C2^3.369C2^4

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

Aliases: C₂⁴.₉C₂³, C₂³.₃₆₉C₂⁴, C₂².₁₂₉2_-⁽¹⁺⁴⁾, C₄⋊C₄.₃₃₂D₄, C₂.₃₃(D₄⋊₆D₄), C₂³⋊Q₈.₅C₂, C₂.₂₉(Q₈⋊₅D₄), (C₂²×C₄).₆₅C₂³, C₂³.Q₈.₅C₂, (C₂×C₄²).₅₁₂C₂², C₂².₂₄₉(C₂²×D₄), (C₂²×Q₈).₄₃₀C₂², C₂³.₇₈C₂³⋊₁₂C₂, C₂³.₈₃C₂³⋊₁₀C₂, C₂³.₆₃C₂³⋊₄₈C₂, C₂³.₆₅C₂³⋊₆₁C₂, C₂.₄₁(C₂².₁₉C₂⁴), C₂⁴.C₂².₁₅C₂, C₂.C₄².₁₂₆C₂², C₂.₃₇(C₂³.₃₆C₂³), C₂.₁₇(C₂².₅₀C₂⁴), C₂.₁₃(C₂².₃₅C₂⁴), (C₄×C₄⋊C₄)⋊₆₂C₂, (C₂×C₄×Q₈)⋊₂₀C₂, (C₂×C₄).₃₄₂(C₂×D₄), (C₂×C₄).₈₅₆(C₄○D₄), (C₂×C₄⋊C₄).₂₄₉C₂², (C₂×C₄²⋊₂C₂).₅C₂, C₂².₂₄₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₄₃C₂², SmallGroup(128,1201)

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

Derived series

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₃₆₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Subgroups: 404 in 235 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂^[×7], C₂, C₄^[×21], C₂²^[×7], C₂²^[×7], C₂×C₄^[×14], C₂×C₄^[×35], Q₈^[×8], C₂³, C₂³^[×7], C₄²^[×10], C₂²⋊C₄^[×14], C₄⋊C₄^[×4], C₄⋊C₄^[×18], C₂²×C₄^[×14], C₂×Q₈^[×6], C₂⁴, C₂.C₄²^[×12], C₂×C₄²^[×5], C₂×C₂²⋊C₄^[×7], C₂×C₄⋊C₄^[×10], C₄×Q₈^[×4], C₄²⋊₂C₂^[×4], C₂²×Q₈, C₄×C₄⋊C₄, C₂³.₆₃C₂³^[×3], C₂⁴.C₂²^[×4], C₂³.₆₅C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₈₃C₂³, C₂×C₄×Q₈, C₂×C₄²⋊₂C₂, C₂³.₃₆₉C₂⁴

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, C₂².₅₀C₂⁴^[×2], C₂³.₃₆₉C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=e²=f²=a, g²=ba=ab, 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 27)(6 28)(7 25)(8 26)(9 19)(10 20)(11 17)(12 18)(21 41)(22 42)(23 43)(24 44)(29 62)(30 63)(31 64)(32 61)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(45 49)(46 50)(47 51)(48 52)

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

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

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

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

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

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

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

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

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

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

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

0	1	0	0	0	0
1	0	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	0	3

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-⁽¹⁺⁴⁾

kernel

C₂³.₃₆₉C₂⁴

C₄×C₄⋊C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³⋊Q₈

C₂³.₇₈C₂³

C₂³.Q₈

C₂³.₈₃C₂³

C₂×C₄×Q₈

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{369}C_2^4

% in TeX

G:=Group("C2^3.369C2^4");

// GroupNames label

G:=SmallGroup(128,1201);

// by ID

G=gap.SmallGroup(128,1201);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,675,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=f^2=a,g^2=b*a=a*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 = C₂³.₃₆₉C₂⁴ order 128 = 2⁷

86^th central stem extension by C₂³ of C₂⁴

G = C23.369C24 order 128 = 27

86th central stem extension by C23 of C24

G = C₂³.₃₆₉C₂⁴ order 128 = 2⁷

86^th central stem extension by C₂³ of C₂⁴