C2^4.338C2^3 - GroupNames

G = C₂⁴.₃₃₈C₂³ order 128 = 2⁷

178^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₃₈C₂³, C₂³.₄₇₄C₂⁴, C₂².₂₅₇2₊¹⁺⁴, C₂²⋊C₄⋊₁₉Q₈, C₄²⋊₈C₄⋊₄₅C₂, C₂³.₂₆(C₂×Q₈), C₂.₃₈(D₄⋊₃Q₈), (C₂×C₄²).₇₀C₂², C₂³⋊Q₈.₁₀C₂, (C₂³×C₄).₄₀₇C₂², (C₂²×C₄).₈₄₄C₂³, C₂³.₇Q₈.₅₄C₂, C₂³.Q₈.₁₅C₂, C₂³.₈Q₈.₃₃C₂, C₂².₁₁₃(C₂²×Q₈), (C₂²×Q₈).₁₄₂C₂², C₂³.₈₁C₂³⋊₄₃C₂, C₂³.₆₃C₂³⋊₉₁C₂, C₂³.₆₇C₂³⋊₆₆C₂, C₂³.₇₈C₂³⋊₁₉C₂, C₂.₂₇(C₂².₃₂C₂⁴), C₂⁴.C₂².₃₃C₂, C₂.₅₅(C₂².₄₅C₂⁴), C₂.C₄².₂₁₀C₂², C₂.₈₅(C₂³.₃₆C₂³), C₂.₂₉(C₂².₄₉C₂⁴), C₂.₃₀(C₂³.₃₇C₂³), (C₄×C₄⋊C₄)⋊₁₀₀C₂, (C₂×C₄).₂₅₆(C₂×Q₈), (C₄×C₂²⋊C₄).₆₅C₂, (C₂×C₄).₃₉₄(C₄○D₄), (C₂×C₄⋊C₄).₃₂₁C₂², C₂².₃₅₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₁₀C₂², SmallGroup(128,1306)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₃₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₃₃₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₃₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₃₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₃₈C₂³

Generators and relations for C₂⁴.₃₃₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=db=bd, g²=c, ab=ba, ac=ca, faf^-1=ad=da, eae^-1=abc, ag=ga, bc=cb, fef^-1=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: 404 in 218 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂⁴.₃₃₈C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, D₄⋊₃Q₈, C₂².₄₉C₂⁴, C₂⁴.₃₃₈C₂³

Smallest permutation representation of C₂⁴.₃₃₈C₂³
►On 64 points

Generators in S₆₄

(2 56)(4 54)(5 61)(6 51)(7 63)(8 49)(9 21)(10 33)(11 23)(12 35)(14 27)(16 25)(17 32)(19 30)(22 40)(24 38)(34 37)(36 39)(41 62)(42 52)(43 64)(44 50)(46 59)(48 57)

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

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

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

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

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

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

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

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

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

C₂⁴.₃₃₈C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₇₈C₂³

C₂³.Q₈

C₂³.₈₁C₂³

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₃₃₈C₂³ ►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	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	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	1	0
0	0	0	0	0	1

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

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

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	3	0
0	0	0	0	0	3

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

C₂⁴.₃₃₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{338}C_2^3

% in TeX

G:=Group("C2^4.338C2^3");

// GroupNames label

G:=SmallGroup(128,1306);

// by ID

G=gap.SmallGroup(128,1306);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,100,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=d*b=b*d,g^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,b*c=c*b,f*e*f^-1=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 = C24.338C23 order 128 = 27

178th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₃₈C₂³ order 128 = 2⁷

178^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²