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## G = C23×F8order 448 = 26·7

### Direct product of C23 and F8

Aliases: C23×F8, C261C7, C252C14, C24⋊(C2×C14), C23⋊(C22×C14), SmallGroup(448,1392)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C23×F8
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g7=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, gdg-1=fe=ef, geg-1=d, gfg-1=e >

Subgroups: 2969 in 463 conjugacy classes, 48 normal (6 characteristic)
C1, C2, C2, C22, C22, C7, C23, C23, C14, C24, C24, C2×C14, C25, C25, F8, C22×C14, C26, C2×F8, C22×F8, C23×F8
Quotients: C1, C2, C22, C7, C23, C14, C2×C14, F8, C22×C14, C2×F8, C22×F8, C23×F8

Smallest permutation representation of C23×F8
On 56 points
Generators in S56
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 45)(9 46)(10 47)(11 48)(12 49)(13 43)(14 44)(22 31)(23 32)(24 33)(25 34)(26 35)(27 29)(28 30)(36 53)(37 54)(38 55)(39 56)(40 50)(41 51)(42 52)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 36)(15 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(43 52)(44 53)(45 54)(46 55)(47 56)(48 50)(49 51)
(1 11)(2 12)(3 13)(4 14)(5 8)(6 9)(7 10)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 40)(23 41)(24 42)(25 36)(26 37)(27 38)(28 39)(29 55)(30 56)(31 50)(32 51)(33 52)(34 53)(35 54)
(1 11)(2 21)(3 42)(4 44)(5 54)(6 29)(7 28)(8 35)(9 55)(10 39)(12 49)(13 24)(14 16)(15 52)(17 37)(18 27)(19 30)(20 48)(22 40)(23 32)(25 53)(26 45)(31 50)(33 43)(34 36)(38 46)(41 51)(47 56)
(1 22)(2 12)(3 15)(4 36)(5 45)(6 55)(7 30)(8 17)(9 29)(10 56)(11 40)(13 43)(14 25)(16 53)(18 38)(19 28)(20 31)(21 49)(23 41)(24 33)(26 54)(27 46)(32 51)(34 44)(35 37)(39 47)(42 52)(48 50)
(1 31)(2 23)(3 13)(4 16)(5 37)(6 46)(7 56)(8 26)(9 18)(10 30)(11 50)(12 41)(14 44)(15 43)(17 54)(19 39)(20 22)(21 32)(24 42)(25 34)(27 55)(28 47)(29 38)(33 52)(35 45)(36 53)(40 48)(49 51)
(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)| (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,45)(9,46)(10,47)(11,48)(12,49)(13,43)(14,44)(22,31)(23,32)(24,33)(25,34)(26,35)(27,29)(28,30)(36,53)(37,54)(38,55)(39,56)(40,50)(41,51)(42,52), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,36)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(43,52)(44,53)(45,54)(46,55)(47,56)(48,50)(49,51), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(29,55)(30,56)(31,50)(32,51)(33,52)(34,53)(35,54), (1,11)(2,21)(3,42)(4,44)(5,54)(6,29)(7,28)(8,35)(9,55)(10,39)(12,49)(13,24)(14,16)(15,52)(17,37)(18,27)(19,30)(20,48)(22,40)(23,32)(25,53)(26,45)(31,50)(33,43)(34,36)(38,46)(41,51)(47,56), (1,22)(2,12)(3,15)(4,36)(5,45)(6,55)(7,30)(8,17)(9,29)(10,56)(11,40)(13,43)(14,25)(16,53)(18,38)(19,28)(20,31)(21,49)(23,41)(24,33)(26,54)(27,46)(32,51)(34,44)(35,37)(39,47)(42,52)(48,50), (1,31)(2,23)(3,13)(4,16)(5,37)(6,46)(7,56)(8,26)(9,18)(10,30)(11,50)(12,41)(14,44)(15,43)(17,54)(19,39)(20,22)(21,32)(24,42)(25,34)(27,55)(28,47)(29,38)(33,52)(35,45)(36,53)(40,48)(49,51), (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( (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,45)(9,46)(10,47)(11,48)(12,49)(13,43)(14,44)(22,