C2^4.347C2^3 - GroupNames

G = C₂⁴.₃₄₇C₂³ order 128 = 2⁷

187^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴.₃₄₇C₂³, C₂³.₄₉₅C₂⁴, C₂².₂₇₇2₊¹⁺⁴, C₂³.₅₈(C₄○D₄), (C₂×C₄²).₇₈C₂², C₂³.₈Q₈⋊₇₄C₂, C₂³.₁₁D₄⋊₅₀C₂, (C₂²×C₄).₁₁₇C₂³, (C₂³×C₄).₁₂₈C₂², C₂⁴.C₂²⋊₉₅C₂, C₂³.₁₀D₄.₂₇C₂, C₂³.₂₃D₄.₄₁C₂, (C₂²×D₄).₁₈₁C₂², C₂³.₈₃C₂³⋊₅₁C₂, C₂.₃₄(C₂².₃₂C₂⁴), C₂⁴.₃C₂².₅₃C₂, C₂³.₆₃C₂³⋊₁₀₀C₂, C₂.₆₃(C₂².₄₅C₂⁴), C₂.C₄².₄₉₄C₂², C₂.₉₇(C₂³.₃₆C₂³), C₂.₂₇(C₂².₅₃C₂⁴), C₂.₇₂(C₂².₄₇C₂⁴), (C₄×C₄⋊C₄)⋊₁₀₈C₂, (C₄×C₂²⋊C₄)⋊₁₈C₂, (C₂×C₄).₄₀₄(C₄○D₄), (C₂×C₄⋊C₄).₃₃₆C₂², C₂².₃₇₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₉₈C₂², SmallGroup(128,1327)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂³ — C₂⁴.₃₄₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂⁴.₃₄₇C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₄₇C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₄₇C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₄₇C₂³

Generators and relations for C₂⁴.₃₄₇C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=bcd, f²=d, g²=cb=bc, eae^-1=gag^-1=ab=ba, ac=ca, ad=da, faf^-1=acd, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, gfg^-1=cdf >

Subgroups: 452 in 227 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂⁴.₃₄₇C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₃C₂⁴, C₂⁴.₃₄₇C₂³

Smallest permutation representation of C₂⁴.₃₄₇C₂³
►On 64 points

Generators in S₆₄

(1 8)(2 61)(3 6)(4 63)(5 42)(7 44)(9 35)(10 37)(11 33)(12 39)(13 34)(14 40)(15 36)(16 38)(17 30)(18 26)(19 32)(20 28)(21 29)(22 25)(23 31)(24 27)(41 64)(43 62)(45 53)(46 59)(47 55)(48 57)(49 56)(50 58)(51 54)(52 60)

(1 41)(2 42)(3 43)(4 44)(5 61)(6 62)(7 63)(8 64)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(45 50)(46 51)(47 52)(48 49)(53 58)(54 59)(55 60)(56 57)

(1 53)(2 54)(3 55)(4 56)(5 46)(6 47)(7 48)(8 45)(9 29)(10 30)(11 31)(12 32)(13 27)(14 28)(15 25)(16 26)(17 37)(18 38)(19 39)(20 40)(21 35)(22 36)(23 33)(24 34)(41 58)(42 59)(43 60)(44 57)(49 63)(50 64)(51 61)(52 62)

(1 60)(2 57)(3 58)(4 59)(5 49)(6 50)(7 51)(8 52)(9 26)(10 27)(11 28)(12 25)(13 30)(14 31)(15 32)(16 29)(17 34)(18 35)(19 36)(20 33)(21 38)(22 39)(23 40)(24 37)(41 55)(42 56)(43 53)(44 54)(45 62)(46 63)(47 64)(48 61)

(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)(57 58 59 60)(61 62 63 64)

(1 7 60 51)(2 52 57 8)(3 5 58 49)(4 50 59 6)(9 17 26 34)(10 35 27 18)(11 19 28 36)(12 33 25 20)(13 38 30 21)(14 22 31 39)(15 40 32 23)(16 24 29 37)(41 63 55 46)(42 47 56 64)(43 61 53 48)(44 45 54 62)

(1 34 58 19)(2 40 59 21)(3 36 60 17)(4 38 57 23)(5 14 51 29)(6 10 52 25)(7 16 49 31)(8 12 50 27)(9 46 28 61)(11 48 26 63)(13 45 32 64)(15 47 30 62)(18 44 33 56)(20 42 35 54)(22 43 37 55)(24 41 39 53)

