C2^4.442C2^3

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

Aliases: C₂⁴.₄₄₂C₂³, C₂³.₆₆₄C₂⁴, C₂².₄₃₇2₊⁽¹⁺⁴⁾, C₂².₃₃₀2_-⁽¹⁺⁴⁾, C₄²⋊₅C₄⋊₃₃C₂, C₂³⋊Q₈.₂₇C₂, C₂³.₁₉₂(C₄○D₄), (C₂³×C₄).₁₆₉C₂², (C₂×C₄²).₆₉₆C₂², (C₂²×C₄).₅₈₃C₂³, C₂³.₈Q₈.₅₉C₂, C₂³.₁₁D₄.₄₇C₂, C₂³.₃₄D₄.₃₂C₂, (C₂²×Q₈).₂₁₅C₂², C₂.₉₁(C₂².₃₂C₂⁴), C₂⁴.C₂².₆₆C₂, C₂³.₈₁C₂³⋊₁₁₆C₂, C₂³.₈₃C₂³⋊₁₀₆C₂, C₂³.₆₇C₂³⋊₁₀₀C₂, C₂.C₄².₃₆₈C₂², C₂.₁₁₆(C₂².₄₅C₂⁴), C₂.₃₈(C₂².₅₇C₂⁴), C₂.₁₀₂(C₂².₄₆C₂⁴), C₂.₁₀₅(C₂².₃₆C₂⁴), (C₂×C₄).₄₅₈(C₄○D₄), (C₂×C₄⋊C₄).₄₇₄C₂², C₂².₅₂₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₁C₂², SmallGroup(128,1496)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₄₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₇C₂³ — C₂⁴.₄₄₂C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₄₂C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₄₂C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₄₂C₂³

Subgroups: 388 in 200 conjugacy classes, 88 normal (22 characteristic)
C₁, C₂^[×3], C₂^[×4], C₂^[×2], C₄^[×16], C₂²^[×3], C₂²^[×4], C₂²^[×10], C₂×C₄^[×4], C₂×C₄^[×44], Q₈^[×4], C₂³, C₂³^[×2], C₂³^[×6], C₄²^[×4], C₂²⋊C₄^[×10], C₄⋊C₄^[×7], C₂²×C₄^[×2], C₂²×C₄^[×12], C₂²×C₄^[×4], C₂×Q₈^[×5], C₂⁴, C₂.C₄²^[×2], C₂.C₄²^[×16], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×2], C₂×C₂²⋊C₄^[×4], C₂×C₄⋊C₄, C₂×C₄⋊C₄^[×6], C₂³×C₄, C₂²×Q₈, C₂³.₃₄D₄, C₄²⋊₅C₄^[×2], C₂³.₈Q₈^[×2], C₂⁴.C₂²^[×2], C₂³.₆₇C₂³^[×2], C₂³⋊Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³^[×2], C₂³.₈₃C₂³^[×2], C₂⁴.₄₄₂C₂³

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, C₂².₃₆C₂⁴^[×2], C₂².₄₅C₂⁴, C₂².₄₆C₂⁴^[×2], C₂².₅₇C₂⁴, C₂⁴.₄₄₂C₂³

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

Smallest permutation representation
►On 64 points

Generators in S₆₄

(2 11)(4 9)(5 21)(6 63)(7 23)(8 61)(13 41)(15 43)(17 62)(18 22)(19 64)(20 24)(25 31)(26 45)(27 29)(28 47)(30 51)(32 49)(33 53)(35 55)(37 59)(39 57)(46 50)(48 52)

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

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

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

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

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

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

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

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

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

2	0	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	3	1
0	0	0	0	0	2

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

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

C₂⁴.₄₄₂C₂³

C₂³.₃₄D₄

C₄²⋊₅C₄

C₂³.₈Q₈

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^4._{442}C_2^3

% in TeX

G:=Group("C2^4.442C2^3");

// GroupNames label

G:=SmallGroup(128,1496);

// by ID

G=gap.SmallGroup(128,1496);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,120,758,723,268,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C₂⁴.₄₄₂C₂³ order 128 = 2⁷

282^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

G = C24.442C23 order 128 = 27

282nd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₄₂C₂³ order 128 = 2⁷

282^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²