C7xC4^2:C4 - GroupNames

G = C₇×C₄²⋊C₄ order 448 = 2⁶·7

Direct product of C₇ and C₄²⋊C₄

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C₇×C₄²⋊C₄, C₄²⋊₂C₂₈, (C₄×C₂₈)⋊₅C₄, (D₄×C₁₄)⋊₄C₄, (C₂×D₄)⋊₂C₂₈, C₂³⋊C₄⋊₂C₁₄, C₂³.₃(C₇×D₄), C₄⋊₁D₄.₂C₁₄, (C₂²×C₁₄).₃D₄, C₁₄.₃₄(C₂³⋊C₄), (D₄×C₁₄).₁₇₆C₂², (C₇×C₂³⋊C₄)⋊₈C₂, (C₂×C₄).₁(C₂×C₂₈), C₂.₈(C₇×C₂³⋊C₄), (C₂×C₂₈).₁₂(C₂×C₄), (C₂×D₄).₃(C₂×C₁₄), (C₇×C₄⋊₁D₄).₉C₂, C₂².₁₂(C₇×C₂²⋊C₄), (C₂×C₁₄).₇₅(C₂²⋊C₄), SmallGroup(448,157)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₇×C₄²⋊C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂×D₄ — D₄×C₁₄ — C₇×C₂³⋊C₄ — C₇×C₄²⋊C₄

Lower central

C₁ — C₂ — C₂² — C₂×C₄ — C₇×C₄²⋊C₄

Upper central

C₁ — C₁₄ — C₂×C₁₄ — D₄×C₁₄ — C₇×C₄²⋊C₄

Generators and relations for C₇×C₄²⋊C₄
G = < a,b,c,d | a⁷=b⁴=c⁴=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c, dcd^-1=b²c >

Subgroups: 242 in 86 conjugacy classes, 26 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₂×D₄, C₂×D₄, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₄⋊₁D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×C₁₄, C₂²×C₁₄, C₄²⋊C₄, C₄×C₂₈, C₇×C₂²⋊C₄, D₄×C₁₄, D₄×C₁₄, C₇×C₂³⋊C₄, C₇×C₄⋊₁D₄, C₇×C₄²⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₇, C₂×C₄, D₄, C₁₄, C₂²⋊C₄, C₂₈, C₂×C₁₄, C₂³⋊C₄, C₂×C₂₈, C₇×D₄, C₄²⋊C₄, C₇×C₂²⋊C₄, C₇×C₂³⋊C₄, C₇×C₄²⋊C₄

Smallest permutation representation of C₇×C₄²⋊C₄
►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)

(8 23 18 56)(9 24 19 50)(10 25 20 51)(11 26 21 52)(12 27 15 53)(13 28 16 54)(14 22 17 55)

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

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

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

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

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

91 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	7A	···	7F	14A	···	14F	14G	···	14L	14M	···	14X	14Y	···	14AD	28A	···	28R	28S	···	28AP
order	1	2	2	2	2	2	4	4	4	4	4	4	4	7	···	7	14	···	14	14	···	14	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	2	4	4	8	4	4	4	8	8	8	8	1	···	1	1	···	1	2	···	2	4	···	4	8	···	8	4	···	4	8	···	8

91 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4	4	4
type	+	+	+								+		+	+
image	C₁	C₂	C₂	C₄	C₄	C₇	C₁₄	C₁₄	C₂₈	C₂₈	D₄	C₇×D₄	C₂³⋊C₄	C₄²⋊C₄	C₇×C₂³⋊C₄	C₇×C₄²⋊C₄
kernel	C₇×C₄²⋊C₄	C₇×C₂³⋊C₄	C₇×C₄⋊₁D₄	C₄×C₂₈	D₄×C₁₄	C₄²⋊C₄	C₂³⋊C₄	C₄⋊₁D₄	C₄²	C₂×D₄	C₂²×C₁₄	C₂³	C₁₄	C₇	C₂	C₁
# reps	1	2	1	2	2	6	12	6	12	12	2	12	1	2	6	12

Matrix representation of C₇×C₄²⋊C₄ ►in GL₄(𝔽₂₉) generated by

20	0	0	0
0	20	0	0
0	0	20	0
0	0	0	20

1	0	0	1
0	1	0	0
0	0	0	28
0	0	1	0

28	28	28	0
2	1	1	1
0	0	0	28
0	0	1	0

1	0	0	0
0	0	0	1
0	1	0	0
27	28	28	28

G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,20,0,0,0,0,20],[1,0,0,0,0,1,0,0,0,0,0,1,1,0,28,0],[28,2,0,0,28,1,0,0,28,1,0,1,0,1,28,0],[1,0,0,27,0,0,1,28,0,0,0,28,0,1,0,28] >;

C₇×C₄²⋊C₄ in GAP, Magma, Sage, TeX

C_7\times C_4^2\rtimes C_4

% in TeX

G:=Group("C7xC4^2:C4");

// GroupNames label

G:=SmallGroup(448,157);

// by ID

G=gap.SmallGroup(448,157);

# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,3538,248,6871,375,14117]);

// Polycyclic

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

// generators/relations

G = C7×C42⋊C4 order 448 = 26·7

Direct product of C7 and C42⋊C4

G = C₇×C₄²⋊C₄ order 448 = 2⁶·7

Direct product of C₇ and C₄²⋊C₄