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## G = C2×C5⋊F7order 420 = 22·3·5·7

### Direct product of C2 and C5⋊F7

Aliases: C2×C5⋊F7, D70⋊C3, C10⋊F7, C701C6, D352C6, C14⋊(C3×D5), C7⋊C32D10, C72(C6×D5), C52(C2×F7), C352(C2×C6), (C2×C7⋊C3)⋊D5, (C10×C7⋊C3)⋊1C2, (C5×C7⋊C3)⋊2C22, SmallGroup(420,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C2×C5⋊F7
 Chief series C1 — C7 — C35 — C5×C7⋊C3 — C5⋊F7 — C2×C5⋊F7
 Lower central C35 — C2×C5⋊F7
 Upper central C1 — C2

Generators and relations for C2×C5⋊F7
G = < a,b,c,d | a2=b5=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Smallest permutation representation of C2×C5⋊F7
On 70 points
Generators in S70
(1 41)(2 42)(3 36)(4 37)(5 38)(6 39)(7 40)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)
(1 34 27 20 13)(2 35 28 21 14)(3 29 22 15 8)(4 30 23 16 9)(5 31 24 17 10)(6 32 25 18 11)(7 33 26 19 12)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)
(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)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)
(2 4 3 7 5 6)(8 33 10 32 14 30)(9 29 12 31 11 35)(13 34)(15 26 17 25 21 23)(16 22 19 24 18 28)(20 27)(36 40 38 39 42 37)(43 68 45 67 49 65)(44 64 47 66 46 70)(48 69)(50 61 52 60 56 58)(51 57 54 59 53 63)(55 62)

G:=sub<Sym(70)| (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70), (1,34,27,20,13)(2,35,28,21,14)(3,29,22,15,8)(4,30,23,16,9)(5,31,24,17,10)(6,32,25,18,11)(7,33,26,19,12)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49), (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)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70), (2,4,3,7,5,6)(8,33,10,32,14,30)(9,29,12,31,11,35)(13,34)(15,26,17,25,21,23)(16,22,19,24,18,28)(20,27)(36,40,38,39,42,37)(43,68,45,67,49,65)(44,64,47,66,46,70)(48,69)(50,61,52,60,56,58)(51,57,54,59,53,63)(55,62)>;

G:=Group( (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70), (1,34,27,20,13)(2,35,28,21,14)(3,29,22,15,8)(4,30,23,16,9)(5,31,24,17,10)(6,32,25,18,11)(7,33,26,19,12)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49), (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)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70), (2,4,3,7,5,6)(8,33,10,32,14,30)(9,29,12,31,11,35)(13,34)(15,26,17,25,21,23)(16,22,19,24,18,28)(20,27)(36,40,38,39,42,37)(43,68,45,67,49,65)(44,64,47,66,46,70)(48,69)(50,61,52,60,56,58)(51,57,54,59,53,63)(55,62) );

G=PermutationGroup([[(1,41),(2,42),(3,36),(4,37),(5,38),(6,39),(7,40),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70)], [(1,34,27,20,13),(2,35,28,21,14),(3,29,22,15,8),(4,30,23,16,9),(5,31,24,17,10),(6,32,25,18,11),(7,33,26,19,12),(36,64,57,50,43),(37,65,58,51,44),(38,66,59,52,45),(39,67,60,53,46),(40,68,61,54,47),(41,69,62,55,48),(42,70,63,56,49)], [(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),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70)], [(2,4,3,7,5,6),(8,33,10,32,14,30),(9,29,12,31,11,35),(13,34),(15,26,17,25,21,23),(16,22,19,24,18,28),(20,27),(36,40,38,39,42,37),(43,68,45,67,49,65),(44,64,47,66,46,70),(48,69),(50,61,52,60,56,58),(51,57,54,59,53,63),(55,62)]])

34 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A 5B 6A 6B 6C 6D 6E 6F 7 10A 10B 14 15A 15B 15C 15D 30A 30B 30C 30D 35A 35B 35C 35D 70A 70B 70C 70D order 1 2 2 2 3 3 5 5 6 6 6 6 6 6 7 10 10 14 15 15 15 15 30 30 30 30 35 35 35 35 70 70 70 70 size 1 1 35 35 7 7 2 2 7 7 35 35 35 35 6 2 2 6 14 14 14 14 14 14 14 14 6 6 6 6 6 6 6 6

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 6 6 type + + + + + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 F7 C2×F7 C5⋊F7 C2×C5⋊F7 kernel C2×C5⋊F7 C5⋊F7 C10×C7⋊C3 D70 D35 C70 C2×C7⋊C3 C7⋊C3 C14 C7 C10 C5 C2 C1 # reps 1 2 1 2 4 2 2 2 4 4 1 1 4 4

Matrix representation of C2×C5⋊F7 in GL6(𝔽211)

 210 0 0 0 0 0 0 210 0 0 0 0 0 0 210 0 0 0 0 0 0 210 0 0 0 0 0 0 210 0 0 0 0 0 0 210
,
 74 181 181 0 181 0 0 74 181 181 0 181 30 30 104 0 0 30 181 0 0 74 181 181 30 0 30 30 104 0 0 30 0 30 30 104
,
 0 1 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 1 210 210 210 210 210 210
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 210 210 210 210 210 210 0 0 0 0 1 0

G:=sub<GL(6,GF(211))| [210,0,0,0,0,0,0,210,0,0,0,0,0,0,210,0,0,0,0,0,0,210,0,0,0,0,0,0,210,0,0,0,0,0,0,210],[74,0,30,181,30,0,181,74,30,0,0,30,181,181,104,0,30,0,0,181,0,74,30,30,181,0,0,181,104,30,0,181,30,181,0,104],[0,0,0,0,0,210,1,0,0,0,0,210,0,1,0,0,0,210,0,0,1,0,0,210,0,0,0,1,0,210,0,0,0,0,1,210],[1,0,0,0,210,0,0,0,0,1,210,0,0,0,0,0,210,0,0,0,1,0,210,0,0,0,0,0,210,1,0,1,0,0,210,0] >;

C2×C5⋊F7 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes F_7
% in TeX

G:=Group("C2xC5:F7");
// GroupNames label

G:=SmallGroup(420,19);
// by ID

G=gap.SmallGroup(420,19);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,963,9004,764]);
// Polycyclic

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

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