C2xD5xC7:C3 - GroupNames

G = C₂×D₅×C₇⋊C₃ order 420 = 2²·3·5·7

Direct product of C₂, D₅ and C₇⋊C₃

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C₂×D₅×C₇⋊C₃, C₇₀⋊₃C₆, (D₅×C₁₄)⋊C₃, C₇⋊₃(C₆×D₅), C₃₅⋊₄(C₂×C₆), (C₇×D₅)⋊₂C₆, C₁₄⋊₂(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 >

C₁

C₂

₅C₂

₇C₃

C₅

C₇

₅C₂²

₇C₆

₃₅C₆

D₅

C₁₀

D₅

C₁₄

₅C₁₄

₇C₁₅

C₇⋊C₃

C₃₅

₃₅C₂×C₆

D₁₀

₅C₂×C₁₄

₇C₃₀

₇C₃×D₅

C₂×C₇⋊C₃

₅C₂×C₇⋊C₃

C₇×D₅

C₇₀

C₇×D₅

C₅×C₇⋊C₃

₇C₆×D₅

₅C₂²×C₇⋊C₃

D₅×C₁₄

D₅×C₇⋊C₃

C₁₀×C₇⋊C₃

C₂×D₅×C₇⋊C₃

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

Export

Subgroup lattice of C₂×D₅×C₇⋊C₃ in TeX

G = C2×D5×C7⋊C3 order 420 = 22·3·5·7

Direct product of C2, D5 and C7⋊C3

G = C₂×D₅×C₇⋊C₃ order 420 = 2²·3·5·7

Direct product of C₂, D₅ and C₇⋊C₃