C2^4.348C2^3 - GroupNames

G = C₂⁴.₃₄₈C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₄₈C₂³, C₂³.₄₉₇C₂⁴, C₂².₂₇₉2₊¹⁺⁴, C₂³.₅₉(C₄○D₄), (C₂×C₄²).₇₉C₂², C₂³.₈Q₈⋊₇₅C₂, C₂³.₁₁D₄⋊₅₁C₂, (C₂²×C₄).₁₁₉C₂³, (C₂³×C₄).₁₂₉C₂², C₂⁴.C₂²⋊₉₆C₂, C₂³.₁₀D₄.₂₈C₂, C₂³.₂₃D₄.₄₂C₂, (C₂²×D₄).₁₈₂C₂², C₂³.₈₃C₂³⋊₅₃C₂, C₂.₃₅(C₂².₃₂C₂⁴), C₂⁴.₃C₂².₅₄C₂, C₂³.₆₃C₂³⋊₁₀₂C₂, C₂.₆₄(C₂².₄₅C₂⁴), C₂.C₄².₄₉₆C₂², C₂.₉₉(C₂³.₃₆C₂³), C₂.₂₉(C₂².₅₃C₂⁴), C₂.₇₄(C₂².₄₇C₂⁴), (C₄×C₄⋊C₄)⋊₁₁₀C₂, (C₄×C₂²⋊C₄)⋊₁₉C₂, (C₂×C₄).₄₀₆(C₄○D₄), (C₂×C₄⋊C₄).₃₃₈C₂², C₂².₃₇₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₈C₂², SmallGroup(128,1329)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₄₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂⁴.₃₄₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₄₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₄₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₄₈C₂³

Generators and relations for C₂⁴.₃₄₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=dc=cd, f²=d, g²=c, eae^-1=gag^-1=ab=ba, ac=ca, ad=da, faf^-1=acd, bc=cb, bd=db, geg^-1=be=eb, bf=fb, bg=gb, 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, C₂⁴.₃₄₈C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₃C₂⁴, C₂⁴.₃₄₈C₂³

Smallest permutation representation of C₂⁴.₃₄₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

C₂⁴.₃₄₈C₂³

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 C₂⁴.₃₄₈C₂³ ►in GL₆(𝔽₅)

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

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	4	0
0	0	0	0	0	4

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

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

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

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

C₂⁴.₃₄₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{348}C_2^3

% in TeX

G:=Group("C2^4.348C2^3");

// GroupNames label

G:=SmallGroup(128,1329);

// by ID

G=gap.SmallGroup(128,1329);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,680,758,723,352,675,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d*c=c*d,f^2=d,g^2=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*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,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 = C24.348C23 order 128 = 27

188th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₄₈C₂³ order 128 = 2⁷

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