C2^3.688C2^4 - GroupNames

G = C₂³.₆₈₈C₂⁴ order 128 = 2⁷

405^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₈₈C₂⁴, C₂⁴.₄₅₃C₂³, C₂².₃₅₁2_-¹⁺⁴, C₂².₄₆₁2₊¹⁺⁴, C₂²⋊C₄⋊₈Q₈, C₂³.₄₄(C₂×Q₈), C₂.₆₂(D₄⋊₃Q₈), C₂³⋊Q₈.₂₉C₂, (C₂×C₄²).₇₁₅C₂², (C₂³×C₄).₄₉₇C₂², (C₂²×C₄).₂₁₅C₂³, C₂³.₈Q₈.₆₅C₂, C₂³.Q₈.₄₁C₂, C₂³.₇Q₈.₇₆C₂, C₂².₁₆₀(C₂²×Q₈), (C₂²×Q₈).₂₂₀C₂², C₂³.₇₈C₂³⋊₆₁C₂, C₂⁴.C₂².₇₇C₂, C₂³.₆₃C₂³⋊₁₈₇C₂, C₂³.₆₇C₂³⋊₁₀₂C₂, C₂³.₈₃C₂³⋊₁₂₂C₂, C₂³.₆₅C₂³⋊₁₅₅C₂, C₂³.₈₁C₂³⋊₁₂₈C₂, C₂.₁₀₅(C₂².₃₂C₂⁴), C₂.C₄².₃₉₂C₂², C₂.₄₂(C₂².₅₆C₂⁴), C₂.₇₁(C₂².₅₀C₂⁴), C₂.₄₁(C₂³.₄₁C₂³), (C₂×C₄).₈₄(C₂×Q₈), (C₂×C₄).₄₇₂(C₄○D₄), (C₂×C₄⋊C₄).₄₉₈C₂², C₂².₅₄₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₄C₂², SmallGroup(128,1520)

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

Derived series

C₁ — C₂³ — C₂³.₆₈₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₈Q₈ — C₂³.₆₈₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₈C₂⁴

Generators and relations for C₂³.₆₈₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=bcd, f²=g²=cb=bc, ab=ba, eae^-1=ac=ca, faf^-1=ad=da, ag=ga, bd=db, fef^-1=be=eb, gfg^-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=bce >

Subgroups: 404 in 208 conjugacy classes, 96 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, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₆₈₈C₂⁴
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂³.₄₁C₂³, D₄⋊₃Q₈, C₂².₅₀C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₈₈C₂⁴

Smallest permutation representation of C₂³.₆₈₈C₂⁴
►On 64 points

Generators in S₆₄

(2 41)(4 43)(5 18)(6 64)(7 20)(8 62)(9 54)(11 56)(14 32)(16 30)(17 51)(19 49)(21 34)(22 57)(23 36)(24 59)(25 46)(27 48)(33 38)(35 40)(37 58)(39 60)(50 61)(52 63)

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₆₈₈C₂⁴	C₂³.₇Q₈	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³⋊Q₈	C₂³.₇₈C₂³	C₂³.Q₈	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂²⋊C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	2	1	2	2	1	1	1	1	1	2	4	8	3	1

Matrix representation of C₂³.₆₈₈C₂⁴ ►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	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	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	3	0	0	0
0	0	0	3	0	0
0	0	0	0	2	4
0	0	0	0	0	3

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

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

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

C₂³.₆₈₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{688}C_2^4

% in TeX

G:=Group("C2^3.688C2^4");

// GroupNames label

G:=SmallGroup(128,1520);

// by ID

G=gap.SmallGroup(128,1520);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.688C24 order 128 = 27

405th central stem extension by C23 of C24

G = C₂³.₆₈₈C₂⁴ order 128 = 2⁷

405^th central stem extension by C₂³ of C₂⁴