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## G = C2×D5×C7⋊C3order 420 = 22·3·5·7

### Direct product of C2, D5 and C7⋊C3

Aliases: C2×D5×C7⋊C3, C703C6, (D5×C14)⋊C3, C73(C6×D5), C354(C2×C6), (C7×D5)⋊2C6, C142(C3×D5), C10⋊(C2×C7⋊C3), C5⋊(C22×C7⋊C3), (C10×C7⋊C3)⋊3C2, (C5×C7⋊C3)⋊4C22, SmallGroup(420,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C2×D5×C7⋊C3
 Chief series C1 — C7 — C35 — C5×C7⋊C3 — D5×C7⋊C3 — C2×D5×C7⋊C3
 Lower central C35 — C2×D5×C7⋊C3
 Upper central C1 — C2

Generators and relations for C2×D5×C7⋊C3
G = < a,b,c,d,e | a2=b5=c2=d7=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Smallest permutation representation of C2×D5×C7⋊C3
On 70 points
Generators in S70
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(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 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)(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 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 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)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)(58 59 61)(60 63 62)(65 66 68)(67 70 69)

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

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

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

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 D5×C7⋊C3 C2×D5×C7⋊C3 kernel C2×D5×C7⋊C3 D5×C7⋊C3 C10×C7⋊C3 D5×C14 C7×D5 C70 C2×C7⋊C3 C7⋊C3 C14 C7 D10 D5 C10 C2 C1 # reps 1 2 1 2 4 2 2 2 4 4 2 4 2 4 4

Matrix representation of C2×D5×C7⋊C3 in GL5(𝔽211)

 210 0 0 0 0 0 210 0 0 0 0 0 210 0 0 0 0 0 210 0 0 0 0 0 210
,
 0 1 0 0 0 210 178 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 210 0 0 0 210 0 0 0 0 0 0 210 0 0 0 0 0 210 0 0 0 0 0 210
,
 1 0 0 0 0 0 1 0 0 0 0 0 20 21 1 0 0 1 0 0 0 0 0 1 0
,
 196 0 0 0 0 0 196 0 0 0 0 0 1 0 0 0 0 190 210 210 0 0 0 1 0

G:=sub<GL(5,GF(211))| [210,0,0,0,0,0,210,0,0,0,0,0,210,0,0,0,0,0,210,0,0,0,0,0,210],[0,210,0,0,0,1,178,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,210,0,0,0,210,0,0,0,0,0,0,210,0,0,0,0,0,210,0,0,0,0,0,210],[1,0,0,0,0,0,1,0,0,0,0,0,20,1,0,0,0,21,0,1,0,0,1,0,0],[196,0,0,0,0,0,196,0,0,0,0,0,1,190,0,0,0,0,210,1,0,0,0,210,0] >;

C2×D5×C7⋊C3 in GAP, Magma, Sage, TeX

C_2\times D_5\times C_7\rtimes C_3
% in TeX

G:=Group("C2xD5xC7:C3");
// GroupNames label

G:=SmallGroup(420,18);
// by ID

G=gap.SmallGroup(420,18);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^7=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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