D7xC4.D4 - GroupNames

G = D₇×C₄.D₄ order 448 = 2⁶·7

Direct product of D₇ and C₄.D₄

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

Aliases: D₇×C₄.D₄, M₄(2)⋊₁₄D₁₄, (C₄×D₇).₁D₄, C₄.₁₄₆(D₄×D₇), C₂₈.₈₉(C₂×D₄), C₂³.₆(C₄×D₇), C₂₈.D₄⋊₃C₂, (D₇×M₄(2))⋊₅C₂, (C₂×C₂₈).₁C₂³, (C₂³×D₇).₂C₄, (C₂×D₄).₁₂₁D₁₄, C₂₈.₄₆D₄⋊₅C₂, C₄.Dic₇⋊₁C₂², (C₂×D₂₈).₃₈C₂², (D₄×C₁₄).₁₁C₂², D₁₄.₁₆(C₂²⋊C₄), (C₇×M₄(2))⋊₁₂C₂², Dic₇.₄(C₂²⋊C₄), (C₂×D₄×D₇).₂C₂, C₇⋊₁(C₂×C₄.D₄), (C₂×C₇⋊D₄).₁C₄, (C₂×C₄×D₇).₁C₂², C₂².₁₄(C₂×C₄×D₇), (C₇×C₄.D₄)⋊₅C₂, C₂.₁₃(D₇×C₂²⋊C₄), (C₂×C₄).₁(C₂²×D₇), C₁₄.₁₂(C₂×C₂²⋊C₄), (C₂²×C₁₄).₆(C₂×C₄), (C₂×C₁₄).₈(C₂²×C₄), (C₂²×D₇).₂(C₂×C₄), (C₂×Dic₇).₁₉(C₂×C₄), SmallGroup(448,278)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₇×C₄.D₄

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₂×C₄×D₇ — C₂×D₄×D₇ — D₇×C₄.D₄

Lower central

C₇ — C₁₄ — C₂×C₁₄ — D₇×C₄.D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₄.D₄

Generators and relations for D₇×C₄.D₄
G = < a,b,c,d,e | a⁷=b²=c⁴=1, d⁴=c², e²=c, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=c^-1, ce=ec, ede^-1=c^-1d³ >

Subgroups: 1228 in 186 conjugacy classes, 53 normal (21 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₇, D₇, C₁₄, C₁₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄.D₄, C₄.D₄, C₂×M₄(2), C₂²×D₄, C₇⋊C₈, C₅₆, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂×C₄.D₄, C₈×D₇, C₈⋊D₇, C₄.Dic₇, C₇×M₄(2), C₂×C₄×D₇, C₂×D₂₈, D₄×D₇, C₂×C₇⋊D₄, D₄×C₁₄, C₂³×D₇, C₂₈.₄₆D₄, C₂₈.D₄, C₇×C₄.D₄, D₇×M₄(2), C₂×D₄×D₇, D₇×C₄.D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₇, C₂²⋊C₄, C₂²×C₄, C₂×D₄, D₁₄, C₄.D₄, C₂×C₂²⋊C₄, C₄×D₇, C₂²×D₇, C₂×C₄.D₄, C₂×C₄×D₇, D₄×D₇, D₇×C₂²⋊C₄, D₇×C₄.D₄

Smallest permutation representation of D₇×C₄.D₄
►On 56 points

Generators in S₅₆

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

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

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

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

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

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

55 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	7A	7B	7C	8A	8B	8C	8D	8E	8F	8G	8H	14A	14B	14C	14D	14E	14F	14G	···	14L	28A	···	28F	56A	···	56L
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	7	7	7	8	8	8	8	8	8	8	8	14	14	14	14	14	14	14	···	14	28	···	28	56	···	56
size	1	1	2	4	4	7	7	14	28	28	2	2	14	14	2	2	2	4	4	4	4	28	28	28	28	2	2	2	4	4	4	8	···	8	4	···	4	8	···	8

55 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

D₇

D₁₄

C₄×D₇

C₄.D₄

D₄×D₇

D₇×C₄.D₄

kernel

D₇×C₄.D₄

C₂₈.₄₆D₄

C₂₈.D₄

C₇×C₄.D₄

D₇×M₄(2)

C₂×D₄×D₇

C₂×C₇⋊D₄

C₂³×D₇

C₄×D₇

C₄.D₄

M₄(2)

C₂×D₄

C₂³

D₇

C₄

C₁

# reps

Matrix representation of D₇×C₄.D₄ ►in GL₆(𝔽₁₁₃)

0	1	0	0	0	0
112	24	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	1	0	0	0	0
1	0	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

112	0	0	0	0	0
0	112	0	0	0	0
0	0	112	111	0	0
0	0	1	1	0	0
0	0	0	18	112	69
0	0	98	0	36	1

98	0	0	0	0	0
0	98	0	0	0	0
0	0	98	0	36	2
0	0	0	0	0	112
0	0	0	0	15	95
0	0	1	1	0	0

98	0	0	0	0	0
0	98	0	0	0	0
0	0	98	0	36	0
0	0	0	0	0	1
0	0	0	69	15	0
0	0	1	1	0	0

G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,1,0,98,0,0,111,1,18,0,0,0,0,0,112,36,0,0,0,0,69,1],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,1,0,0,0,0,0,1,0,0,36,0,15,0,0,0,2,112,95,0],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,1,0,0,0,0,69,1,0,0,36,0,15,0,0,0,0,1,0,0] >;

D₇×C₄.D₄ in GAP, Magma, Sage, TeX

D_7\times C_4.D_4

% in TeX

G:=Group("D7xC4.D4");

// GroupNames label

G:=SmallGroup(448,278);

// by ID

G=gap.SmallGroup(448,278);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,58,570,136,438,18822]);

// Polycyclic

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

// generators/relations

G = D7×C4.D4 order 448 = 26·7

Direct product of D7 and C4.D4

G = D₇×C₄.D₄ order 448 = 2⁶·7

Direct product of D₇ and C₄.D₄