C2^3.708C2^4 - GroupNames

G = C₂³.₇₀₈C₂⁴ order 128 = 2⁷

425^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉₉C₂³, C₂³.₇₀₈C₂⁴, C₂².₃₆₈2_-¹⁺⁴, C₂².₄₈₁2₊¹⁺⁴, C₂³⋊Q₈⋊₅₉C₂, (C₂²×C₄).₆₁₄C₂³, (C₂×C₄²).₁₁₆C₂², C₂³.₁₁D₄⋊₁₂₅C₂, C₂³.₁₀D₄.₇₀C₂, (C₂²×D₄).₂₉₀C₂², (C₂²×Q₈).₂₂₈C₂², C₂⁴.C₂²⋊₁₇₇C₂, C₂⁴.₃C₂².₇₉C₂, C₂³.₆₃C₂³⋊₁₉₆C₂, C₂³.₆₇C₂³⋊₁₀₅C₂, C₂³.₆₅C₂³⋊₁₆₃C₂, C₂.₁₁₂(C₂².₃₂C₂⁴), C₂.C₄².₄₁₂C₂², C₂.₄₉(C₂².₄₉C₂⁴), C₂.₇₇(C₂².₅₀C₂⁴), C₂.₄₇(C₂².₅₆C₂⁴), C₂.₄₉(C₂².₅₃C₂⁴), C₂.₁₁₉(C₂².₃₃C₂⁴), (C₂×C₄).₂₄₉(C₄○D₄), (C₂×C₄⋊C₄).₅₁₈C₂², C₂².₅₆₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₃₁C₂², SmallGroup(128,1540)

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

Derived series

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂³.₇₀₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Generators and relations for C₂³.₇₀₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=b, f²=cb=bc, g²=ba=ab, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 452 in 214 conjugacy classes, 88 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×D₄, C₂²×Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₇₀₈C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂².₅₃C₂⁴, C₂².₅₆C₂⁴, C₂³.₇₀₈C₂⁴

Smallest permutation representation of C₂³.₇₀₈C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4R	4S	4T	4U	4V
order	1	2	···	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₇₀₈C₂⁴

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₇₀₈C₂⁴ ►in GL₆(𝔽₅)

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	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

3	1	0	0	0	0
0	2	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	0	2

4	0	0	0	0	0
1	1	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	1	0

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

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

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

C₂³.₇₀₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{708}C_2^4

% in TeX

G:=Group("C2^3.708C2^4");

// GroupNames label

G:=SmallGroup(128,1540);

// by ID

G=gap.SmallGroup(128,1540);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,520,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.708C24 order 128 = 27

425th central stem extension by C23 of C24

G = C₂³.₇₀₈C₂⁴ order 128 = 2⁷

425^th central stem extension by C₂³ of C₂⁴