G:=sub<Sym(64)| (1,8)(2,61)(3,6)(4,63)(5,42)(7,44)(9,35)(10,37)(11,33)(12,39)(13,34)(14,40)(15,36)(16,38)(17,30)(18,26)(19,32)(20,28)(21,29)(22,25)(23,31)(24,27)(41,64)(43,62)(45,53)(46,59)(47,55)(48,57)(49,56)(50,58)(51,54)(52,60), (1,41)(2,42)(3,43)(4,44)(5,61)(6,62)(7,63)(8,64)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21)(25,30)(26,31)(27,32)(28,29)(33,38)(34,39)(35,40)(36,37)(45,50)(46,51)(47,52)(48,49)(53,58)(54,59)(55,60)(56,57), (1,53)(2,54)(3,55)(4,56)(5,46)(6,47)(7,48)(8,45)(9,29)(10,30)(11,31)(12,32)(13,27)(14,28)(15,25)(16,26)(17,37)(18,38)(19,39)(20,40)(21,35)(22,36)(23,33)(24,34)(41,58)(42,59)(43,60)(44,57)(49,63)(50,64)(51,61)(52,62), (1,60)(2,57)(3,58)(4,59)(5,49)(6,50)(7,51)(8,52)(9,26)(10,27)(11,28)(12,25)(13,30)(14,31)(15,32)(16,29)(17,34)(18,35)(19,36)(20,33)(21,38)(22,39)(23,40)(24,37)(41,55)(42,56)(43,53)(44,54)(45,62)(46,63)(47,64)(48,61), (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)(57,58,59,60)(61,62,63,64), (1,7,60,51)(2,52,57,8)(3,5,58,49)(4,50,59,6)(9,17,26,34)(10,35,27,18)(11,19,28,36)(12,33,25,20)(13,38,30,21)(14,22,31,39)(15,40,32,23)(16,24,29,37)(41,63,55,46)(42,47,56,64)(43,61,53,48)(44,45,54,62), (1,34,58,19)(2,40,59,21)(3,36,60,17)(4,38,57,23)(5,14,51,29)(6,10,52,25)(7,16,49,31)(8,12,50,27)(9,46,28,61)(11,48,26,63)(13,45,32,64)(15,47,30,62)(18,44,33,56)(20,42,35,54)(22,43,37,55)(24,41,39,53)>;

G:=Group( (1,8)(2,61)(3,6)(4,63)(5,42)(7,44)(9,35)(10,37)(11,33)(12,39)(13,34)(14,40)(15,36)(16,38)(17,30)(18,26)(19,32)(20,28)(21,29)(22,25)(23,31)(24,27)(41,64)(43,62)(45,53)(46,59)(47,55)(48,57)(49,56)(50,58)(51,54)(52,60), (1,41)(2,42)(3,43)(4,44)(5,61)(6,62)(7,63)(8,64)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21)(25,30)(26,31)(27,32)(28,29)(33,38)(34,39)(35,40)(36,37)(45,50)(46,51)(47,52)(48,49)(53,58)(54,59)(55,60)(56,57), (1,53)(2,54)(3,55)(4,56)(5,46)(6,47)(7,48)(8,45)(9,29)(10,30)(11,31)(12,32)(13,27)(14,28)(15,25)(16,26)(17,37)(18,38)(19,39)(20,40)(21,35)(22,36)(23,33)(24,34)(41,58)(42,59)(43,60)(44,57)(49,63)(50,64)(51,61)(52,62), (1,60)(2,57)(3,58)(4,59)(5,49)(6,50)(7,51)(8,52)(9,26)(10,27)(11,28)(12,25)(13,30)(14,31)(15,32)(16,29)(17,34)(18,35)(19,36)(20,33)(21,38)(22,39)(23,40)(24,37)(41,55)(42,56)(43,53)(44,54)(45,62)(46,63)(47,64)(48,61), (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)(57,58,59,60)(61,62,63,64), (1,7,60,51)(2,52,57,8)(3,5,58,49)(4,50,59,6)(9,17,26,34)(10,35,27,18)(11,19,28,36)(12,33,25,20)(13,38,30,21)(14,22,31,39)(15,40,32,23)(16,24,29,37)(41,63,55,46)(42,47,56,64)(43,61,53,48)(44,45,54,62), (1,34,58,19)(2,40,59,21)(3,36,60,17)(4,38,57,23)(5,14,51,29)(6,10,52,25)(7,16,49,31)(8,12,50,27)(9,46,28,61)(11,48,26,63)(13,45,32,64)(15,47,30,62)(18,44,33,56)(20,42,35,54)(22,43,37,55)(24,41,39,53) );

