C2^4.332C2^3 - GroupNames

G = C₂⁴.₃₃₂C₂³ order 128 = 2⁷

172^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₃₂C₂³, C₂³.₄₆₀C₂⁴, C₂².₁₈₇2_-¹⁺⁴, C₂².₂₄₅2₊¹⁺⁴, C₄⋊C₄⋊₂₉D₄, C₂³⋊Q₈⋊₁₈C₂, C₂.₈₀(D₄⋊₅D₄), C₂.₄₁(Q₈⋊₅D₄), C₂³.₅₃(C₄○D₄), C₂³.₁₁D₄⋊₄₅C₂, (C₂×C₄²).₅₆₃C₂², (C₂²×C₄).₅₄₂C₂³, (C₂³×C₄).₄₀₁C₂², C₂².₃₁₁(C₂²×D₄), C₂⁴.C₂²⋊₈₆C₂, C₂³.₁₀D₄.₂₁C₂, C₂³.₂₃D₄.₃₅C₂, (C₂²×D₄).₁₇₄C₂², (C₂²×Q₈).₁₃₈C₂², C₂³.₆₇C₂³⋊₆₄C₂, C₂³.₆₃C₂³⋊₈₇C₂, C₂³.₈₁C₂³⋊₄₁C₂, C₂⁴.₃C₂².₄₇C₂, C₂.₄₉(C₂².₄₅C₂⁴), C₂.C₄².₁₉₇C₂², C₂.₄₀(C₂².₂₆C₂⁴), C₂.₃₉(C₂².₅₀C₂⁴), C₂.₂₇(C₂².₃₃C₂⁴), C₂.₈₂(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₉₅C₂, (C₂×C₄).₈₂(C₂×D₄), (C₄×C₂²⋊C₄)⋊₉₀C₂, (C₂×C₂²⋊Q₈)⋊₂₃C₂, (C₂×C₄²⋊₂C₂)⋊₁₂C₂, (C₂×C₄).₃₉₁(C₄○D₄), (C₂×C₄⋊C₄).₈₇₄C₂², C₂².₃₃₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₈₃C₂², SmallGroup(128,1292)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₃₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₃₂C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₃₂C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₃₂C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₃₂C₂³

Generators and relations for C₂⁴.₃₃₂C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=b, f²=dc=cd, g²=cb=bc, ab=ba, eae^-1=ac=ca, faf^-1=ad=da, ag=ga, bd=db, fef^-1=geg^-1=be=eb, bf=fb, bg=gb, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 500 in 259 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂×C₂²⋊Q₈, C₂×C₄²⋊₂C₂, C₂⁴.₃₃₂C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂².₃₃C₂⁴, D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₄₅C₂⁴, C₂².₅₀C₂⁴, C₂⁴.₃₃₂C₂³

Smallest permutation representation of C₂⁴.₃₃₂C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

C₂⁴.₃₃₂C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₂²⋊Q₈

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₃₂C₂³ ►in GL₆(𝔽₅)

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

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

0	4	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	2	1
0	0	0	0	0	3

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

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

C₂⁴.₃₃₂C₂³ in GAP, Magma, Sage, TeX

C_2^4._{332}C_2^3

% in TeX

G:=Group("C2^4.332C2^3");

// GroupNames label

G:=SmallGroup(128,1292);

// by ID

G=gap.SmallGroup(128,1292);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,680,758,723,675,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b,f^2=d*c=c*d,g^2=c*b=b*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*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 = C24.332C23 order 128 = 27

172nd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₃₂C₂³ order 128 = 2⁷

172^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²