C2^4.393C2^3 - GroupNames

G = C₂⁴.₃₉₃C₂³ order 128 = 2⁷

233^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₉₃C₂³, C₂³.₅₈₆C₂⁴, C₂².₂₆₇2_-¹⁺⁴, C₂².₃₆₀2₊¹⁺⁴, (C₂×D₄).₁₃₉D₄, C₂³.₆₄(C₂×D₄), C₂³⋊Q₈⋊₄₂C₂, C₂.₉₁(D₄⋊₅D₄), C₂.₅₉(D₄⋊₆D₄), C₂³.₇Q₈⋊₈₃C₂, C₂³.₂₃D₄⋊₈₄C₂, C₂³.₁₁D₄⋊₇₈C₂, C₂³.₁₀D₄⋊₈₁C₂, C₂³.₃₄D₄⋊₄₅C₂, C₂.₄₃(C₂³⋊₃D₄), (C₂×C₄²).₆₄₁C₂², (C₂²×C₄).₅₅₆C₂³, (C₂³×C₄).₄₅₂C₂², C₂³.₈Q₈⋊₁₀₀C₂, C₂².₃₉₅(C₂²×D₄), (C₂²×D₄).₂₂₅C₂², (C₂²×Q₈).₁₇₉C₂², C₂⁴.C₂²⋊₁₂₄C₂, C₂³.₇₈C₂³⋊₄₁C₂, C₂.₆₃(C₂².₃₂C₂⁴), C₂³.₆₃C₂³⋊₁₂₈C₂, C₂.C₄².₂₉₃C₂², C₂.₁₁(C₂².₅₆C₂⁴), C₂.₆₉(C₂².₃₆C₂⁴), C₂.₃₄(C₂².₄₉C₂⁴), (C₂×C₄).₄₁₇(C₂×D₄), (C₂×C₄.₄D₄)⋊₂₈C₂, (C₂×C₄⋊D₄).₄₄C₂, (C₂×C₄).₄₁₈(C₄○D₄), (C₂×C₄⋊C₄).₄₀₀C₂², C₂².₄₄₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₅₃C₂², SmallGroup(128,1418)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₉₃C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₃₉₃C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₉₃C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₉₃C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₉₃C₂³

Generators and relations for C₂⁴.₃₉₃C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=g²=1, e²=c, f²=b, eae^-1=ab=ba, faf^-1=ac=ca, ad=da, gag=abc, bc=cb, bd=db, geg=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 612 in 281 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂×C₄⋊D₄, C₂×C₄.₄D₄, C₂⁴.₃₉₃C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₂C₂⁴, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₉C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₃₉₃C₂³

Smallest permutation representation of C₂⁴.₃₉₃C₂³
►On 64 points

Generators in S₆₄

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

(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	4A	···	4N	4O	···	4S
order	1	2	···	2	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₃₉₃C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂×C₄⋊D₄

C₂×C₄.₄D₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₃₉₃C₂³ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	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	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

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

1	0	0	0	0	0
0	4	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

4	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	1
0	0	0	0	1	0

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

C₂⁴.₃₉₃C₂³ in GAP, Magma, Sage, TeX

C_2^4._{393}C_2^3

% in TeX

G:=Group("C2^4.393C2^3");

// GroupNames label

G:=SmallGroup(128,1418);

// by ID

G=gap.SmallGroup(128,1418);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.393C23 order 128 = 27

233rd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₉₃C₂³ order 128 = 2⁷

233^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²