C7xD4:4D4 - GroupNames

G = C₇×D₄⋊₄D₄ order 448 = 2⁶·7

Direct product of C₇ and D₄⋊₄D₄

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

Aliases: C₇×D₄⋊₄D₄, 2₊¹⁺⁴⋊₁C₁₄, D₄⋊₄(C₇×D₄), C₄≀C₂⋊₁C₁₄, Q₈⋊₄(C₇×D₄), (C₇×D₄)⋊₂₂D₄, (C₇×Q₈)⋊₂₂D₄, C₄⋊₁D₄⋊₁C₁₄, C₈⋊C₂²⋊₁C₁₄, C₄²⋊₂(C₂×C₁₄), C₄.₂₇(D₄×C₁₄), C₂³.₅(C₇×D₄), (C₄×C₂₈)⋊₃₅C₂², C₄.D₄⋊₁C₁₄, C₂₈.₃₈₈(C₂×D₄), (C₂²×C₁₄).₅D₄, (D₄×C₁₄)⋊₂₈C₂², M₄(2)⋊₁(C₂×C₁₄), C₂².₁₄(D₄×C₁₄), C₁₄.₁₀₀C₂²≀C₂, (C₂×C₂₈).₆₀₉C₂³, (C₇×2₊¹⁺⁴)⋊₇C₂, (C₇×M₄(2))⋊₁₇C₂², (C₇×C₄≀C₂)⋊₉C₂, (C₂×D₄)⋊₂(C₂×C₁₄), (C₇×C₈⋊C₂²)⋊₈C₂, (C₇×C₄⋊₁D₄)⋊₁₁C₂, C₄○D₄.₁(C₂×C₁₄), (C₇×C₄.D₄)⋊₇C₂, C₂.₁₄(C₇×C₂²≀C₂), (C₂×C₁₄).₄₀₉(C₂×D₄), (C₂×C₄).₄(C₂²×C₁₄), (C₇×C₄○D₄).₃₁C₂², SmallGroup(448,861)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₇×D₄⋊₄D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂×C₂₈ — D₄×C₁₄ — C₇×C₈⋊C₂² — C₇×D₄⋊₄D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₇×D₄⋊₄D₄

Upper central

C₁ — C₁₄ — C₂×C₂₈ — C₇×D₄⋊₄D₄

Generators and relations for C₇×D₄⋊₄D₄
G = < a,b,c,d,e | a⁷=b⁴=c²=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, dcd^-1=b^-1c, ece=bc, ede=d^-1 >

Subgroups: 370 in 168 conjugacy classes, 54 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₁₄, C₁₄, C₄², M₄(2), D₈, SD₁₆, C₂×D₄, C₂×D₄, C₄○D₄, C₄○D₄, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₄.D₄, C₄≀C₂, C₄⋊₁D₄, C₈⋊C₂², 2₊¹⁺⁴, C₅₆, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₇×Q₈, C₂²×C₁₄, C₂²×C₁₄, D₄⋊₄D₄, C₄×C₂₈, C₇×M₄(2), C₇×D₈, C₇×SD₁₆, D₄×C₁₄, D₄×C₁₄, C₇×C₄○D₄, C₇×C₄○D₄, C₇×C₄.D₄, C₇×C₄≀C₂, C₇×C₄⋊₁D₄, C₇×C₈⋊C₂², C₇×2₊¹⁺⁴, C₇×D₄⋊₄D₄
Quotients: C₁, C₂, C₂², C₇, D₄, C₂³, C₁₄, C₂×D₄, C₂×C₁₄, C₂²≀C₂, C₇×D₄, C₂²×C₁₄, D₄⋊₄D₄, D₄×C₁₄, C₇×C₂²≀C₂, C₇×D₄⋊₄D₄

Smallest permutation representation of C₇×D₄⋊₄D₄
►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 43 35 38)(2 44 29 39)(3 45 30 40)(4 46 31 41)(5 47 32 42)(6 48 33 36)(7 49 34 37)(8 23 21 56)(9 24 15 50)(10 25 16 51)(11 26 17 52)(12 27 18 53)(13 28 19 54)(14 22 20 55)

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

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

(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)

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

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

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

112 conjugacy classes

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

112 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+				+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₇	C₁₄	C₁₄	C₁₄	C₁₄	C₁₄	D₄	D₄	D₄	C₇×D₄	C₇×D₄	C₇×D₄	D₄⋊₄D₄	C₇×D₄⋊₄D₄
kernel	C₇×D₄⋊₄D₄	C₇×C₄.D₄	C₇×C₄≀C₂	C₇×C₄⋊₁D₄	C₇×C₈⋊C₂²	C₇×2₊¹⁺⁴	D₄⋊₄D₄	C₄.D₄	C₄≀C₂	C₄⋊₁D₄	C₈⋊C₂²	2₊¹⁺⁴	C₇×D₄	C₇×Q₈	C₂²×C₁₄	D₄	Q₈	C₂³	C₇	C₁
# reps	1	1	2	1	2	1	6	6	12	6	12	6	2	2	2	12	12	12	2	12

Matrix representation of C₇×D₄⋊₄D₄ ►in GL₄(𝔽₁₁₃) generated by

30	0	0	0
0	30	0	0
0	0	30	0
0	0	0	30

0	1	27	50
112	0	14	36
0	0	112	1
0	0	111	1

86	0	0	88
99	0	1	93
1	1	0	77
2	0	0	27

112	0	27	86
0	112	14	99
0	0	112	1
0	0	111	1

0	1	0	50
1	0	0	63
0	0	112	1
0	0	0	1

G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,0,0,1,0,0,0,27,14,112,111,50,36,1,1],[86,99,1,2,0,0,1,0,0,1,0,0,88,93,77,27],[112,0,0,0,0,112,0,0,27,14,112,111,86,99,1,1],[0,1,0,0,1,0,0,0,0,0,112,0,50,63,1,1] >;

C₇×D₄⋊₄D₄ in GAP, Magma, Sage, TeX

C_7\times D_4\rtimes_4D_4

% in TeX

G:=Group("C7xD4:4D4");

// GroupNames label

G:=SmallGroup(448,861);

// by ID

G=gap.SmallGroup(448,861);

# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2438,9804,4911,2468,172,7068]);

// Polycyclic

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

// generators/relations

G = C7×D4⋊4D4 order 448 = 26·7

Direct product of C7 and D4⋊4D4

G = C₇×D₄⋊₄D₄ order 448 = 2⁶·7

Direct product of C₇ and D₄⋊₄D₄