C2^4.340C2^3

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

Aliases: C₂⁴.₃₄₀C₂³, C₂³.₄₇₆C₂⁴, C₂².₂₅₉2₊⁽¹⁺⁴⁾, C₄²⋊₈C₄⋊₄₇C₂, C₂³.₅₅(C₄○D₄), (C₂×C₄²).₇₁C₂², C₂³.₈Q₈⋊₇₀C₂, C₂³.Q₈⋊₃₂C₂, C₂³.₁₁D₄⋊₄₇C₂, C₂³.₃₄D₄⋊₃₈C₂, (C₂³×C₄).₁₂₂C₂², (C₂²×C₄).₁₀₇C₂³, C₂³.₈₄C₂³⋊₅C₂, C₂⁴.C₂²⋊₉₀C₂, C₂³.₁₀D₄.₂₄C₂, C₂³.₂₃D₄.₃₈C₂, (C₂²×D₄).₁₇₆C₂², C₂³.₆₃C₂³⋊₉₃C₂, C₂.₂₈(C₂².₃₂C₂⁴), C₂.₅₇(C₂².₄₅C₂⁴), C₂.C₄².₄₉₁C₂², C₂.₆₂(C₂².₄₇C₂⁴), C₂.₈₇(C₂³.₃₆C₂³), (C₄×C₂²⋊C₄)⋊₁₆C₂, (C₂×C₄).₁₅₄(C₄○D₄), (C₂×C₄⋊C₄).₃₂₃C₂², C₂².₃₅₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₉₁C₂², SmallGroup(128,1308)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₄₀C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Subgroups: 468 in 234 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×4], C₄^[×16], C₂²^[×7], C₂²^[×20], C₂×C₄^[×6], C₂×C₄^[×44], D₄^[×4], C₂³, C₂³^[×4], C₂³^[×12], C₄²^[×4], C₂²⋊C₄^[×17], C₄⋊C₄^[×7], C₂²×C₄^[×13], C₂²×C₄^[×9], C₂×D₄^[×5], C₂⁴^[×2], C₂.C₄²^[×14], C₂×C₄²^[×3], C₂×C₂²⋊C₄^[×10], C₂×C₄⋊C₄^[×5], C₂³×C₄^[×2], C₂²×D₄, C₄×C₂²⋊C₄^[×2], C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄^[×2], C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×2], C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₄C₂³, C₂⁴.₃₄₀C₂³

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×10], C₂⁴, C₂×C₄○D₄^[×5], 2₊⁽¹⁺⁴⁾^[×2], C₂³.₃₆C₂³^[×2], C₂².₃₂C₂⁴, C₂².₄₅C₂⁴^[×2], C₂².₄₇C₂⁴^[×2], C₂⁴.₃₄₀C₂³

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

Smallest permutation representation
►On 64 points

Generators in S₆₄

(2 42)(4 44)(5 63)(6 8)(7 61)(10 15)(12 13)(17 24)(18 20)(19 22)(21 23)(25 30)(27 32)(33 35)(34 37)(36 39)(38 40)(45 47)(46 49)(48 51)(50 52)(54 59)(56 57)(62 64)

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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

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

4	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	1	0
0	0	0	0	0	1

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

0	2	0	0	0	0
3	0	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	0
0	0	0	0	0	2

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

kernel

C₂⁴.₃₄₀C₂³

C₄×C₂²⋊C₄

C₂³.₃₄D₄

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^4._{340}C_2^3

% in TeX

G:=Group("C2^4.340C2^3");

// GroupNames label

G:=SmallGroup(128,1308);

// by ID

G=gap.SmallGroup(128,1308);

# by ID

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

// Polycyclic

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

// generators/relations

G = C₂⁴.₃₄₀C₂³ order 128 = 2⁷

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

G = C24.340C23 order 128 = 27

180th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₄₀C₂³ order 128 = 2⁷

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