,55)(33,56)(34,50)(35,51) );
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)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,28),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,55),(16,56),(17,50),(18,51),(19,52),(20,53),(21,54),(29,41),(30,42),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,54),(9,55),(10,56),(11,50),(12,51),(13,52),(14,53),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,54),(9,55),(10,56),(11,50),(12,51),(13,52),(14,53),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,55),(23,56),(24,50),(25,51),(26,52),(27,53),(28,54),(29,41),(30,42),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,11),(2,12),(3,13),(4,14),(5,8),(6,9),(7,10),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,43),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,36),(29,52),(30,53),(31,54),(32,55),(33,56),(34,50),(35,51)]])
C7×C22≀C2 is a maximal subgroup of
C24⋊Dic7 C24⋊D14 C24.56D14 C24.32D14 C24⋊2D14 C24⋊3D14 C24.33D14 C24.34D14 C24.35D14 C24⋊4D14 C24.36D14 C7×D42
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 4A | 4B | 4C | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 14BC | ··· | 14BH | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | C7×D4 |
| kernel | C7×C22≀C2 | C7×C22⋊C4 | D4×C14 | C23×C14 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C14 | C22 |
| # reps | 1 | 3 | 3 | 1 | 6 | 18 | 18 | 6 | 6 | 36 |
Matrix representation of C7×C22≀C2 ►in GL4(𝔽29) generated by
| 24 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 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 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 28 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [24,0,0,0,0,24,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,1,0,0,0,0,1,27,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,28,0,0,0,28,1] >;
C7×C22≀C2 in GAP, Magma, Sage, TeX
C_7\times C_2^2\wr C_2
% in TeX
G:=Group("C7xC2^2wrC2"); // GroupNames label
G:=SmallGroup(224,155);
// by ID
G=gap.SmallGroup(224,155);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,2090]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations