C2^3.384C2^4

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

Aliases: C₂³.₃₈₄C₂⁴, C₂⁴.₅₇₆C₂³, C₂².₁₃₈2_-⁽¹⁺⁴⁾, C₂³.₆₁₃(C₂×D₄), (C₂²×C₄).₃₈₁D₄, (C₂×C₄²).₄₀C₂², (C₂²×C₄).₆₉C₂³, C₂³.₃₀₆(C₄○D₄), (C₂³×C₄).₃₇₀C₂², C₂².₂₆₄(C₂²×D₄), C₂³.₈Q₈.₁₉C₂, C₂³.₃₄D₄.₁₄C₂, C₂³.₆₃C₂³⋊₅₉C₂, C₂³.₈₃C₂³⋊₁₆C₂, C₂³.₈₁C₂³⋊₂₃C₂, C₂.₅₅(C₂².₁₉C₂⁴), C₂.C₄².₁₃₇C₂², C₂².₆₃(C₂².D₄), C₂.₃₂(C₂².₄₆C₂⁴), C₂.₁₅(C₂³.₃₈C₂³), (C₂×C₄).₃₄₈(C₂×D₄), (C₂²×C₄⋊C₄).₃₆C₂, (C₂×C₄).₃₇₄(C₄○D₄), (C₂×C₄⋊C₄).₂₅₅C₂², C₂².₂₆₁(C₂×C₄○D₄), (C₂×C₄²⋊C₂).₄₁C₂, C₂.₂₉(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₄₆₂C₂², SmallGroup(128,1216)

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

Derived series

C₁ — C₂³ — C₂³.₃₈₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₄⋊C₄ — C₂³.₃₈₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₈₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₈₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₈₄C₂⁴

Subgroups: 436 in 256 conjugacy classes, 104 normal (18 characteristic)
C₁, C₂^[×3], C₂^[×4], C₂^[×4], C₄^[×18], C₂², C₂²^[×10], C₂²^[×12], C₂×C₄^[×8], C₂×C₄^[×58], C₂³, C₂³^[×6], C₂³^[×4], C₄²^[×4], C₂²⋊C₄^[×8], C₄⋊C₄^[×20], C₂²×C₄^[×2], C₂²×C₄^[×16], C₂²×C₄^[×16], C₂⁴, C₂.C₄²^[×16], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×4], C₂×C₄⋊C₄^[×2], C₂×C₄⋊C₄^[×8], C₂×C₄⋊C₄^[×4], C₄²⋊C₂^[×4], C₂³×C₄^[×3], C₂³.₃₄D₄, C₂³.₃₄D₄^[×2], C₂³.₈Q₈^[×2], C₂³.₆₃C₂³^[×4], C₂³.₈₁C₂³^[×2], C₂³.₈₃C₂³^[×2], C₂²×C₄⋊C₄, C₂×C₄²⋊C₂, C₂³.₃₈₄C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×8], C₂⁴, C₂².D₄^[×4], C₂²×D₄, C₂×C₄○D₄^[×4], 2_-⁽¹⁺⁴⁾^[×2], C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂³.₃₈C₂³, C₂².₄₆C₂⁴^[×4], C₂³.₃₈₄C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=c, e²=f²=a, ab=ba, ac=ca, ede^-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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	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	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	3	0
0	0	0	0	0	3

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

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

1	0	0	0	0	0
0	4	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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-⁽¹⁺⁴⁾

kernel

C₂³.₃₈₄C₂⁴

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂²×C₄⋊C₄

C₂×C₄²⋊C₂

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{384}C_2^4

% in TeX

G:=Group("C2^3.384C2^4");

// GroupNames label

G:=SmallGroup(128,1216);

// by ID

G=gap.SmallGroup(128,1216);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,100,675]);

// Polycyclic

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

// generators/relations

G = C₂³.₃₈₄C₂⁴ order 128 = 2⁷

101^st central stem extension by C₂³ of C₂⁴

G = C23.384C24 order 128 = 27

101st central stem extension by C23 of C24

G = C₂³.₃₈₄C₂⁴ order 128 = 2⁷

101^st central stem extension by C₂³ of C₂⁴