C2^4.314C2^3

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

Aliases: C₂⁴.₃₁₄C₂³, C₂³.₄₂₆C₂⁴, C₂².₂₁₈2₊⁽¹⁺⁴⁾, C₄²⋊₄C₄⋊₂₀C₂, (C₂×C₄²).₅₅C₂², (C₂²×C₄).₈₈C₂³, C₂³.Q₈⋊₂₅C₂, C₂³.₁₄₈(C₄○D₄), C₂³.₃₄D₄⋊₃₄C₂, C₂³.₁₁D₄⋊₃₈C₂, (C₂³×C₄).₁₀₉C₂², C₂⁴.C₂²⋊₇₄C₂, C₂³.₂₃D₄.₃₂C₂, C₂³.₁₀D₄.₁₆C₂, (C₂²×D₄).₁₅₈C₂², C₂³.₆₃C₂³⋊₇₈C₂, C₂³.₈₃C₂³⋊₃₂C₂, C₂³.₆₅C₂³⋊₈₁C₂, C₂⁴.₃C₂².₄₁C₂, C₂.₄₀(C₂².₄₅C₂⁴), C₂.C₄².₅₄₂C₂², C₂.₂₀(C₂².₄₉C₂⁴), C₂.₂₂(C₂².₃₄C₂⁴), C₂.₄₈(C₂².₄₇C₂⁴), C₂.₆₉(C₂³.₃₆C₂³), (C₄×C₂²⋊C₄)⋊₈₁C₂, (C₂×C₄).₅₂₂(C₄○D₄), (C₂×C₄⋊C₄).₂₈₈C₂², C₂².₃₀₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₁C₂², SmallGroup(128,1258)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₁₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₁₄C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₁₄C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₁₄C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₁₄C₂³

Subgroups: 452 in 227 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×3], C₄^[×17], C₂²^[×7], C₂²^[×17], C₂×C₄^[×8], C₂×C₄^[×39], D₄^[×4], C₂³, C₂³^[×2], C₂³^[×13], C₄²^[×7], C₂²⋊C₄^[×17], C₄⋊C₄^[×8], C₂²×C₄^[×13], C₂²×C₄^[×5], C₂×D₄^[×5], C₂⁴^[×2], C₂.C₄²^[×12], C₂×C₄²^[×4], C₂×C₂²⋊C₄^[×11], C₂×C₄⋊C₄^[×6], C₂³×C₄, C₂²×D₄, C₄²⋊₄C₄, C₄×C₂²⋊C₄, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂²^[×4], C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂⁴.₃₁₄C₂³

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×10], C₂⁴, C₂×C₄○D₄^[×5], 2₊⁽¹⁺⁴⁾^[×2], C₂³.₃₆C₂³^[×2], C₂².₃₄C₂⁴, C₂².₄₅C₂⁴^[×2], C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂⁴.₃₁₄C₂³

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

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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

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

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

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

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

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

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

C₂⁴.₃₁₄C₂³

C₄²⋊₄C₄

C₄×C₂²⋊C₄

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^4._{314}C_2^3

% in TeX

G:=Group("C2^4.314C2^3");

// GroupNames label

G:=SmallGroup(128,1258);

// by ID

G=gap.SmallGroup(128,1258);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,675,136]);

// Polycyclic

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

// generators/relations

G = C₂⁴.₃₁₄C₂³ order 128 = 2⁷

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

G = C24.314C23 order 128 = 27

154th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₁₄C₂³ order 128 = 2⁷

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