C2^3.617C2^4 - GroupNames

G = C₂³.₆₁₇C₂⁴ order 128 = 2⁷

334^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₁₇C₂⁴, C₂⁴.₄₁₅C₂³, C₂².₃₉₁2₊¹⁺⁴, C₂².₂₉₂2_-¹⁺⁴, (C₂×D₄).₁₄₄D₄, C₂³.₇₃(C₂×D₄), C₂.₆₈(D₄⋊₆D₄), C₂³.₇Q₈⋊₉₈C₂, C₂³.₁₁D₄⋊₉₇C₂, C₂.₅₀(C₂³⋊₃D₄), (C₂³×C₄).₄₇₁C₂², (C₂²×C₄).₅₆₀C₂³, (C₂×C₄²).₆₆₈C₂², C₂³.₈Q₈⋊₁₁₇C₂, C₂².₄₂₆(C₂²×D₄), C₂³.₂₃D₄.₅₆C₂, (C₂²×D₄).₂₅₀C₂², (C₂²×Q₈).₁₉₄C₂², C₂³.₇₈C₂³⋊₄₉C₂, C₂³.₆₃C₂³⋊₁₄₄C₂, C₂.₈₂(C₂².₄₅C₂⁴), C₂.C₄².₃₂₃C₂², C₂.₈₂(C₂².₃₆C₂⁴), C₂.₂₀(C₂².₅₇C₂⁴), (C₂×C₄).₄₂₇(C₂×D₄), (C₂×C₄).₄₃₄(C₄○D₄), (C₂×C₄⋊C₄).₄₃₀C₂², (C₂×C₄.₄D₄).₃₃C₂, C₂².₄₇₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₈₂C₂², (C₂×C₂².D₄).₂₈C₂, SmallGroup(128,1449)

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

Derived series

C₁ — C₂³ — C₂³.₆₁₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₈Q₈ — C₂³.₆₁₇C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₁₇C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₁₇C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₁₇C₂⁴

Generators and relations for C₂³.₆₁₇C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=g²=1, e²=b, f²=ba=ab, ac=ca, ede^-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 516 in 256 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂×C₂².D₄, C₂×C₄.₄D₄, C₂³.₆₁₇C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₆C₂⁴, D₄⋊₆D₄, C₂².₄₅C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₁₇C₂⁴

Smallest permutation representation of C₂³.₆₁₇C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

(2 24)(4 22)(6 10)(8 12)(14 26)(16 28)(17 19)(18 61)(20 63)(30 38)(32 40)(33 35)(34 45)(36 47)(41 59)(42 44)(43 57)(46 48)(49 51)(50 55)(52 53)(54 56)(58 60)(62 64)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₁₇C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂×C₂².D₄

C₂×C₄.₄D₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₁₇C₂⁴ ►in GL₆(𝔽₅)

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

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

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

0	4	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

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

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	0
0	0	0	0	3	4

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

C₂³.₆₁₇C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{617}C_2^4

% in TeX

G:=Group("C2^3.617C2^4");

// GroupNames label

G:=SmallGroup(128,1449);

// by ID

G=gap.SmallGroup(128,1449);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.617C24 order 128 = 27

334th central stem extension by C23 of C24

G = C₂³.₆₁₇C₂⁴ order 128 = 2⁷

334^th central stem extension by C₂³ of C₂⁴