C2^3.600C2^4 - GroupNames

G = C₂³.₆₀₀C₂⁴ order 128 = 2⁷

317^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₆₂C₂³, C₂³.₆₀₀C₂⁴, C₂².₂₇₈2_-¹⁺⁴, C₂².₃₇₄2₊¹⁺⁴, C₂²⋊C₄.₁₇D₄, C₂³⋊Q₈⋊₄₆C₂, C₂³.₂₁₅(C₂×D₄), C₂.₁₀₅(D₄⋊₅D₄), C₂³.Q₈⋊₆₂C₂, C₂³.₃₄D₄⋊₄₉C₂, C₂³.₁₁D₄⋊₈₇C₂, (C₂×C₄²).₆₅₂C₂², (C₂²×C₄).₈₇₇C₂³, (C₂³×C₄).₄₆₂C₂², C₂².₄₀₉(C₂²×D₄), C₂³.₂₃D₄.₅₃C₂, C₂³.₁₀D₄.₄₃C₂, (C₂²×D₄).₂₃₆C₂², (C₂²×Q₈).₁₈₆C₂², C₂³.₇₈C₂³⋊₄₆C₂, C₂⁴.C₂²⋊₁₃₁C₂, C₂³.₆₇C₂³⋊₈₂C₂, C₂³.₆₃C₂³⋊₁₃₆C₂, C₂.C₄².₃₀₆C₂², C₂.₅₅(C₂².₅₀C₂⁴), C₂.₁₆(C₂².₅₆C₂⁴), C₂.₆₁(C₂³.₃₈C₂³), C₂.₇₄(C₂².₃₆C₂⁴), (C₂×C₄).₁₀₂(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₄₁C₂, (C₂×C₄).₄₂₇(C₄○D₄), (C₂×C₄⋊C₄).₄₁₃C₂², (C₂×C₄.₄D₄).₃₁C₂, C₂².₄₆₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₆₆C₂², SmallGroup(128,1432)

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

Derived series

C₁ — C₂³ — C₂³.₆₀₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂³.₆₀₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₀₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₀₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₀₀C₂⁴

Generators and relations for C₂³.₆₀₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=b, e²=ba=ab, f²=a, ac=ca, ede^-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

Subgroups: 516 in 253 conjugacy classes, 96 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, C₂²⋊Q₈, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂×C₂²⋊Q₈, C₂×C₄.₄D₄, C₂³.₆₀₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₈C₂³, C₂².₃₆C₂⁴, D₄⋊₅D₄, C₂².₅₀C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₀₀C₂⁴

Smallest permutation representation of C₂³.₆₀₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4P	4Q	···	4U
order	1	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	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

C₂³.₆₀₀C₂⁴

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂×C₂²⋊Q₈

C₂×C₄.₄D₄

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₀₀C₂⁴ ►in GL₆(𝔽₅)

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

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

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

0	4	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	4
0	0	0	0	4	0

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

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

C₂³.₆₀₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{600}C_2^4

% in TeX

G:=Group("C2^3.600C2^4");

// GroupNames label

G:=SmallGroup(128,1432);

// by ID

G=gap.SmallGroup(128,1432);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,344,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.600C24 order 128 = 27

317th central stem extension by C23 of C24

G = C₂³.₆₀₀C₂⁴ order 128 = 2⁷

317^th central stem extension by C₂³ of C₂⁴