G=PermutationGroup([[(1,8),(2,61),(3,6),(4,63),(5,42),(7,44),(9,35),(10,37),(11,33),(12,39),(13,34),(14,40),(15,36),(16,38),(17,30),(18,26),(19,32),(20,28),(21,29),(22,25),(23,31),(24,27),(41,64),(43,62),(45,53),(46,59),(47,55),(48,57),(49,56),(50,58),(51,54),(52,60)], [(1,41),(2,42),(3,43),(4,44),(5,61),(6,62),(7,63),(8,64),(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21),(25,30),(26,31),(27,32),(28,29),(33,38),(34,39),(35,40),(36,37),(45,50),(46,51),(47,52),(48,49),(53,58),(54,59),(55,60),(56,57)], [(1,53),(2,54),(3,55),(4,56),(5,46),(6,47),(7,48),(8,45),(9,29),(10,30),(11,31),(12,32),(13,27),(14,28),(15,25),(16,26),(17,37),(18,38),(19,39),(20,40),(21,35),(22,36),(23,33),(24,34),(41,58),(42,59),(43,60),(44,57),(49,63),(50,64),(51,61),(52,62)], [(1,60),(2,57),(3,58),(4,59),(5,49),(6,50),(7,51),(8,52),(9,26),(10,27),(11,28),(12,25),(13,30),(14,31),(15,32),(16,29),(17,34),(18,35),(19,36),(20,33),(21,38),(22,39),(23,40),(24,37),(41,55),(42,56),(43,53),(44,54),(45,62),(46,63),(47,64),(48,61)], [(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),(57,58,59,60),(61,62,63,64)], [(1,7,60,51),(2,52,57,8),(3,5,58,49),(4,50,59,6),(9,17,26,34),(10,35,27,18),(11,19,28,36),(12,33,25,20),(13,38,30,21),(14,22,31,39),(15,40,32,23),(16,24,29,37),(41,63,55,46),(42,47,56,64),(43,61,53,48),(44,45,54,62)], [(1,34,58,19),(2,40,59,21),(3,36,60,17),(4,38,57,23),(5,14,51,29),(6,10,52,25),(7,16,49,31),(8,12,50,27),(9,46,28,61),(11,48,26,63),(13,45,32,64),(15,47,30,62),(18,44,33,56),(20,42,35,54),(22,43,37,55),(24,41,39,53)]])

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4H	4I	···	4X	4Y	4Z	4AA
order	1	2	···	2	2	2	2	4	···	4	4	···	4	4	4	4
size	1	1	···	1	4	4	8	2	···	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₃₄₇C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₄₇C₂³ ►in GL₆(𝔽₅)

3	4	0	0	0	0
3	2	0	0	0	0
0	0	0	3	0	0
0	0	2	0	0	0
0	0	0	0	2	4
0	0	0	0	3	3

1	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	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

4	0	0	0	0	0
0	4	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

1	3	0	0	0	0
1	4	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	2	4
0	0	0	0	0	3

3	0	0	0	0	0
3	2	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	4	0
0	0	0	0	0	4

3	4	0	0	0	0
3	2	0	0	0	0
0	0	0	4	0	0
0	0	1	0	0	0
0	0	0	0	2	0
0	0	0	0	3	3

G:=sub<GL(6,GF(5))| [3,3,0,0,0,0,4,2,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,2,3,0,0,0,0,4,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,4,3],[3,3,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,3,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,3,0,0,0,0,0,3] >;

C₂⁴.₃₄₇C₂³ in GAP, Magma, Sage, TeX

C_2^4._{347}C_2^3

% in TeX

G:=Group("C2^4.347C2^3");

// GroupNames label

G:=SmallGroup(128,1327);

// by ID

G=gap.SmallGroup(128,1327);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,792,758,723,352,675,136]);

// Polycyclic

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

// generators/relations

G = C24.347C23 order 128 = 27

187th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₄₇C₂³ order 128 = 2⁷

187^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²