31)(23,32)(24,33)(25,34)(26,35)(27,29)(28,30)(36,53)(37,54)(38,55)(39,56)(40,50)(41,51)(42,52), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,36)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(43,52)(44,53)(45,54)(46,55)(47,56)(48,50)(49,51), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(29,55)(30,56)(31,50)(32,51)(33,52)(34,53)(35,54), (1,11)(2,21)(3,42)(4,44)(5,54)(6,29)(7,28)(8,35)(9,55)(10,39)(12,49)(13,24)(14,16)(15,52)(17,37)(18,27)(19,30)(20,48)(22,40)(23,32)(25,53)(26,45)(31,50)(33,43)(34,36)(38,46)(41,51)(47,56), (1,22)(2,12)(3,15)(4,36)(5,45)(6,55)(7,30)(8,17)(9,29)(10,56)(11,40)(13,43)(14,25)(16,53)(18,38)(19,28)(20,31)(21,49)(23,41)(24,33)(26,54)(27,46)(32,51)(34,44)(35,37)(39,47)(42,52)(48,50), (1,31)(2,23)(3,13)(4,16)(5,37)(6,46)(7,56)(8,26)(9,18)(10,30)(11,50)(12,41)(14,44)(15,43)(17,54)(19,39)(20,22)(21,32)(24,42)(25,34)(27,55)(28,47)(29,38)(33,52)(35,45)(36,53)(40,48)(49,51), (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([[(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,45),(9,46),(10,47),(11,48),(12,49),(13,43),(14,44),(22,31),(23,32),(24,33),(25,34),(26,35),(27,29),(28,30),(36,53),(37,54),(38,55),(39,56),(40,50),(41,51),(42,52)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,36),(15,33),(16,34),(17,35),(18,29),(19,30),(20,31),(21,32),(43,52),(44,53),(45,54),(46,55),(47,56),(48,50),(49,51)], [(1,11),(2,12),(3,13),(4,14),(5,8),(6,9),(7,10),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,40),(23,41),(24,42),(25,36),(26,37),(27,38),(28,39),(29,55),(30,56),(31,50),(32,51),(33,52),(34,53),(35,54)], [(1,11),(2,21),(3,42),(4,44),(5,54),(6,29),(7,28),(8,35),(9,55),(10,39),(12,49),(13,24),(14,16),(15,52),(17,37),(18,27),(19,30),(20,48),(22,40),(23,32),(25,53),(26,45),(31,50),(33,43),(34,36),(38,46),(41,51),(47,56)], [(1,22),(2,12),(3,15),(4,36),(5,45),(6,55),(7,30),(8,17),(9,29),(10,56),(11,40),(13,43),(14,25),(16,53),(18,38),(19,28),(20,31),(21,49),(23,41),(24,33),(26,54),(27,46),(32,51),(34,44),(35,37),(39,47),(42,52),(48,50)], [(1,31),(2,23),(3,13),(4,16),(5,37),(6,46),(7,56),(8,26),(9,18),(10,30),(11,50),(12,41),(14,44),(15,43),(17,54),(19,39),(20,22),(21,32),(24,42),(25,34),(27,55),(28,47),(29,38),(33,52),(35,45),(36,53),(40,48),(49,51)], [(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 ··· 2G 2H ··· 2O 7A ··· 7F 14A ··· 14AP order 1 2 ··· 2 2 ··· 2 7 ··· 7 14 ··· 14 size 1 1 ··· 1 7 ··· 7 8 ··· 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 7 7 type + + + + image C1 C2 C7 C14 F8 C2×F8 kernel C23×F8 C22×F8 C26 C25 C23 C22 # reps 1 7 6 42 1 7

Matrix representation of C23×F8 in GL9(ℤ)

 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

G:=sub<GL(9,Integers())| [-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C23×F8 in GAP, Magma, Sage, TeX

C_2^3\times F_8
% in TeX

G:=Group("C2^3xF8");
// GroupNames label

G:=SmallGroup(448,1392);
// by ID

G=gap.SmallGroup(448,1392);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,2,2,515,1202,1742]);
// Polycyclic

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

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