C7xC4.D4 - GroupNames

G = C₇xC₄.D₄ order 224 = 2⁵·7

Direct product of C₇ and C₄.D₄

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₇xC₄.D₄, C₂³.C₂₈, C₂₈.₅₈D₄, M₄(2):₃C₁₄, C₄.₉(C₇xD₄), (D₄xC₁₄).₈C₂, (C₂xD₄).₂C₁₄, (C₇xM₄(2)):₉C₂, C₂².₃(C₂xC₂₈), (C₂²xC₁₄).₁C₄, (C₂xC₂₈).₅₉C₂², C₁₄.₂₂(C₂²:C₄), (C₂xC₄).₁(C₂xC₁₄), C₂.₄(C₇xC₂²:C₄), (C₂xC₁₄).₂₀(C₂xC₄), SmallGroup(224,49)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₇xC₄.D₄

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xC₂₈ — C₇xM₄(2) — C₇xC₄.D₄

Lower central

C₁ — C₂ — C₂² — C₇xC₄.D₄

Upper central

C₁ — C₁₄ — C₂xC₂₈ — C₇xC₄.D₄

Generators and relations for C₇xC₄.D₄
G = < a,b,c,d | a⁷=b⁴=1, c⁴=b², d²=b, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, bd=db, dcd^-1=b^-1c³ >

Subgroups: 84 in 46 conjugacy classes, 24 normal (12 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₇, C₂xC₄, D₄, C₁₄, C₂²:C₄, C₂₈, C₂xC₁₄, C₄.D₄, C₂xC₂₈, C₇xD₄, C₇xC₂²:C₄, C₇xC₄.D₄

C₁

C₂

₂C₂

₄C₂

C₇

C₄

C₂²

₂C₂²

₄C₂²

C₁₄

₂C₁₄

₄C₁₄

C₂³

C₂xC₄

₂D₄

₂C₈

₂D₄

₂C₈

C₂xC₁₄

C₂₈

₂C₂xC₁₄

₄C₂xC₁₄

C₂xD₄

M₄(2)

C₂²xC₁₄

C₂xC₂₈

₂C₇xD₄

₂C₅₆

₂C₇xD₄

C₄.D₄

D₄xC₁₄

C₇xM₄(2)

C₇xC₄.D₄

Smallest permutation representation of C₇xC₄.D₄
►On 56 points

Generators in S₅₆

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

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

(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)

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

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

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

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

C₇xC₄.D₄ is a maximal subgroup of C₂³.₃D₂₈ C₂³.₄D₂₈ M₄(2).₁₉D₁₄ D₂₈.₁D₄ D₂₈:₁D₄ D₂₈.₂D₄ D₂₈.₃D₄

77 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	7A	···	7F	8A	8B	8C	8D	14A	···	14F	14G	···	14L	14M	···	14X	28A	···	28L	56A	···	56X
order	1	2	2	2	2	4	4	7	···	7	8	8	8	8	14	···	14	14	···	14	14	···	14	28	···	28	56	···	56
size	1	1	2	4	4	2	2	1	···	1	4	4	4	4	1	···	1	2	···	2	4	···	4	2	···	2	4	···	4

77 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+						+		+
image	C₁	C₂	C₂	C₄	C₇	C₁₄	C₁₄	C₂₈	D₄	C₇xD₄	C₄.D₄	C₇xC₄.D₄
kernel	C₇xC₄.D₄	C₇xM₄(2)	D₄xC₁₄	C₂²xC₁₄	C₄.D₄	M₄(2)	C₂xD₄	C₂³	C₂₈	C₄	C₇	C₁
# reps	1	2	1	4	6	12	6	24	2	12	1	6

Matrix representation of C₇xC₄.D₄ ►in GL₆(F₁₁₃)

106	0	0	0	0	0
0	106	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

112	0	0	0	0	0
0	112	0	0	0	0
0	0	0	1	0	0
0	0	112	0	0	0
0	0	0	0	0	1
0	0	0	0	112	0

1	0	0	0	0	0
112	112	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	112	0	0	0
0	0	0	1	0	0

1	2	0	0	0	0
112	112	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	112	0	0	0

G:=sub<GL(6,GF(113))| [106,0,0,0,0,0,0,106,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[1,112,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[1,112,0,0,0,0,2,112,0,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₇xC₄.D₄ in GAP, Magma, Sage, TeX

C_7\times C_4.D_4

% in TeX

G:=Group("C7xC4.D4");

// GroupNames label

G:=SmallGroup(224,49);

// by ID

G=gap.SmallGroup(224,49);

# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,3363,2530,88]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₇xC₄.D₄ in TeX

G = C7xC4.D4 order 224 = 25·7

Direct product of C7 and C4.D4

G = C₇xC₄.D₄ order 224 = 2⁵·7

Direct product of C₇ and C₄.D₄