C7xC2wrC2^2 - GroupNames

G = C₇×C₂≀C₂² order 448 = 2⁶·7

Direct product of C₇ and C₂≀C₂²

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

Aliases: C₇×C₂≀C₂², 2₊¹⁺⁴⋊₂C₁₄, C₂³⋊(C₇×D₄), (C₂×C₂₈)⋊₄D₄, C₂³⋊C₄⋊₃C₁₄, C₂⁴⋊₂(C₂×C₁₄), (C₂²×C₁₄)⋊₁D₄, C₂²≀C₂⋊₁C₁₄, (C₂³×C₁₄)⋊₁C₂², C₂².₁₈(D₄×C₁₄), C₁₄.₁₀₄C₂²≀C₂, C₂³.₃(C₂²×C₁₄), (C₇×2₊¹⁺⁴)⋊₈C₂, (D₄×C₁₄).₁₈₃C₂², (C₂²×C₁₄).₈₂C₂³, (C₂×C₄)⋊(C₇×D₄), (C₇×C₂³⋊C₄)⋊₉C₂, C₂²⋊C₄⋊₁(C₂×C₁₄), (C₂×D₄).₈(C₂×C₁₄), (C₇×C₂²≀C₂)⋊₁₁C₂, C₂.₁₈(C₇×C₂²≀C₂), (C₂×C₁₄).₄₁₃(C₂×D₄), (C₇×C₂²⋊C₄)⋊₃₆C₂², SmallGroup(448,865)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₇×C₂≀C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₁₄ — C₂³×C₁₄ — C₇×C₂²≀C₂ — C₇×C₂≀C₂²

Lower central

C₁ — C₂ — C₂³ — C₇×C₂≀C₂²

Upper central

C₁ — C₁₄ — C₂²×C₁₄ — C₇×C₂≀C₂²

Generators and relations for C₇×C₂≀C₂²
G = < a,b,c,d,e,f | a⁷=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=fbf=bcd, ece^-1=cd=dc, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 450 in 198 conjugacy classes, 54 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₁₄, C₁₄, C₂²⋊C₄, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂⁴, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₂²≀C₂, 2₊¹⁺⁴, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂≀C₂², C₇×C₂²⋊C₄, C₇×C₂²⋊C₄, D₄×C₁₄, D₄×C₁₄, C₇×C₄○D₄, C₂³×C₁₄, C₇×C₂³⋊C₄, C₇×C₂²≀C₂, C₇×2₊¹⁺⁴, C₇×C₂≀C₂²
Quotients: C₁, C₂, C₂², C₇, D₄, C₂³, C₁₄, C₂×D₄, C₂×C₁₄, C₂²≀C₂, C₇×D₄, C₂²×C₁₄, C₂≀C₂², D₄×C₁₄, C₇×C₂²≀C₂, C₇×C₂≀C₂²

Smallest permutation representation of C₇×C₂≀C₂²
►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 14)(2 8)(3 9)(4 10)(5 11)(6 12)(7 13)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 32)(23 33)(24 34)(25 35)(26 29)(27 30)(28 31)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 50)

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

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

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

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

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

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

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

112 conjugacy classes

class	1	2A	2B	2C	2D	2E	···	2I	4A	4B	4C	4D	4E	4F	7A	···	7F	14A	···	14F	14G	···	14X	14Y	···	14BB	28A	···	28R	28S	···	28AJ
order	1	2	2	2	2	2	···	2	4	4	4	4	4	4	7	···	7	14	···	14	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	2	2	2	4	···	4	4	4	4	8	8	8	1	···	1	1	···	1	2	···	2	4	···	4	4	···	4	8	···	8

112 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+					+	+			+
image	C₁	C₂	C₂	C₂	C₇	C₁₄	C₁₄	C₁₄	D₄	D₄	C₇×D₄	C₇×D₄	C₂≀C₂²	C₇×C₂≀C₂²
kernel	C₇×C₂≀C₂²	C₇×C₂³⋊C₄	C₇×C₂²≀C₂	C₇×2₊¹⁺⁴	C₂≀C₂²	C₂³⋊C₄	C₂²≀C₂	2₊¹⁺⁴	C₂×C₂₈	C₂²×C₁₄	C₂×C₄	C₂³	C₇	C₁
# reps	1	3	3	1	6	18	18	6	3	3	18	18	2	12

Matrix representation of C₇×C₂≀C₂² ►in GL₆(𝔽₂₉)

16	0	0	0	0	0
0	16	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	28	0	0	0	0
28	0	0	0	0	0
0	0	0	0	1	0
0	0	15	26	1	2
0	0	1	0	0	0
0	0	22	25	23	3

28	0	0	0	0	0
0	28	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	15	26	1	2
0	0	23	23	0	28

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

1	0	0	0	0	0
0	28	0	0	0	0
0	0	14	3	28	27
0	0	0	0	1	0
0	0	1	0	0	0
0	0	25	6	9	15

28	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	14	3	28	27
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,15,1,22,0,0,0,26,0,25,0,0,1,1,0,23,0,0,0,2,0,3],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,15,23,0,0,1,0,26,23,0,0,0,0,1,0,0,0,0,0,2,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,14,0,1,25,0,0,3,0,0,6,0,0,28,1,0,9,0,0,27,0,0,15],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,14,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,27,1] >;

C₇×C₂≀C₂² in GAP, Magma, Sage, TeX

C_7\times C_2\wr C_2^2

% in TeX

G:=Group("C7xC2wrC2^2");

// GroupNames label

G:=SmallGroup(448,865);

// by ID

G=gap.SmallGroup(448,865);

# by ID

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

// Polycyclic

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

// generators/relations

G = C7×C2≀C22 order 448 = 26·7

Direct product of C7 and C2≀C22

G = C₇×C₂≀C₂² order 448 = 2⁶·7

Direct product of C₇ and C₂≀C₂²