C2^3.668C2^4 - GroupNames

G = C₂³.₆₆₈C₂⁴ order 128 = 2⁷

385^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₆₈C₂⁴, C₂⁴.₄₄₄C₂³, C₂².₄₄₁2₊¹⁺⁴, C₂².₃₃₄2_-¹⁺⁴, C₂²⋊C₄.₂Q₈, C₄²⋊₈C₄⋊₆₄C₂, C₂³.₄₁(C₂×Q₈), C₂.₅₈(D₄⋊₃Q₈), (C₂²×C₄).₂₀₉C₂³, (C₂³×C₄).₄₉₀C₂², (C₂×C₄²).₇₀₀C₂², C₂³.₇Q₈.₇₃C₂, C₂³.₄Q₈.₂₅C₂, C₂³.Q₈.₃₆C₂, C₂³.₈Q₈.₆₁C₂, C₂².₁₅₆(C₂²×Q₈), C₂.₉₃(C₂².₃₂C₂⁴), C₂⁴.C₂².₆₈C₂, C₂³.₆₅C₂³⋊₁₄₄C₂, C₂³.₆₃C₂³⋊₁₇₃C₂, C₂³.₈₁C₂³⋊₁₁₈C₂, C₂.C₄².₃₇₂C₂², C₂.₃₉(C₂³.₄₁C₂³), C₂.₅₈(C₂².₃₄C₂⁴), C₂.₃₆(C₂².₅₆C₂⁴), C₂.₁₀₆(C₂².₄₆C₂⁴), (C₂×C₄).₈₀(C₂×Q₈), (C₂×C₄).₄₆₂(C₄○D₄), (C₂×C₄⋊C₄).₄₇₈C₂², C₂².₅₂₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₂C₂², SmallGroup(128,1500)

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

Derived series

C₁ — C₂³ — C₂³.₆₆₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₆₆₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₆₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₆₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₆₈C₂⁴

Generators and relations for C₂³.₆₆₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=ba=ab, f²=b, g²=a, ede^-1=ad=da, geg^-1=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^-1=abd, fg=gf >

Subgroups: 388 in 204 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₆₆₈C₂⁴
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂³.₄₁C₂³, C₂².₄₆C₂⁴, D₄⋊₃Q₈, C₂².₅₆C₂⁴, C₂³.₆₆₈C₂⁴

Smallest permutation representation of C₂³.₆₆₈C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₆₆₈C₂⁴	C₂³.₇Q₈	C₄²⋊₈C₄	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂³.Q₈	C₂³.₈₁C₂³	C₂³.₄Q₈	C₂²⋊C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	1	2	1	2	2	1	4	1	4	8	3	1

Matrix representation of C₂³.₆₆₈C₂⁴ ►in GL₆(𝔽₅)

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

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	2
0	0	0	0	2	0

3	0	0	0	0	0
0	3	0	0	0	0
0	0	0	2	0	0
0	0	3	0	0	0
0	0	0	0	2	0
0	0	0	0	0	3

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

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

C₂³.₆₆₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{668}C_2^4

% in TeX

G:=Group("C2^3.668C2^4");

// GroupNames label

G:=SmallGroup(128,1500);

// by ID

G=gap.SmallGroup(128,1500);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,184,1571,346,80]);

// 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*a=a*b,f^2=b,g^2=a,e*d*e^-1=a*d=d*a,g*e*g^-1=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^-1=a*b*d,f*g=g*f>;

// generators/relations

G = C23.668C24 order 128 = 27

385th central stem extension by C23 of C24

G = C₂³.₆₆₈C₂⁴ order 128 = 2⁷

385^th central stem extension by C₂³ of C₂⁴