C2^4.569C2^3 - GroupNames

G = C₂⁴.₅₆₉C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₆₉C₂³, C₂³.₃₄₂C₂⁴, C₂².₁₁₁2_-¹⁺⁴, C₂².₁₅₁2₊¹⁺⁴, C₄⋊C₄.₃₂₄D₄, C₄²⋊₈C₄⋊₂₄C₂, (C₂²×C₄).₅₈Q₈, C₂³.₉₇(C₂×Q₈), C₄.₁₈(C₂²⋊Q₈), C₂.₃₅(D₄⋊₅D₄), C₂.₁₉(Q₈⋊₅D₄), (C₂²×C₄).₆₀C₂³, C₂²⋊₂(C₄².C₂), C₂³.₃₂₈(C₄○D₄), C₂².₇₃(C₂²×Q₈), (C₂×C₄²).₄₈₅C₂², (C₂³×C₄).₃₅₅C₂², C₂².₂₂₂(C₂²×D₄), C₂³.₇Q₈.₄₀C₂, C₂³.₈Q₈.₁₄C₂, C₂³.₈₃C₂³⋊₆C₂, C₂³.₈₁C₂³⋊₁₁C₂, C₂³.₆₅C₂³⋊₅₁C₂, C₂.C₄².₅₃₉C₂², C₂.₁₇(C₂².₄₇C₂⁴), C₂.₁₀(C₂³.₄₁C₂³), C₂.₂₀(C₂².₄₆C₂⁴), (C₂×C₄).₁₂₄(C₂×Q₈), (C₂×C₄².C₂)⋊₃C₂, C₂.₂₁(C₂×C₂²⋊Q₈), C₂.₈(C₂×C₄².C₂), (C₂×C₄).₁₀₇₇(C₂×D₄), (C₂²×C₄⋊C₄).₃₄C₂, (C₄×C₂²⋊C₄).₄₂C₂, (C₂×C₄).₈₀₈(C₄○D₄), (C₂×C₄⋊C₄).₂₂₄C₂², C₂².₂₁₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₅₂C₂², SmallGroup(128,1174)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₆₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₅₆₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₆₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₆₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₆₉C₂³

Generators and relations for C₂⁴.₅₆₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=b, e²=g²=a, f²=ba=ab, ac=ca, ede^-1=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 452 in 260 conjugacy classes, 116 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄².C₂, C₂³×C₄, C₂³×C₄, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₆₅C₂³, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂²×C₄⋊C₄, C₂×C₄².C₂, C₂⁴.₅₆₉C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₄².C₂, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂²⋊Q₈, C₂×C₄².C₂, C₂³.₄₁C₂³, D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂⁴.₅₆₉C₂³

Smallest permutation representation of C₂⁴.₅₆₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	-			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	C₄○D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₅₆₉C₂³	C₄×C₂²⋊C₄	C₂³.₇Q₈	C₄²⋊₈C₄	C₂³.₈Q₈	C₂³.₆₅C₂³	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂²×C₄⋊C₄	C₂×C₄².C₂	C₄⋊C₄	C₂²×C₄	C₂×C₄	C₂³	C₂²	C₂²
# reps	1	1	3	1	2	2	2	2	1	1	4	4	4	8	1	1

Matrix representation of C₂⁴.₅₆₉C₂³ ►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	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

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

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

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

0	1	0	0	0	0
4	0	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	2

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

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

C₂⁴.₅₆₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{569}C_2^3

% in TeX

G:=Group("C2^4.569C2^3");

// GroupNames label

G:=SmallGroup(128,1174);

// by ID

G=gap.SmallGroup(128,1174);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,268,675,80]);

// Polycyclic

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

// generators/relations

G = C24.569C23 order 128 = 27

50th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₆₉C₂³ order 128 = 2⁷

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