C2^4.443C2^3

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

Aliases: C₂⁴.₄₄₃C₂³, C₂³.₆₆₅C₂⁴, C₂².₃₃₁2_-⁽¹⁺⁴⁾, C₂².₄₃₈2₊⁽¹⁺⁴⁾, C₄²⋊₈C₄⋊₆₃C₂, C₂³.₁₉₃(C₄○D₄), (C₂²×C₄).₅₈₄C₂³, (C₂³×C₄).₁₇₀C₂², (C₂×C₄²).₆₉₇C₂², C₂³.₈Q₈.₆₀C₂, C₂³.Q₈.₃₅C₂, C₂³.₃₄D₄.₃₃C₂, C₂³.₁₁D₄.₄₈C₂, C₂³.₈₄C₂³⋊₁₃C₂, C₂.₉₂(C₂².₃₂C₂⁴), C₂⁴.C₂².₆₇C₂, C₂³.₈₃C₂³⋊₁₀₇C₂, C₂³.₆₅C₂³⋊₁₄₃C₂, C₂³.₈₁C₂³⋊₁₁₇C₂, C₂³.₆₃C₂³⋊₁₇₂C₂, C₂.C₄².₃₆₉C₂², C₂.₃₉(C₂².₅₇C₂⁴), C₂.₉₂(C₂².₃₃C₂⁴), C₂.₁₀₆(C₂².₃₆C₂⁴), C₂.₁₀₁(C₂².₄₇C₂⁴), C₂.₁₀₃(C₂².₄₆C₂⁴), (C₂×C₄).₄₅₉(C₄○D₄), (C₂×C₄⋊C₄).₄₇₅C₂², C₂².₅₂₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₁C₂², SmallGroup(128,1497)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₈Q₈ — C₂⁴.₄₄₃C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×2], C₄^[×16], C₂²^[×7], C₂²^[×10], C₂×C₄^[×4], C₂×C₄^[×44], C₂³, C₂³^[×2], C₂³^[×6], C₄²^[×3], C₂²⋊C₄^[×10], C₄⋊C₄^[×13], C₂²×C₄^[×14], C₂²×C₄^[×4], C₂⁴, C₂.C₄²^[×16], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×6], C₂×C₄⋊C₄^[×10], C₂³×C₄, C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₈Q₈^[×2], C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×2], C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³^[×2], C₂³.₈₃C₂³, C₂³.₈₄C₂³, C₂⁴.₄₄₃C₂³

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×2], 2_-⁽¹⁺⁴⁾^[×2], C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴^[×2], C₂².₄₇C₂⁴, C₂².₅₇C₂⁴, C₂⁴.₄₄₃C₂³

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=g²=cb=bc, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Smallest permutation representation
►On 64 points

Generators in S₆₄

(2 24)(4 22)(5 36)(6 39)(7 34)(8 37)(10 50)(12 52)(14 54)(16 56)(17 45)(18 30)(19 47)(20 32)(26 42)(28 44)(29 57)(31 59)(33 63)(35 61)(38 62)(40 64)(46 58)(48 60)

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

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

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

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

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

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

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

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	1	4	0	0
0	0	0	0	1	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	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

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

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

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	2

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

2_-⁽¹⁺⁴⁾

kernel

C₂⁴.₄₄₃C₂³

C₂³.₃₄D₄

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^4._{443}C_2^3

% in TeX

G:=Group("C2^4.443C2^3");

// GroupNames label

G:=SmallGroup(128,1497);

// by ID

G=gap.SmallGroup(128,1497);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C₂⁴.₄₄₃C₂³ order 128 = 2⁷

283^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

G = C24.443C23 order 128 = 27

283rd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₄₃C₂³ order 128 = 2⁷

283^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²