C2^3xC7:C3 - GroupNames

G = C₂³×C₇⋊C₃ order 168 = 2³·3·7

Direct product of C₂³ and C₇⋊C₃

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

Aliases: C₂³×C₇⋊C₃, (C₂×C₁₄)⋊₈C₆, C₁₄⋊₂(C₂×C₆), C₇⋊₂(C₂²×C₆), (C₂²×C₁₄)⋊₂C₃, SmallGroup(168,51)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₇ — C₂³×C₇⋊C₃

Chief series

C₁ — C₇ — C₇⋊C₃ — C₂×C₇⋊C₃ — C₂²×C₇⋊C₃ — C₂³×C₇⋊C₃

Lower central

C₇ — C₂³×C₇⋊C₃

Upper central

C₁ — C₂³

Generators and relations for C₂³×C₇⋊C₃
G = < a,b,c,d,e | a²=b²=c²=d⁷=e³=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d⁴ >

Subgroups: 160 in 64 conjugacy classes, 48 normal (6 characteristic)
C₁, C₂, C₃, C₂², C₆, C₇, C₂³, C₂×C₆, C₁₄, C₇⋊C₃, C₂²×C₆, C₂×C₁₄, C₂×C₇⋊C₃, C₂²×C₁₄, C₂²×C₇⋊C₃, C₂³×C₇⋊C₃
Quotients: C₁, C₂, C₃, C₂², C₆, C₂³, C₂×C₆, C₇⋊C₃, C₂²×C₆, C₂×C₇⋊C₃, C₂²×C₇⋊C₃, C₂³×C₇⋊C₃

Smallest permutation representation of C₂³×C₇⋊C₃
►On 56 points

Generators in S₅₆

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

(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)

(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)

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

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

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

C₂³×C₇⋊C₃ is a maximal subgroup of C₂³.₂F₇

40 conjugacy classes

class	1	2A	···	2G	3A	3B	6A	···	6N	7A	7B	14A	···	14N
order	1	2	···	2	3	3	6	···	6	7	7	14	···	14
size	1	1	···	1	7	7	7	···	7	3	3	3	···	3

40 irreducible representations

dim	1	1	1	1	3	3
type	+	+
image	C₁	C₂	C₃	C₆	C₇⋊C₃	C₂×C₇⋊C₃
kernel	C₂³×C₇⋊C₃	C₂²×C₇⋊C₃	C₂²×C₁₄	C₂×C₁₄	C₂³	C₂²
# reps	1	7	2	14	2	14

Matrix representation of C₂³×C₇⋊C₃ ►in GL₆(𝔽₄₃)

1	0	0	0	0	0
0	42	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

42	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	0
0	0	0	0	0	1

1	0	0	0	0	0
0	42	0	0	0	0
0	0	42	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	24	25	1
0	0	0	1	0	0
0	0	0	0	1	0

36	0	0	0	0	0
0	36	0	0	0	0
0	0	36	0	0	0
0	0	0	1	0	0
0	0	0	18	42	42
0	0	0	0	1	0

G:=sub<GL(6,GF(43))| [1,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,1,0,0,0,0,25,0,1,0,0,0,1,0,0],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,18,0,0,0,0,0,42,1,0,0,0,0,42,0] >;

C₂³×C₇⋊C₃ in GAP, Magma, Sage, TeX

C_2^3\times C_7\rtimes C_3

% in TeX

G:=Group("C2^3xC7:C3");

// GroupNames label

G:=SmallGroup(168,51);

// by ID

G=gap.SmallGroup(168,51);

# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,169]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^7=e^3=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^4>;

// generators/relations

G = C23×C7⋊C3 order 168 = 23·3·7

Direct product of C23 and C7⋊C3

G = C₂³×C₇⋊C₃ order 168 = 2³·3·7

Direct product of C₂³ and C₇⋊C₃