C2^4.384C2^3 - GroupNames

G = C₂⁴.₃₈₄C₂³ order 128 = 2⁷

224^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₈₄C₂³, C₂³.₅₇₅C₂⁴, C₂².₂₆₀2_-¹⁺⁴, C₂².₃₄₉2₊¹⁺⁴, C₂.₃₉D₄², C₂²⋊C₄⋊₁₂D₄, C₂³.₆₁(C₂×D₄), C₂³⋊₂D₄⋊₃₆C₂, C₂³.Q₈⋊₄₉C₂, C₂³.₈Q₈⋊₉₅C₂, (C₂³×C₄).₄₄₅C₂², (C₂²×C₄).₈₆₄C₂³, C₂².₃₈₄(C₂²×D₄), (C₂²×D₄).₂₁₆C₂², C₂.₉(C₂².₅₄C₂⁴), C₂.C₄².₂₈₆C₂², C₂.₃₈(C₂².₃₁C₂⁴), (C₂×C₄⋊D₄)⋊₃₁C₂, (C₂×C₄).₄₁₄(C₂×D₄), (C₂×C₄⋊C₄).₃₉₃C₂², (C₂×C₂²⋊C₄).₂₄₆C₂², SmallGroup(128,1407)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₈₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄⋊D₄ — C₂⁴.₃₈₄C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₈₄C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₈₄C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₈₄C₂³

Generators and relations for C₂⁴.₃₈₄C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=1, g²=b, eae=gag^-1=ab=ba, ac=ca, faf=ad=da, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=bce, fg=gf >

Subgroups: 916 in 402 conjugacy classes, 112 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₈Q₈, C₂³⋊₂D₄, C₂³⋊₂D₄, C₂³.Q₈, C₂×C₄⋊D₄, C₂⁴.₃₈₄C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₁C₂⁴, D₄², C₂².₅₄C₂⁴, C₂⁴.₃₈₄C₂³

Smallest permutation representation of C₂⁴.₃₈₄C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

(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)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	···	2M	2N	2O	4A	···	4L	4M	4N	4O	4P
order	1	2	···	2	2	···	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	···	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₃₈₄C₂³	C₂³.₈Q₈	C₂³⋊₂D₄	C₂³.Q₈	C₂×C₄⋊D₄	C₂²⋊C₄	C₂²	C₂²
# reps	1	3	4	2	6	12	3	1

Matrix representation of C₂⁴.₃₈₄C₂³ ►in GL₆(ℤ)

-1	-1	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
0	0	0	0	1	-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	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	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	-1	0
0	0	0	0	0	-1

1	0	0	0	0	0
-2	-1	0	0	0	0
0	0	-1	1	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	1	0	0	0
0	0	2	-1	0	0
0	0	0	0	1	-2
0	0	0	0	0	-1

-1	-1	0	0	0	0
2	1	0	0	0	0
0	0	-1	0	0	0
0	0	-2	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[1,-2,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[-1,2,0,0,0,0,-1,1,0,0,0,0,0,0,-1,-2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂⁴.₃₈₄C₂³ in GAP, Magma, Sage, TeX

C_2^4._{384}C_2^3

% in TeX

G:=Group("C2^4.384C2^3");

// GroupNames label

G:=SmallGroup(128,1407);

// by ID

G=gap.SmallGroup(128,1407);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24.384C23 order 128 = 27

224th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₈₄C₂³ order 128 = 2⁷

224^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²