C2^3.374C2^4 - GroupNames

G = C₂³.₃₇₄C₂⁴ order 128 = 2⁷

91^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₇₄C₂⁴, C₂⁴.₂₉₁C₂³, C₂².₁₇₉2₊¹⁺⁴, C₂².₁₃₂2_-¹⁺⁴, C₂²⋊C₄.₁₃₂D₄, C₂³.₁₇₈(C₂×D₄), C₂.₃₅(D₄⋊₆D₄), C₂.₅₆(D₄⋊₅D₄), (C₂×C₄²).₃₃C₂², C₂³.Q₈⋊₂₀C₂, C₂³.₈Q₈⋊₅₅C₂, C₂³.₁₁D₄⋊₂₀C₂, (C₂²×C₄).₅₁₈C₂³, (C₂³×C₄).₃₆₂C₂², C₂³.₁₀D₄.₉C₂, C₂².₂₅₄(C₂²×D₄), C₂⁴.C₂²⋊₅₅C₂, C₂³.₂₃D₄.₂₁C₂, (C₂²×D₄).₁₄₀C₂², (C₂²×Q₈).₁₁₂C₂², C₂³.₆₃C₂³⋊₅₂C₂, C₂³.₇₈C₂³⋊₁₃C₂, C₂³.₆₇C₂³⋊₄₆C₂, C₂.₄₆(C₂².₁₉C₂⁴), C₂.C₄².₁₃₀C₂², C₂.₁₈(C₂².₅₀C₂⁴), C₂.₄₀(C₂³.₃₆C₂³), C₂.₁₀(C₂².₄₉C₂⁴), C₂.₂₅(C₂².₃₆C₂⁴), (C₄×C₄⋊C₄)⋊₆₃C₂, (C₂×C₄).₉₀₁(C₂×D₄), (C₂×C₄²⋊C₂)⋊₂₄C₂, (C₂×C₄).₃₆₉(C₄○D₄), (C₂×C₄⋊C₄).₂₅₂C₂², (C₂×C₄.₄D₄).₂₂C₂, C₂².₂₅₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₅₈C₂², SmallGroup(128,1206)

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

Derived series

C₁ — C₂³ — C₂³.₃₇₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₃₇₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₇₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₇₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₇₄C₂⁴

Generators and relations for C₂³.₃₇₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=cb=bc, f²=b, g²=c, eae^-1=gag^-1=ab=ba, faf^-1=ac=ca, ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 500 in 263 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₄×C₄⋊C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂×C₄²⋊C₂, C₂×C₄.₄D₄, C₂³.₃₇₄C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂³.₃₇₄C₂⁴

Smallest permutation representation of C₂³.₃₇₄C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₃₇₄C₂⁴

C₄×C₄⋊C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂×C₄²⋊C₂

C₂×C₄.₄D₄

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₃₇₄C₂⁴ ►in GL₆(𝔽₅)

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

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	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

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

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

3	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	0	1
0	0	0	0	4	0

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

C₂³.₃₇₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{374}C_2^4

% in TeX

G:=Group("C2^3.374C2^4");

// GroupNames label

G:=SmallGroup(128,1206);

// by ID

G=gap.SmallGroup(128,1206);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,100,675,136]);

// Polycyclic

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

// generators/relations

G = C23.374C24 order 128 = 27

91st central stem extension by C23 of C24

G = C₂³.₃₇₄C₂⁴ order 128 = 2⁷

91^st central stem extension by C₂³ of C₂⁴