C2^3.417C2^4 - GroupNames

G = C₂³.₄₁₇C₂⁴ order 128 = 2⁷

134^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₄₁₇C₂⁴, C₂⁴.₃₁₀C₂³, C₂².₁₆₀2_-¹⁺⁴, C₂².₂₁₁2₊¹⁺⁴, C₄²⋊₈C₄⋊₃₆C₂, C₂³.₄₄(C₄○D₄), (C₂²×C₄).₈₅C₂³, C₂³.₄Q₈.₅C₂, (C₂³×C₄).₁₀₆C₂², (C₂×C₄²).₅₃₂C₂², C₂³.₈Q₈.₂₄C₂, C₂³.₁₁D₄.₁₂C₂, C₂³.₆₃C₂³⋊₇₁C₂, C₂.₃₄(C₂².₄₅C₂⁴), C₂.C₄².₁₆₅C₂², C₂.₂₂(C₂².₃₃C₂⁴), C₂.₄₉(C₂².₄₆C₂⁴), C₂.₁₅(C₂².₅₃C₂⁴), C₂.₆₀(C₂³.₃₆C₂³), (C₄×C₂²⋊C₄).₅₆C₂, (C₂×C₄).₁₃₈(C₄○D₄), (C₂×C₄⋊C₄).₂₈₀C₂², C₂².₂₉₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₆₆C₂², SmallGroup(128,1249)

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

Derived series

C₁ — C₂³ — C₂³.₄₁₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₄₁₇C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₁₇C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₁₇C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₁₇C₂⁴

Generators and relations for C₂³.₄₁₇C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=b, g²=c, eae^-1=ab=ba, ac=ca, faf^-1=ad=da, 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, df=fd, dg=gd, fg=gf >

Subgroups: 372 in 206 conjugacy classes, 92 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊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, C₂³×C₄, C₄×C₂²⋊C₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂³.₁₁D₄, C₂³.₁₁D₄, C₂³.₄Q₈, C₂³.₄₁₇C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₃C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₅₃C₂⁴, C₂³.₄₁₇C₂⁴

Smallest permutation representation of C₂³.₄₁₇C₂⁴
►On 64 points

Generators in S₆₄

(2 12)(4 10)(5 40)(6 8)(7 38)(14 42)(16 44)(17 19)(18 48)(20 46)(22 50)(24 52)(26 54)(28 56)(29 31)(30 60)(32 58)(33 35)(34 62)(36 64)(37 39)(45 47)(57 59)(61 63)

(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 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 63)(34 64)(35 61)(36 62)

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+	+			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₄₁₇C₂⁴	C₄×C₂²⋊C₄	C₄²⋊₈C₄	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂³.₁₁D₄	C₂³.₄Q₈	C₂×C₄	C₂³	C₂²	C₂²
# reps	1	2	2	1	6	3	1	16	4	1	1

Matrix representation of C₂³.₄₁₇C₂⁴ ►in GL₆(𝔽₅)

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

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

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

C₂³.₄₁₇C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{417}C_2^4

% in TeX

G:=Group("C2^3.417C2^4");

// GroupNames label

G:=SmallGroup(128,1249);

// by ID

G=gap.SmallGroup(128,1249);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,100,675,136]);

// Polycyclic

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

// generators/relations

G = C23.417C24 order 128 = 27

134th central stem extension by C23 of C24

G = C₂³.₄₁₇C₂⁴ order 128 = 2⁷

134^th central stem extension by C₂³ of C₂⁴