C2^3.697C2^4

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

Aliases: C₂⁴.₉₄C₂³, C₂³.₆₉₇C₂⁴, C₂².₄₇₀2₊⁽¹⁺⁴⁾, C₂³⋊₂D₄.₃₄C₂, C₂³.₁₀₃(C₄○D₄), (C₂²×C₄).₆₀₇C₂³, (C₂×C₄²).₇₂₀C₂², (C₂³×C₄).₁₇₇C₂², C₂³.₈Q₈⋊₁₄₀C₂, C₂³.₁₁D₄⋊₁₂₂C₂, C₂³.₁₀D₄⋊₁₀₄C₂, C₂³.₂₃D₄⋊₁₀₇C₂, C₂⁴.₃C₂²⋊₉₅C₂, (C₂²×D₄).₂₈₅C₂², C₂⁴.C₂²⋊₁₇₃C₂, C₂³.₆₃C₂³⋊₁₉₂C₂, C₂.₁₀₆(C₂².₃₂C₂⁴), C₂.₃₈(C₂².₅₄C₂⁴), C₂.C₄².₄₀₁C₂², C₂.₁₂₂(C₂².₄₅C₂⁴), C₂.₆₅(C₂².₃₄C₂⁴), C₂.₄₄(C₂².₅₃C₂⁴), C₂.₁₁₉(C₂².₄₇C₂⁴), (C₂×C₄).₂₃₈(C₄○D₄), (C₂×C₄⋊C₄).₅₀₇C₂², C₂².₅₅₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₉C₂², SmallGroup(128,1529)

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

Derived series

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₉₇C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Subgroups: 548 in 242 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×4], C₄^[×14], C₂²^[×7], C₂²^[×24], C₂×C₄^[×4], C₂×C₄^[×38], D₄^[×12], C₂³, C₂³^[×2], C₂³^[×20], C₄²^[×2], C₂²⋊C₄^[×17], C₄⋊C₄^[×7], C₂²×C₄^[×12], C₂²×C₄^[×3], C₂×D₄^[×13], C₂⁴^[×3], C₂.C₄²^[×10], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×14], C₂×C₄⋊C₄^[×5], C₂³×C₄, C₂²×D₄^[×3], C₂³.₈Q₈, C₂³.₂₃D₄^[×2], C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×2], C₂⁴.₃C₂²^[×2], C₂³⋊₂D₄, C₂³.₁₀D₄^[×2], C₂³.₁₁D₄^[×3], C₂³.₆₉₇C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×4], C₂².₃₂C₂⁴^[×2], C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₃C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₉₇C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=ca=ac, e²=a, f²=abc, ab=ba, 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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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	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	1	0
0	0	0	0	0	1

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

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

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

4	3	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	2
0	0	0	0	0	4

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

kernel

C₂³.₆₉₇C₂⁴

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{697}C_2^4

% in TeX

G:=Group("C2^3.697C2^4");

// GroupNames label

G:=SmallGroup(128,1529);

// by ID

G=gap.SmallGroup(128,1529);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,604,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c*a=a*c,e^2=a,f^2=a*b*c,a*b=b*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 = C₂³.₆₉₇C₂⁴ order 128 = 2⁷

414^th central stem extension by C₂³ of C₂⁴

G = C23.697C24 order 128 = 27

414th central stem extension by C23 of C24

G = C₂³.₆₉₇C₂⁴ order 128 = 2⁷

414^th central stem extension by C₂³ of C₂⁴