C2^3.422C2^4 - GroupNames

G = C₂³.₄₂₂C₂⁴ order 128 = 2⁷

139^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₄₂₂C₂⁴, C₂⁴.₃₁₂C₂³, C₂².₂₁₅2₊¹⁺⁴, C₂²⋊C₄.₉Q₈, C₂³.₁₉(C₂×Q₈), C₂.₂₈(D₄⋊₃Q₈), C₂³.₄Q₈.₇C₂, C₂².₉₅(C₂²×Q₈), (C₂³×C₄).₃₈₆C₂², (C₂²×C₄).₅₃₂C₂³, (C₂×C₄²).₅₃₇C₂², C₂³.Q₈.₁₀C₂, C₂³.₇Q₈.₅₀C₂, C₂³.₈Q₈.₂₅C₂, C₂³.₆₅C₂³⋊₇₉C₂, C₂³.₆₃C₂³⋊₇₅C₂, C₂³.₈₁C₂³⋊₃₀C₂, C₂⁴.C₂².₂₅C₂, C₂.₃₈(C₂².₄₅C₂⁴), C₂.C₄².₁₇₀C₂², C₂.₄₆(C₂².₄₇C₂⁴), C₂.₂₄(C₂³.₃₇C₂³), C₂.₂₁(C₂².₃₄C₂⁴), C₂.₆₅(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₈₁C₂, (C₂×C₄).₄₈(C₂×Q₈), (C₄×C₂²⋊C₄).₅₇C₂, (C₂×C₄).₁₄₁(C₄○D₄), (C₂×C₄⋊C₄).₂₈₅C₂², C₂².₂₉₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₀₃C₂², SmallGroup(128,1254)

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

Derived series

C₁ — C₂³ — C₂³.₄₂₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₄₂₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₂₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₂₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₂₂C₂⁴

Generators and relations for C₂³.₄₂₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=ca=ac, e²=a, g²=ba=ab, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 388 in 214 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₄₂₂C₂⁴
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, D₄⋊₃Q₈, C₂³.₄₂₂C₂⁴

Smallest permutation representation of C₂³.₄₂₂C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

(2 27)(4 25)(5 39)(6 44)(7 37)(8 42)(9 54)(11 56)(14 57)(16 59)(17 33)(18 45)(19 35)(20 47)(21 41)(22 38)(23 43)(24 40)(30 49)(32 51)(34 63)(36 61)(46 64)(48 62)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

Q₈

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₄₂₂C₂⁴

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₂₂C₂⁴ ►in 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	4	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	4	0	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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

3	0	0	0	0	0
0	2	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

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],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂³.₄₂₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{422}C_2^4

% in TeX

G:=Group("C2^3.422C2^4");

// GroupNames label

G:=SmallGroup(128,1254);

// by ID

G=gap.SmallGroup(128,1254);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,675,192]);

// Polycyclic

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

// generators/relations

G = C23.422C24 order 128 = 27

139th central stem extension by C23 of C24

G = C₂³.₄₂₂C₂⁴ order 128 = 2⁷

139^th central stem extension by C₂³ of C₂⁴