D7xC4wrC2 - GroupNames

G = D₇xC₄wrC₂ order 448 = 2⁶·7

Direct product of D₇ and C₄wrC₂

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₇xC₄wrC₂, C₄²:₃₁D₁₄, M₄(2):₁₅D₁₄, D₄:₇(C₄xD₇), (D₄xD₇):₃C₄, Q₈:₇(C₄xD₇), (Q₈xD₇):₃C₄, D₂₈:₅(C₂xC₄), D₄:₂D₇:₃C₄, (C₄xC₂₈):₉C₂², Q₈:₂D₇:₃C₄, (C₄xD₇).₃₂D₄, C₄.₂₀₀(D₄xD₇), (D₇xC₄²):₁C₂, D₂₈:₄C₄:₅C₂, Dic₁₄:C₄:₃C₂, Dic₁₄:₅(C₂xC₄), C₄oD₄.₁₈D₁₄, C₂₈.₃₅₉(C₂xD₄), (D₇xM₄(2)):₈C₂, C₂².₂₇(D₄xD₇), D₄:₂Dic₇:₁C₂, C₂₈.₁₇(C₂²xC₄), C₄oD₂₈.₉C₂², (C₂²xD₇).₈₀D₄, C₄.Dic₇:₂C₂², (C₂xC₂₈).₂₆₀C₂³, (C₂xDic₇).₁₅₉D₄, (C₄xDic₇):₆₁C₂², D₁₄.₂₀(C₂²:C₄), (C₇xM₄(2)):₁₃C₂², Dic₇.₁₀(C₂²:C₄), C₇:₁(C₂xC₄wrC₂), (C₇xC₄wrC₂):₅C₂, C₄.₁₇(C₂xC₄xD₇), (C₇xD₄):₅(C₂xC₄), (C₇xQ₈):₅(C₂xC₄), (D₇xC₄oD₄).₁C₂, (C₄xD₇).₁₆(C₂xC₄), (C₂xC₁₄).₂₄(C₂xD₄), C₂.₂₅(D₇xC₂²:C₄), C₁₄.₂₄(C₂xC₂²:C₄), (C₇xC₄oD₄).₁C₂², (C₂xC₄xD₇).₂₃₀C₂², (C₂xC₄).₃₆₇(C₂²xD₇), SmallGroup(448,354)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — D₇xC₄wrC₂

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂xC₂₈ — C₂xC₄xD₇ — D₇xC₄oD₄ — D₇xC₄wrC₂

Lower central

C₇ — C₁₄ — C₂₈ — D₇xC₄wrC₂

Upper central

C₁ — C₄ — C₂xC₄ — C₄wrC₂

Generators and relations for D₇xC₄wrC₂
G = < a,b,c,d,e | a⁷=b²=c⁴=d²=e⁴=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c^-1, ce=ec, ede^-1=c^-1d >

Subgroups: 892 in 170 conjugacy classes, 53 normal (51 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, D₇, D₇, C₁₄, C₁₄, C₄², C₄², C₂xC₈, M₄(2), M₄(2), C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂xC₁₄, C₂xC₁₄, C₄wrC₂, C₄wrC₂, C₂xC₄², C₂xM₄(2), C₂xC₄oD₄, C₇:C₈, C₅₆, Dic₁₄, Dic₁₄, C₄xD₇, C₄xD₇, D₂₈, D₂₈, C₂xDic₇, C₂xDic₇, C₇:D₄, C₂xC₂₈, C₂xC₂₈, C₇xD₄, C₇xD₄, C₇xQ₈, C₂²xD₇, C₂²xD₇, C₂xC₄wrC₂, C₈xD₇, C₈:D₇, C₄.Dic₇, C₄xDic₇, C₄xDic₇, C₄xC₂₈, C₇xM₄(2), C₂xC₄xD₇, C₂xC₄xD₇, C₄oD₂₈, C₄oD₂₈, D₄xD₇, D₄xD₇, D₄:₂D₇, D₄:₂D₇, Q₈xD₇, Q₈:₂D₇, C₇xC₄oD₄, Dic₁₄:C₄, D₂₈:₄C₄, D₄:₂Dic₇, C₇xC₄wrC₂, D₇xC₄², D₇xM₄(2), D₇xC₄oD₄, D₇xC₄wrC₂
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, D₇, C₂²:C₄, C₂²xC₄, C₂xD₄, D₁₄, C₄wrC₂, C₂xC₂²:C₄, C₄xD₇, C₂²xD₇, C₂xC₄wrC₂, C₂xC₄xD₇, D₄xD₇, D₇xC₂²:C₄, D₇xC₄wrC₂

Smallest permutation representation of D₇xC₄wrC₂
►On 56 points

Generators in S₅₆

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

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

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

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

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	···	4G	4H	4I	4J	4K	···	4O	4P	7A	7B	7C	8A	8B	8C	8D	14A	14B	14C	14D	14E	14F	14G	14H	14I	28A	···	28F	28G	···	28U	28V	28W	28X	56A	···	56F
order	1	2	2	2	2	2	2	2	4	4	4	···	4	4	4	4	4	···	4	4	7	7	7	8	8	8	8	14	14	14	14	14	14	14	14	14	28	···	28	28	···	28	28	28	28	56	···	56
size	1	1	2	4	7	7	14	28	1	1	2	···	2	4	7	7	14	···	14	28	2	2	2	4	4	28	28	2	2	2	4	4	4	8	8	8	2	···	2	4	···	4	8	8	8	8	···	8

70 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+					+	+	+	+	+	+	+				+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₄	D₄	D₇	D₁₄	D₁₄	D₁₄	C₄wrC₂	C₄xD₇	C₄xD₇	D₄xD₇	D₄xD₇	D₇xC₄wrC₂
kernel	D₇xC₄wrC₂	Dic₁₄:C₄	D₂₈:₄C₄	D₄:₂Dic₇	C₇xC₄wrC₂	D₇xC₄²	D₇xM₄(2)	D₇xC₄oD₄	D₄xD₇	D₄:₂D₇	Q₈xD₇	Q₈:₂D₇	C₄xD₇	C₂xDic₇	C₂²xD₇	C₄wrC₂	C₄²	M₄(2)	C₄oD₄	D₇	D₄	Q₈	C₄	C₂²	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	1	1	3	3	3	3	8	6	6	3	3	12

Matrix representation of D₇xC₄wrC₂ ►in GL₄(F₁₁₃) generated by

104	1	0	0
41	33	0	0
0	0	1	0
0	0	0	1

33	112	0	0
71	80	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	1	0	0
0	0	98	0
0	0	28	15

112	0	0	0
0	112	0	0
0	0	1	106
0	0	0	112

1	0	0	0
0	1	0	0
0	0	112	0
0	0	30	15

G:=sub<GL(4,GF(113))| [104,41,0,0,1,33,0,0,0,0,1,0,0,0,0,1],[33,71,0,0,112,80,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,28,0,0,0,15],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,106,112],[1,0,0,0,0,1,0,0,0,0,112,30,0,0,0,15] >;

D₇xC₄wrC₂ in GAP, Magma, Sage, TeX

D_7\times C_4\wr C_2

% in TeX

G:=Group("D7xC4wrC2");

// GroupNames label

G:=SmallGroup(448,354);

// by ID

G=gap.SmallGroup(448,354);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,58,136,851,438,102,18822]);

// Polycyclic

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

// generators/relations

G = D7xC4wrC2 order 448 = 26·7

Direct product of D7 and C4wrC2

G = D₇xC₄wrC₂ order 448 = 2⁶·7

Direct product of D₇ and C₄wrC₂