C4oD4xC7:C3 - GroupNames

G = C₄○D₄×C₇⋊C₃ order 336 = 2⁴·3·7

Direct product of C₄○D₄ and C₇⋊C₃

direct product, metabelian, supersoluble, monomial

Aliases: C₄○D₄×C₇⋊C₃, (C₂×C₂₈)⋊₅C₆, (C₇×D₄)⋊₆C₆, (C₇×Q₈)⋊₁₀C₆, C₂₈.₂₁(C₂×C₆), C₁₄.₁₄(C₂²×C₆), (D₄×C₇⋊C₃)⋊₅C₂, D₄⋊₂(C₂×C₇⋊C₃), C₇⋊₄(C₃×C₄○D₄), Q₈⋊₃(C₂×C₇⋊C₃), (Q₈×C₇⋊C₃)⋊₅C₂, (C₇×C₄○D₄)⋊₂C₃, C₄.₅(C₂²×C₇⋊C₃), C₂.₄(C₂³×C₇⋊C₃), C₂².(C₂²×C₇⋊C₃), (C₂×C₁₄).₁₂(C₂×C₆), (C₄×C₇⋊C₃).₂₁C₂², (C₂×C₇⋊C₃).₁₄C₂³, (C₂²×C₇⋊C₃).₁₂C₂², (C₂×C₄×C₇⋊C₃)⋊₅C₂, (C₂×C₄)⋊₃(C₂×C₇⋊C₃), SmallGroup(336,167)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₄○D₄×C₇⋊C₃

Chief series

C₁ — C₇ — C₁₄ — C₂×C₇⋊C₃ — C₂²×C₇⋊C₃ — C₂×C₄×C₇⋊C₃ — C₄○D₄×C₇⋊C₃

Lower central

C₇ — C₁₄ — C₄○D₄×C₇⋊C₃

Upper central

C₁ — C₄ — C₄○D₄

Generators and relations for C₄○D₄×C₇⋊C₃
G = < a,b,c,d,e | a⁴=c²=d⁷=e³=1, b²=a², ab=ba, ac=ca, ad=da, ae=ea, cbc=a²b, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d⁴ >

Subgroups: 230 in 80 conjugacy classes, 51 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₆, C₇, C₂×C₄, D₄, Q₈, C₁₂, C₂×C₆, C₁₄, C₁₄, C₄○D₄, C₇⋊C₃, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂₈, C₂₈, C₂×C₁₄, C₂×C₇⋊C₃, C₂×C₇⋊C₃, C₃×C₄○D₄, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₄×C₇⋊C₃, C₄×C₇⋊C₃, C₂²×C₇⋊C₃, C₇×C₄○D₄, C₂×C₄×C₇⋊C₃, D₄×C₇⋊C₃, Q₈×C₇⋊C₃, C₄○D₄×C₇⋊C₃
Quotients: C₁, C₂, C₃, C₂², C₆, C₂³, C₂×C₆, C₄○D₄, C₇⋊C₃, C₂²×C₆, C₂×C₇⋊C₃, C₃×C₄○D₄, C₂²×C₇⋊C₃, C₂³×C₇⋊C₃, C₄○D₄×C₇⋊C₃

Smallest permutation representation of C₄○D₄×C₇⋊C₃
►On 56 points

Generators in S₅₆

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

(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 43 22 50)(16 44 23 51)(17 45 24 52)(18 46 25 53)(19 47 26 54)(20 48 27 55)(21 49 28 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 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 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)| (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56), (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)(29,43)(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), (1,2,3,4,5,6,7)(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( (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56), (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)(29,43)(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), (1,2,3,4,5,6,7)(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([[(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,43,22,50),(16,44,23,51),(17,45,24,52),(18,46,25,53),(19,47,26,54),(20,48,27,55),(21,49,28,56)], [(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),(29,43),(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)], [(1,2,3,4,5,6,7),(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	3A	3B	4A	4B	4C	4D	4E	6A	6B	6C	···	6H	7A	7B	12A	12B	12C	12D	12E	···	12J	14A	14B	14C	···	14H	28A	28B	28C	28D	28E	···	28J
order	1	2	2	2	2	3	3	4	4	4	4	4	6	6	6	···	6	7	7	12	12	12	12	12	···	12	14	14	14	···	14	28	28	28	28	28	···	28
size	1	1	2	2	2	7	7	1	1	2	2	2	7	7	14	···	14	3	3	7	7	7	7	14	···	14	3	3	6	···	6	3	3	3	3	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	C₂	C₃	C₆	C₆	C₆	C₄○D₄	C₃×C₄○D₄	C₇⋊C₃	C₂×C₇⋊C₃	C₂×C₇⋊C₃	C₂×C₇⋊C₃	C₄○D₄×C₇⋊C₃
kernel	C₄○D₄×C₇⋊C₃	C₂×C₄×C₇⋊C₃	D₄×C₇⋊C₃	Q₈×C₇⋊C₃	C₇×C₄○D₄	C₂×C₂₈	C₇×D₄	C₇×Q₈	C₇⋊C₃	C₇	C₄○D₄	C₂×C₄	D₄	Q₈	C₁
# reps	1	3	3	1	2	6	6	2	2	4	2	6	6	2	4

Matrix representation of C₄○D₄×C₇⋊C₃ ►in GL₅(𝔽₃₃₇)

189	0	0	0	0
0	189	0	0	0
0	0	336	0	0
0	0	0	336	0
0	0	0	0	336

189	0	0	0	0
0	148	0	0	0
0	0	336	0	0
0	0	0	336	0
0	0	0	0	336

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

1	0	0	0	0
0	1	0	0	0
0	0	336	124	1
0	0	0	124	1
0	0	336	125	1

1	0	0	0	0
0	1	0	0	0
0	0	125	1	213
0	0	1	0	0
0	0	1	1	212

G:=sub<GL(5,GF(337))| [189,0,0,0,0,0,189,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[189,0,0,0,0,0,148,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[0,1,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,336,0,0,124,124,125,0,0,1,1,1],[1,0,0,0,0,0,1,0,0,0,0,0,125,1,1,0,0,1,0,1,0,0,213,0,212] >;

C₄○D₄×C₇⋊C₃ in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_7\rtimes C_3

% in TeX

G:=Group("C4oD4xC7:C3");

// GroupNames label

G:=SmallGroup(336,167);

// by ID

G=gap.SmallGroup(336,167);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,506,455]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^7=e^3=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*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 = C4○D4×C7⋊C3 order 336 = 24·3·7

Direct product of C4○D4 and C7⋊C3

G = C₄○D₄×C₇⋊C₃ order 336 = 2⁴·3·7

Direct product of C₄○D₄ and C₇⋊C₃