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## G = C23×F7order 336 = 24·3·7

### Direct product of C23 and F7

Aliases: C23×F7, C7⋊C3⋊C24, C7⋊(C23×C6), C14⋊(C22×C6), D7⋊(C22×C6), D144(C2×C6), (C23×D7)⋊2C3, (C22×C14)⋊3C6, (C22×D7)⋊6C6, (C2×C7⋊C3)⋊C23, (C2×C14)⋊5(C2×C6), (C23×C7⋊C3)⋊2C2, (C22×C7⋊C3)⋊3C22, SmallGroup(336,216)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C23×F7
 Chief series C1 — C7 — C7⋊C3 — F7 — C2×F7 — C22×F7 — C23×F7
 Lower central C7 — C23×F7
 Upper central C1 — C23

Generators and relations for C23×F7
G = < a,b,c,d,e | a2=b2=c2=d7=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 976 in 268 conjugacy classes, 150 normal (8 characteristic)
C1, C2, C2, C3, C22, C22, C6, C7, C23, C23, C2×C6, D7, C14, C24, C7⋊C3, C22×C6, D14, C2×C14, F7, C2×C7⋊C3, C23×C6, C22×D7, C22×C14, C2×F7, C22×C7⋊C3, C23×D7, C22×F7, C23×C7⋊C3, C23×F7
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, F7, C23×C6, C2×F7, C22×F7, C23×F7

Smallest permutation representation of C23×F7
On 56 points
Generators in S56
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(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 43)(2 46 3 49 5 48)(4 45 7 47 6 44)(8 50)(9 53 10 56 12 55)(11 52 14 54 13 51)(15 29)(16 32 17 35 19 34)(18 31 21 33 20 30)(22 36)(23 39 24 42 26 41)(25 38 28 40 27 37)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 6A ··· 6AD 7 14A ··· 14G order 1 2 ··· 2 2 ··· 2 3 3 6 ··· 6 7 14 ··· 14 size 1 1 ··· 1 7 ··· 7 7 7 7 ··· 7 6 6 ··· 6

56 irreducible representations

 dim 1 1 1 1 1 1 6 6 type + + + + + image C1 C2 C2 C3 C6 C6 F7 C2×F7 kernel C23×F7 C22×F7 C23×C7⋊C3 C23×D7 C22×D7 C22×C14 C23 C22 # reps 1 14 1 2 28 2 1 7

Matrix representation of C23×F7 in GL8(𝔽43)

 42 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 0 0 0 0 0 0 0 1
,
 42 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 42
,
 42 0 0 0 0 0 0 0 0 42 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 42 42 42 42 42 42 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
,
 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 0 0 0 0 0 42 0 0 0 0 0 42 0 0 0 0 0 42 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 42 0

G:=sub<GL(8,GF(43))| [42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42],[42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,1,0,0,0,0,0,0,42,0,1,0,0,0,0,0,42,0,0,1,0,0,0,0,42,0,0,0,1,0,0,0,42,0,0,0,0,1,0,0,42,0,0,0,0,0],[7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,42,0,0,0,1,0,0,0,0,0,0,42,1,0,0,0,0,0,0,0,1,0,0,0,0,0,42,0,1,0,0,0,0,0,0,0,1,42,0,0,0,42,0,0,1,0] >;

C23×F7 in GAP, Magma, Sage, TeX

C_2^3\times F_7
% in TeX

G:=Group("C2^3xF7");
// GroupNames label

G:=SmallGroup(336,216);
// by ID

G=gap.SmallGroup(336,216);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,10373,461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^7=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations

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