C2^3.708C2^4

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

Aliases: C₂⁴.₉₉C₂³, C₂³.₇₀₈C₂⁴, C₂².₄₈₁2₊⁽¹⁺⁴⁾, C₂².₃₆₈2_-⁽¹⁺⁴⁾, C₂³⋊Q₈⋊₅₉C₂, (C₂×C₄²).₁₁₆C₂², (C₂²×C₄).₆₁₄C₂³, C₂³.₁₁D₄⋊₁₂₅C₂, C₂³.₁₀D₄.₇₀C₂, (C₂²×D₄).₂₉₀C₂², (C₂²×Q₈).₂₂₈C₂², C₂⁴.C₂²⋊₁₇₇C₂, C₂⁴.₃C₂².₇₉C₂, C₂³.₆₅C₂³⋊₁₆₃C₂, C₂³.₆₇C₂³⋊₁₀₅C₂, C₂³.₆₃C₂³⋊₁₉₆C₂, C₂.₁₁₂(C₂².₃₂C₂⁴), C₂.C₄².₄₁₂C₂², C₂.₄₉(C₂².₄₉C₂⁴), C₂.₄₇(C₂².₅₆C₂⁴), C₂.₇₇(C₂².₅₀C₂⁴), C₂.₄₉(C₂².₅₃C₂⁴), C₂.₁₁₉(C₂².₃₃C₂⁴), (C₂×C₄).₂₄₉(C₄○D₄), (C₂×C₄⋊C₄).₅₁₈C₂², C₂².₅₆₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₃₁C₂², SmallGroup(128,1540)

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

Derived series

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂³.₇₀₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₇₀₈C₂⁴

Subgroups: 452 in 214 conjugacy classes, 88 normal (30 characteristic)
C₁, C₂^[×7], C₂^[×2], C₄^[×16], C₂²^[×7], C₂²^[×14], C₂×C₄^[×6], C₂×C₄^[×36], D₄^[×4], Q₈^[×4], C₂³, C₂³^[×14], C₄²^[×3], C₂²⋊C₄^[×16], C₄⋊C₄^[×9], C₂²×C₄^[×3], C₂²×C₄^[×10], C₂×D₄^[×4], C₂×Q₈^[×4], C₂⁴^[×2], C₂.C₄²^[×12], C₂×C₄², C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×12], C₂×C₄⋊C₄^[×4], C₂×C₄⋊C₄^[×2], C₂²×D₄, C₂²×Q₈, C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×4], C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³⋊Q₈^[×2], C₂³.₁₀D₄^[×2], C₂³.₁₁D₄^[×2], C₂³.₇₀₈C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×3], 2_-⁽¹⁺⁴⁾, C₂².₃₂C₂⁴^[×2], C₂².₃₃C₂⁴, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂².₅₃C₂⁴, C₂².₅₆C₂⁴, C₂³.₇₀₈C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=b, f²=cb=bc, g²=ba=ab, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ 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	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,1,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4R	4S	4T	4U	4V
order	1	2	···	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

2_-⁽¹⁺⁴⁾

kernel

C₂³.₇₀₈C₂⁴

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{708}C_2^4

% in TeX

G:=Group("C2^3.708C2^4");

// GroupNames label

G:=SmallGroup(128,1540);

// by ID

G=gap.SmallGroup(128,1540);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,520,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C₂³.₇₀₈C₂⁴ order 128 = 2⁷

425^th central stem extension by C₂³ of C₂⁴

G = C23.708C24 order 128 = 27

425th central stem extension by C23 of C24

G = C₂³.₇₀₈C₂⁴ order 128 = 2⁷

425^th central stem extension by C₂³ of C₂⁴