C2^3.375C2^4

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

Aliases: C₂³.₃₇₅C₂⁴, C₂⁴.₂₉₂C₂³, C₂².₁₃₃2_-⁽¹⁺⁴⁾, C₄⋊C₄.₃₃₄D₄, C₂.₃₆(D₄⋊₆D₄), C₂.₃₀(Q₈⋊₅D₄), (C₂×C₄²).₃₄C₂², C₂³.Q₈.₆C₂, C₂³.₁₄₂(C₄○D₄), (C₂³×C₄).₃₆₃C₂², (C₂²×C₄).₅₁₉C₂³, C₂³.₁₁D₄.₄C₂, C₂².₂₅₅(C₂²×D₄), C₂³.₈Q₈.₁₇C₂, C₂³.₈₃C₂³⋊₁₃C₂, C₂³.₆₃C₂³⋊₅₃C₂, C₂³.₈₁C₂³⋊₁₉C₂, C₂³.₆₅C₂³⋊₆₄C₂, C₂.₄₇(C₂².₁₉C₂⁴), C₂⁴.C₂².₁₆C₂, C₂.C₄².₁₃₁C₂², C₂.₂₇(C₂².₄₆C₂⁴), C₂.₁₄(C₂².₃₅C₂⁴), C₂.₄₁(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₆₄C₂, (C₂×C₄).₃₄₄(C₂×D₄), (C₂×C₄².C₂)⋊₇C₂, (C₂×C₄).₃₇₀(C₄○D₄), (C₂×C₄⋊C₄).₈₅₂C₂², C₂².₂₅₂(C₂×C₄○D₄), (C₂×C₄²⋊C₂).₃₉C₂, (C₂×C₂²⋊C₄).₁₄₅C₂², SmallGroup(128,1207)

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

Derived series

C₁ — C₂³ — C₂³.₃₇₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₃₇₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₇₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₇₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₇₅C₂⁴

Subgroups: 404 in 238 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×2], C₄^[×20], C₂²^[×7], C₂²^[×10], C₂×C₄^[×12], C₂×C₄^[×44], C₂³, C₂³^[×2], C₂³^[×6], C₄²^[×8], C₂²⋊C₄^[×12], C₄⋊C₄^[×4], C₄⋊C₄^[×22], C₂²×C₄^[×14], C₂²×C₄^[×6], C₂⁴, C₂.C₄²^[×12], C₂×C₄²^[×4], C₂×C₂²⋊C₄^[×6], C₂×C₄⋊C₄^[×12], C₄²⋊C₂^[×4], C₄².C₂^[×4], C₂³×C₄, C₄×C₄⋊C₄, C₂³.₈Q₈^[×2], C₂³.₆₃C₂³^[×3], C₂⁴.C₂²^[×2], C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₄²⋊C₂, C₂×C₄².C₂, C₂³.₃₇₅C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×8], C₂⁴, C₂²×D₄, C₂×C₄○D₄^[×4], 2_-⁽¹⁺⁴⁾^[×2], C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₃₅C₂⁴, D₄⋊₆D₄, Q₈⋊₅D₄, C₂².₄₆C₂⁴^[×2], C₂³.₃₇₅C₂⁴

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

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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	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

1	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	3	0
0	0	0	0	2	2

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

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

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

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

type

image

C₁

C₂

D₄

C₄○D₄

2_-⁽¹⁺⁴⁾

kernel

C₂³.₃₇₅C₂⁴

C₄×C₄⋊C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄²⋊C₂

C₂×C₄².C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{375}C_2^4

% in TeX

G:=Group("C2^3.375C2^4");

// GroupNames label

G:=SmallGroup(128,1207);

// by ID

G=gap.SmallGroup(128,1207);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,675,136]);

// Polycyclic

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

// generators/relations

G = C₂³.₃₇₅C₂⁴ order 128 = 2⁷

92^nd central stem extension by C₂³ of C₂⁴

G = C23.375C24 order 128 = 27

92nd central stem extension by C23 of C24

G = C₂³.₃₇₅C₂⁴ order 128 = 2⁷

92^nd central stem extension by C₂³ of C₂⁴