C2^4.568C2^3 - GroupNames

G = C₂⁴.₅₆₈C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₆₈C₂³, C₂³.₃₄₀C₂⁴, C₂².₁₄₉2₊¹⁺⁴, C₂².₁₀₉2_-¹⁺⁴, C₂.₁₆D₄², C₂.₁₁(D₄×Q₈), C₄⋊C₄.₃₂₂D₄, C₂²⋊C₄⋊₁₃Q₈, C₄⋊₁(C₂²⋊Q₈), C₂²⋊₂(C₄⋊Q₈), (C₂²×C₄)⋊₁₀Q₈, C₂²⋊C₄.₇₁D₄, C₄²⋊₉C₄⋊₁₇C₂, C₂³.₉₆(C₂×Q₈), C₂³.₄₂₅(C₂×D₄), C₂.₂₂(D₄⋊₆D₄), C₂.₁₁(D₄⋊₃Q₈), C₂².₇₁(C₂²×Q₈), (C₂²×C₄).₈₀₂C₂³, (C₂³×C₄).₃₅₃C₂², (C₂×C₄²).₄₈₃C₂², C₂².₂₂₀(C₂²×D₄), C₂³.₇Q₈.₃₈C₂, C₂³.₈Q₈.₁₃C₂, C₂³.₇₈C₂³⋊₈C₂, (C₂²×Q₈).₁₀₂C₂², C₂³.₆₅C₂³⋊₄₉C₂, C₂³.₈₁C₂³⋊₁₀C₂, C₂.C₄².₉₈C₂², C₂.₈(C₂³.₄₁C₂³), (C₂×C₄⋊Q₈)⋊₈C₂, C₂.₉(C₂×C₄⋊Q₈), (C₂×C₄).₅₂(C₂×D₄), (C₂×C₄).₂₈(C₂×Q₈), C₂.₁₉(C₂×C₂²⋊Q₈), (C₄×C₂²⋊C₄).₄₀C₂, (C₂²×C₄⋊C₄).₃₃C₂, (C₂×C₂²⋊Q₈).₂₇C₂, (C₂×C₄).₈₀₇(C₄○D₄), (C₂×C₄⋊C₄).₈₄₈C₂², C₂².₂₁₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₉₅C₂², SmallGroup(128,1172)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₆₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₅₆₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₆₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₆₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₆₈C₂³

Generators and relations for C₂⁴.₅₆₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=c, g²=b, ab=ba, ac=ca, faf^-1=ad=da, ae=ea, ag=ga, bc=cb, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 532 in 304 conjugacy classes, 132 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄⋊Q₈, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₄²⋊₉C₄, C₂³.₈Q₈, C₂³.₆₅C₂³, C₂³.₇₈C₂³, C₂³.₈₁C₂³, C₂²×C₄⋊C₄, C₂×C₂²⋊Q₈, C₂×C₄⋊Q₈, C₂⁴.₅₆₈C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂²⋊Q₈, C₂×C₄⋊Q₈, C₂³.₄₁C₂³, D₄², D₄⋊₆D₄, D₄×Q₈, D₄⋊₃Q₈, C₂⁴.₅₆₈C₂³

Smallest permutation representation of C₂⁴.₅₆₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	-	+	-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	D₄	Q₈	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₅₆₈C₂³	C₄×C₂²⋊C₄	C₂³.₇Q₈	C₄²⋊₉C₄	C₂³.₈Q₈	C₂³.₆₅C₂³	C₂³.₇₈C₂³	C₂³.₈₁C₂³	C₂²×C₄⋊C₄	C₂×C₂²⋊Q₈	C₂×C₄⋊Q₈	C₂²⋊C₄	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	1	1	2	2	2	2	1	2	1	4	4	4	4	4	1	1

Matrix representation of C₂⁴.₅₆₈C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

C₂⁴.₅₆₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{568}C_2^3

% in TeX

G:=Group("C2^4.568C2^3");

// GroupNames label

G:=SmallGroup(128,1172);

// by ID

G=gap.SmallGroup(128,1172);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,184,675,80]);

// Polycyclic

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

// generators/relations

G = C24.568C23 order 128 = 27

49th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₆₈C₂³ order 128 = 2⁷

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