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## G = C7×C2≀C22order 448 = 26·7

### Direct product of C7 and C2≀C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C7×C2≀C22
 Chief series C1 — C2 — C22 — C23 — C22×C14 — C23×C14 — C7×C22≀C2 — C7×C2≀C22
 Lower central C1 — C2 — C23 — C7×C2≀C22
 Upper central C1 — C14 — C22×C14 — C7×C2≀C22

Generators and relations for C7×C2≀C22
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 450 in 198 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, C28, C2×C14, C2×C14, C23⋊C4, C22≀C2, 2+ 1+4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22×C14, C2≀C22, C7×C22⋊C4, C7×C22⋊C4, D4×C14, D4×C14, C7×C4○D4, C23×C14, C7×C23⋊C4, C7×C22≀C2, C7×2+ 1+4, C7×C2≀C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, C2≀C22, D4×C14, C7×C22≀C2, C7×C2≀C22

Smallest permutation representation of C7×C2≀C22
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 14)(2 8)(3 9)(4 10)(5 11)(6 12)(7 13)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 32)(23 33)(24 34)(25 35)(26 29)(27 30)(28 31)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 50)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(15 55)(16 56)(17 50)(18 51)(19 52)(20 53)(21 54)(36 46)(37 47)(38 48)(39 49)(40 43)(41 44)(42 45)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 43)(7 44)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 36)
(1 14 34 50)(2 8 35 51)(3 9 29 52)(4 10 30 53)(5 11 31 54)(6 12 32 55)(7 13 33 56)(15 40 22 43)(16 41 23 44)(17 42 24 45)(18 36 25 46)(19 37 26 47)(20 38 27 48)(21 39 28 49)
(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 50)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)

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), (1,14)(2,8)(3,9)(4,10)(5,11)(6,12)(7,13)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,50), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,55)(16,56)(17,50)(18,51)(19,52)(20,53)(21,54)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36), (1,14,34,50)(2,8,35,51)(3,9,29,52)(4,10,30,53)(5,11,31,54)(6,12,32,55)(7,13,33,56)(15,40,22,43)(16,41,23,44)(17,42,24,45)(18,36,25,46)(19,37,26,47)(20,38,27,48)(21,39,28,49), (8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,50)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)>;

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), (1,14)(2,8)(3,9)(4,10)(5,11)(6,12)(7,13)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,50), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,55)(16,56)(17,50)(18,51)(19,52)(20,53)(21,54)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36), (1,14,34,50)(2,8,35,51)(3,9,29,52)(4,10,30,53)(5,11,31,54)(6,12,32,55)(7,13,33,56)(15,40,22,43)(16,41,23,44)(17,42,24,45)(18,36,25,46)(19,37,26,47)(20,38,27,48)(21,39,28,49), (8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,50)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28) );

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)], [(1,14),(2,8),(3,9),(4,10),(5,11),(6,12),(7,13),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,32),(23,33),(24,34),(25,35),(26,29),(27,30),(28,31),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,50)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,25),(9,26),(10,27),(11,28),(12,22),(13,23),(14,24),(15,55),(16,56),(17,50),(18,51),(19,52),(20,53),(21,54),(36,46),(37,47),(38,48),(39,49),(40,43),(41,44),(42,45)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,43),(7,44),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,55),(23,56),(24,50),(25,51),(26,52),(27,53),(28,54),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,36)], [(1,14,34,50),(2,8,35,51),(3,9,29,52),(4,10,30,53),(5,11,31,54),(6,12,32,55),(7,13,33,56),(15,40,22,43),(16,41,23,44),(17,42,24,45),(18,36,25,46),(19,37,26,47),(20,38,27,48),(21,39,28,49)], [(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,50),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)]])

112 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2I 4A 4B 4C 4D 4E 4F 7A ··· 7F 14A ··· 14F 14G ··· 14X 14Y ··· 14BB 28A ··· 28R 28S ··· 28AJ order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 4 ··· 4 4 4 4 8 8 8 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 D4 C7×D4 C7×D4 C2≀C22 C7×C2≀C22 kernel C7×C2≀C22 C7×C23⋊C4 C7×C22≀C2 C7×2+ 1+4 C2≀C22 C23⋊C4 C22≀C2 2+ 1+4 C2×C28 C22×C14 C2×C4 C23 C7 C1 # reps 1 3 3 1 6 18 18 6 3 3 18 18 2 12

Matrix representation of C7×C2≀C22 in GL6(𝔽29)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 28 0 0 0 0 28 0 0 0 0 0 0 0 0 0 1 0 0 0 15 26 1 2 0 0 1 0 0 0 0 0 22 25 23 3
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 15 26 1 2 0 0 23 23 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 28 0 0 0 0 0 0 14 3 28 27 0 0 0 0 1 0 0 0 1 0 0 0 0 0 25 6 9 15
,
 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 14 3 28 27 0 0 0 0 0 1

G:=sub<GL(6,GF(29))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,15,1,22,0,0,0,26,0,25,0,0,1,1,0,23,0,0,0,2,0,3],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,15,23,0,0,1,0,26,23,0,0,0,0,1,0,0,0,0,0,2,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,14,0,1,25,0,0,3,0,0,6,0,0,28,1,0,9,0,0,27,0,0,15],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,14,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,27,1] >;

C7×C2≀C22 in GAP, Magma, Sage, TeX

C_7\times C_2\wr C_2^2
% in TeX

G:=Group("C7xC2wrC2^2");
// GroupNames label

G:=SmallGroup(448,865);
// by ID

G=gap.SmallGroup(448,865);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2438,2468,7068]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=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,b*d=d*b,e*b*e^-1=f*b*f=b*c*d,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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