C2^4.567C2^3 - GroupNames

G = C₂⁴.₅₆₇C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₆₇C₂³, C₂³.₃₃₈C₂⁴, C₂².₁₀₇2_-¹⁺⁴, C₂².₁₄₇2₊¹⁺⁴, (C₂²×C₄)⋊₈Q₈, C₂³.₉₅(C₂×Q₈), C₂²⋊C₄.₁₂₆D₄, C₂³.₄₂₄(C₂×D₄), C₂.₃₃(D₄⋊₅D₄), C₂.₂₁(D₄⋊₆D₄), (C₂²×C₄).₅₉C₂³, C₂².₉(C₂²⋊Q₈), C₂³.₃₀₄(C₄○D₄), C₂².₆₉(C₂²×Q₈), (C₂×C₄²).₄₈₁C₂², (C₂³×C₄).₃₅₁C₂², C₂².₂₁₈(C₂²×D₄), C₂³.₇Q₈.₃₆C₂, C₂³.₈Q₈.₁₂C₂, C₂³.₇₈C₂³⋊₇C₂, C₂³.₈₁C₂³⋊₉C₂, C₂³.₈₃C₂³⋊₅C₂, C₂³.₃₄D₄.₁₃C₂, (C₂²×Q₈).₁₀₀C₂², C₂³.₆₃C₂³⋊₃₁C₂, C₂³.₆₅C₂³⋊₄₇C₂, C₂³.₆₇C₂³⋊₃₉C₂, C₂.C₄².₉₆C₂², C₂.₁₅(C₂².₄₅C₂⁴), C₂.₆(C₂³.₄₁C₂³), C₂.₁₂(C₂³.₃₇C₂³), C₂.₁₈(C₂².₄₆C₂⁴), (C₂×C₄).₃₂₂(C₂×D₄), (C₂×C₄).₁₆₄(C₂×Q₈), C₂.₁₇(C₂×C₂²⋊Q₈), (C₂×C₂²⋊Q₈).₂₆C₂, (C₂×C₄).₃₆₁(C₄○D₄), (C₂×C₄⋊C₄).₂₂₁C₂², C₂².₂₁₅(C₂×C₄○D₄), (C₂×C₄²⋊C₂).₃₆C₂, (C₂×C₂²⋊C₄).₄₉₃C₂², (C₂×C₂.C₄²).₂₅C₂, SmallGroup(128,1170)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₆₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂⁴.₅₆₇C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₆₇C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₆₇C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₆₇C₂³

Generators and relations for C₂⁴.₅₆₇C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=c, f²=b, eae^-1=ab=ba, ac=ca, ad=da, af=fa, ag=ga, bc=cb, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 468 in 264 conjugacy classes, 112 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂×C₂.C₄², C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₄²⋊C₂, C₂×C₂²⋊Q₈, C₂⁴.₅₆₇C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂²⋊Q₈, C₂³.₃₇C₂³, C₂³.₄₁C₂³, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂⁴.₅₆₇C₂³

Smallest permutation representation of C₂⁴.₅₆₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4V	4W	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	-			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	C₄○D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₅₆₇C₂³	C₂×C₂.C₄²	C₂³.₇Q₈	C₂³.₃₄D₄	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³.₇₈C₂³	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂×C₄²⋊C₂	C₂×C₂²⋊Q₈	C₂²⋊C₄	C₂²×C₄	C₂×C₄	C₂³	C₂²	C₂²
# reps	1	1	1	1	2	2	1	1	1	2	1	1	1	4	4	8	4	1	1

Matrix representation of C₂⁴.₅₆₇C₂³ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	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	4

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	4	0
0	0	0	0	0	4

4	0	0	0	0	0
0	4	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

3	0	0	0	0	0
0	2	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

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	2	0
0	0	0	0	0	3

0	1	0	0	0	0
4	0	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	3

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

C₂⁴.₅₆₇C₂³ in GAP, Magma, Sage, TeX

C_2^4._{567}C_2^3

% in TeX

G:=Group("C2^4.567C2^3");

// GroupNames label

G:=SmallGroup(128,1170);

// by ID

G=gap.SmallGroup(128,1170);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,184,675]);

// Polycyclic

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

// generators/relations

G = C24.567C23 order 128 = 27

48th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₆₇C₂³ order 128 = 2⁷

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