C2^3.385C2^4 - GroupNames

G = C₂³.₃₈₅C₂⁴ order 128 = 2⁷

102^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₈₅C₂⁴, C₂⁴.₂₉₈C₂³, C₂².₁₈₇2₊¹⁺⁴, C₂².₁₃₉2_-¹⁺⁴, (C₂²×C₄).₃₈₂D₄, C₂³.₁₈₂(C₂×D₄), C₂³.₃₉(C₄○D₄), (C₂×C₄²).₄₁C₂², C₂³.₇Q₈⋊₅₃C₂, C₂³.₁₁D₄⋊₂₆C₂, (C₂²×C₄).₅₂₁C₂³, (C₂³×C₄).₃₇₁C₂², C₂².₂₆₅(C₂²×D₄), C₂⁴.C₂²⋊₆₂C₂, C₂³.₁₀D₄.₁₂C₂, (C₂²×D₄).₅₂₃C₂², C₂³.₆₃C₂³⋊₆₀C₂, C₂³.₆₅C₂³⋊₆₇C₂, C₂.₅₆(C₂².₁₉C₂⁴), C₄.₁₂₄(C₂².D₄), C₂.C₄².₁₃₈C₂², C₂.₂₀(C₂².₅₀C₂⁴), C₂.₃₃(C₂².₄₆C₂⁴), C₂.₃₀(C₂².₄₇C₂⁴), C₂.₁₀(C₂².₃₁C₂⁴), C₂.₁₂(C₂².₄₉C₂⁴), (C₂×C₄×D₄).₅₅C₂, (C₂×C₄).₁₁₉₁(C₂×D₄), (C₂×C₄²⋊C₂)⋊₂₉C₂, (C₂×C₄).₁₂₁(C₄○D₄), (C₂×C₄⋊C₄).₂₅₆C₂², C₂².₂₆₂(C₂×C₄○D₄), C₂.₃₀(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₁₅₁C₂², SmallGroup(128,1217)

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

Derived series

C₁ — C₂³ — C₂³.₃₈₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₂³.₃₈₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₈₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₈₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₈₅C₂⁴

Generators and relations for C₂³.₃₈₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=d, g²=b, ab=ba, eae^-1=ac=ca, faf=ad=da, ag=ga, bc=cb, bd=db, fef=geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 500 in 270 conjugacy classes, 104 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂×C₄²⋊C₂, C₂×C₄×D₄, C₂³.₃₈₅C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂².₃₁C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂³.₃₈₅C₂⁴

Smallest permutation representation of C₂³.₃₈₅C₂⁴
►On 64 points

Generators in S₆₄

(2 52)(4 50)(5 64)(6 8)(7 62)(10 24)(12 22)(14 28)(16 26)(17 19)(18 30)(20 32)(29 31)(33 35)(34 38)(36 40)(37 39)(42 56)(44 54)(45 47)(46 58)(48 60)(57 59)(61 63)

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

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

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

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

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

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

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₃₈₅C₂⁴

C₂³.₇Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄²⋊C₂

C₂×C₄×D₄

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₈₅C₂⁴ ►in GL₆(𝔽₅)

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	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	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

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

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

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

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

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

C₂³.₃₈₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{385}C_2^4

% in TeX

G:=Group("C2^3.385C2^4");

// GroupNames label

G:=SmallGroup(128,1217);

// by ID

G=gap.SmallGroup(128,1217);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f=a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f=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 = C23.385C24 order 128 = 27

102nd central stem extension by C23 of C24

G = C₂³.₃₈₅C₂⁴ order 128 = 2⁷

102^nd central stem extension by C₂³ of C₂⁴