C2^2:C4xC7:C3 - GroupNames

G = C₂²⋊C₄×C₇⋊C₃ order 336 = 2⁴·3·7

Direct product of C₂²⋊C₄ and C₇⋊C₃

direct product, metabelian, supersoluble, monomial

Aliases: C₂²⋊C₄×C₇⋊C₃, (C₂×C₂₈)⋊₇C₆, (C₂×C₁₄)⋊₅C₁₂, C₁₄.₁₂(C₃×D₄), C₁₄.₁₁(C₂×C₁₂), (C₂²×C₁₄).₅C₆, (C₇×C₂²⋊C₄)⋊C₃, C₂.₁(D₄×C₇⋊C₃), C₇⋊₃(C₃×C₂²⋊C₄), C₂²⋊₂(C₄×C₇⋊C₃), (C₂²×C₇⋊C₃)⋊₃C₄, (C₂×C₇⋊C₃).₁₂D₄, C₂³.₂(C₂×C₇⋊C₃), (C₂³×C₇⋊C₃).₃C₂, (C₂×C₁₄).₁₅(C₂×C₆), C₂².₃(C₂²×C₇⋊C₃), (C₂²×C₇⋊C₃).₁₄C₂², (C₂×C₄×C₇⋊C₃)⋊₇C₂, C₂.₃(C₂×C₄×C₇⋊C₃), (C₂×C₄)⋊₁(C₂×C₇⋊C₃), (C₂×C₇⋊C₃).₁₁(C₂×C₄), SmallGroup(336,49)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂²⋊C₄×C₇⋊C₃

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₇⋊C₃ — C₂³×C₇⋊C₃ — C₂²⋊C₄×C₇⋊C₃

Lower central

C₇ — C₁₄ — C₂²⋊C₄×C₇⋊C₃

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for C₂²⋊C₄×C₇⋊C₃
G = < a,b,c,d,e | a²=b²=c⁴=d⁷=e³=1, cac^-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d⁴ >

Subgroups: 230 in 68 conjugacy classes, 33 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₇, C₂×C₄, C₂³, C₁₂, C₂×C₆, C₁₄, C₁₄, C₁₄, C₂²⋊C₄, C₇⋊C₃, C₂×C₁₂, C₂²×C₆, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₇⋊C₃, C₂×C₇⋊C₃, C₂×C₇⋊C₃, C₃×C₂²⋊C₄, C₂×C₂₈, C₂²×C₁₄, C₄×C₇⋊C₃, C₂²×C₇⋊C₃, C₂²×C₇⋊C₃, C₂²×C₇⋊C₃, C₇×C₂²⋊C₄, C₂×C₄×C₇⋊C₃, C₂³×C₇⋊C₃, C₂²⋊C₄×C₇⋊C₃
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₁₂, C₂×C₆, C₂²⋊C₄, C₇⋊C₃, C₂×C₁₂, C₃×D₄, C₂×C₇⋊C₃, C₃×C₂²⋊C₄, C₄×C₇⋊C₃, C₂²×C₇⋊C₃, C₂×C₄×C₇⋊C₃, D₄×C₇⋊C₃, C₂²⋊C₄×C₇⋊C₃

Smallest permutation representation of C₂²⋊C₄×C₇⋊C₃
►On 56 points

Generators in S₅₆

(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 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	6A	···	6F	6G	6H	6I	6J	7A	7B	12A	···	12H	14A	···	14F	14G	14H	14I	14J	28A	···	28H
order	1	2	2	2	2	2	3	3	4	4	4	4	6	···	6	6	6	6	6	7	7	12	···	12	14	···	14	14	14	14	14	28	···	28
size	1	1	1	1	2	2	7	7	2	2	2	2	7	···	7	14	14	14	14	3	3	14	···	14	3	···	3	6	6	6	6	6	···	6

50 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	3	3	3	3	6
type	+	+	+						+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	D₄	C₃×D₄	C₇⋊C₃	C₂×C₇⋊C₃	C₂×C₇⋊C₃	C₄×C₇⋊C₃	D₄×C₇⋊C₃
kernel	C₂²⋊C₄×C₇⋊C₃	C₂×C₄×C₇⋊C₃	C₂³×C₇⋊C₃	C₇×C₂²⋊C₄	C₂²×C₇⋊C₃	C₂×C₂₈	C₂²×C₁₄	C₂×C₁₄	C₂×C₇⋊C₃	C₁₄	C₂²⋊C₄	C₂×C₄	C₂³	C₂²	C₂
# reps	1	2	1	2	4	4	2	8	2	4	2	4	2	8	4

Matrix representation of C₂²⋊C₄×C₇⋊C₃ ►in GL₇(𝔽₃₃₇)

1	0	0	0	0	0	0
314	336	0	0	0	0	0
0	0	336	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

336	0	0	0	0	0	0
0	336	0	0	0	0	0
0	0	336	0	0	0	0
0	0	0	336	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

336	249	0	0	0	0	0
0	1	0	0	0	0	0
0	0	0	1	0	0	0
0	0	336	0	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	212	213	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	124	336	336
0	0	0	0	0	1	0

G:=sub<GL(7,GF(337))| [1,314,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[336,0,0,0,0,0,0,249,1,0,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,212,1,0,0,0,0,0,213,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,124,0,0,0,0,0,0,336,1,0,0,0,0,0,336,0] >;

C₂²⋊C₄×C₇⋊C₃ in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_7\rtimes C_3

% in TeX

G:=Group("C2^2:C4xC7:C3");

// GroupNames label

G:=SmallGroup(336,49);

// by ID

G=gap.SmallGroup(336,49);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,881]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^7=e^3=1,c*a*c^-1=a*b=b*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 = C22⋊C4×C7⋊C3 order 336 = 24·3·7

Direct product of C22⋊C4 and C7⋊C3

G = C₂²⋊C₄×C₇⋊C₃ order 336 = 2⁴·3·7

Direct product of C₂²⋊C₄ and C₇⋊C₃