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## G = C2×C4×F8order 448 = 26·7

### Direct product of C2×C4 and F8

Aliases: C2×C4×F8, C24⋊C28, C25.C14, (C24×C4)⋊C7, C23⋊(C2×C28), C22.(C2×F8), (C22×F8).C2, (C23×C4)⋊2C14, C2.1(C22×F8), C24.1(C2×C14), (C2×F8).1C22, SmallGroup(448,1362)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C2×C4×F8
 Chief series C1 — C23 — C24 — C2×F8 — C22×F8 — C2×C4×F8
 Lower central C23 — C2×C4×F8
 Upper central C1 — C2×C4

Generators and relations for C2×C4×F8
G = < a,b,c,d,e,f | a2=b4=c2=d2=e2=f7=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=ed=de, fdf-1=c, fef-1=d >

Subgroups: 753 in 127 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, C14, C22×C4, C24, C24, C24, C28, C2×C14, C23×C4, C23×C4, C25, C2×C28, F8, C24×C4, C2×F8, C2×F8, C4×F8, C22×F8, C2×C4×F8
Quotients: C1, C2, C4, C22, C7, C2×C4, C14, C28, C2×C14, C2×C28, F8, C2×F8, C4×F8, C22×F8, C2×C4×F8

Smallest permutation representation of C2×C4×F8
On 56 points
Generators in S56
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(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)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)
(1 15 43 24)(2 16 44 25)(3 17 45 26)(4 18 46 27)(5 19 47 28)(6 20 48 22)(7 21 49 23)(8 37 56 34)(9 38 50 35)(10 39 51 29)(11 40 52 30)(12 41 53 31)(13 42 54 32)(14 36 55 33)
(1 38)(2 44)(3 30)(4 31)(5 42)(7 49)(8 56)(9 24)(10 51)(11 17)(12 18)(13 28)(15 50)(16 25)(19 54)(21 23)(26 52)(27 53)(29 39)(32 47)(34 37)(35 43)(40 45)(41 46)
(1 43)(2 39)(3 45)(4 31)(5 32)(6 36)(9 50)(10 25)(11 52)(12 18)(13 19)(14 22)(15 24)(16 51)(17 26)(20 55)(27 53)(28 54)(29 44)(30 40)(33 48)(35 38)(41 46)(42 47)
(2 44)(3 40)(4 46)(5 32)(6 33)(7 37)(8 23)(10 51)(11 26)(12 53)(13 19)(14 20)(16 25)(17 52)(18 27)(21 56)(22 55)(28 54)(29 39)(30 45)(31 41)(34 49)(36 48)(42 47)
(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)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 14A ··· 14R 28A ··· 28X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 1 1 7 7 7 7 1 1 1 1 7 7 7 7 8 ··· 8 8 ··· 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 7 7 7 7 type + + + + + + image C1 C2 C2 C4 C7 C14 C14 C28 F8 C2×F8 C2×F8 C4×F8 kernel C2×C4×F8 C4×F8 C22×F8 C2×F8 C24×C4 C23×C4 C25 C24 C2×C4 C4 C22 C2 # reps 1 2 1 4 6 12 6 24 1 2 1 4

Matrix representation of C2×C4×F8 in GL8(𝔽29)

 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 12 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 17
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0

G:=sub<GL(8,GF(29))| [1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

C2×C4×F8 in GAP, Magma, Sage, TeX

C_2\times C_4\times F_8
% in TeX

G:=Group("C2xC4xF8");
// GroupNames label

G:=SmallGroup(448,1362);
// by ID

G=gap.SmallGroup(448,1362);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,2,2,204,998,2371,3450]);
// Polycyclic

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

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