C2^4.448C2^3 - GroupNames

G = C₂⁴.₄₄₈C₂³ order 128 = 2⁷

288^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₄₄₈C₂³, C₂³.₆₈₀C₂⁴, C₂².₄₅₃2₊¹⁺⁴, (C₂×D₄).₁₉Q₈, C₂³.₄₂(C₂×Q₈), C₂.₆₀(D₄⋊₃Q₈), C₂³.Q₈⋊₈₄C₂, C₂³.₄Q₈⋊₆₁C₂, (C₂×C₄²).₇₀₉C₂², (C₂²×C₄).₂₁₃C₂³, (C₂³×C₄).₁₇₃C₂², C₂³.₈Q₈⋊₁₃₄C₂, C₂³.₇Q₈⋊₁₀₉C₂, C₂.₁₇(C₂³⋊₂Q₈), C₂².₁₅₈(C₂²×Q₈), C₂³.₂₃D₄.₇₁C₂, (C₂²×D₄).₂₇₇C₂², C₂⁴.₃C₂².₇₃C₂, C₂³.₈₃C₂³⋊₁₁₆C₂, C₂³.₆₃C₂³⋊₁₈₁C₂, C₂³.₆₅C₂³⋊₁₅₂C₂, C₂³.₈₁C₂³⋊₁₂₃C₂, C₂.₁₀₀(C₂².₃₂C₂⁴), C₂.₃₂(C₂².₅₄C₂⁴), C₂.C₄².₃₈₄C₂², C₂.₆₁(C₂².₃₄C₂⁴), C₂.₁₀₉(C₂².₄₇C₂⁴), (C₂×C₄).₈₂(C₂×Q₈), (C₂×C₄).₂₂₇(C₄○D₄), (C₂×C₄⋊C₄).₄₉₀C₂², C₂².₅₄₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₆C₂², SmallGroup(128,1512)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₄₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₈Q₈ — C₂⁴.₄₄₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₄₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₄₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₄₈C₂³

Generators and relations for C₂⁴.₄₄₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=abc, e²=a, g²=ba=ab, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 484 in 232 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂⁴.₄₄₈C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂³⋊₂Q₈, C₂².₄₇C₂⁴, D₄⋊₃Q₈, C₂².₅₄C₂⁴, C₂⁴.₄₄₈C₂³

Smallest permutation representation of C₂⁴.₄₄₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

Q₈

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₄₄₈C₂³

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.Q₈

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₄₄₈C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

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

C₂⁴.₄₄₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{448}C_2^3

% in TeX

G:=Group("C2^4.448C2^3");

// GroupNames label

G:=SmallGroup(128,1512);

// by ID

G=gap.SmallGroup(128,1512);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=a*b*c,e^2=a,g^2=b*a=a*b,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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 = C24.448C23 order 128 = 27

288th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₄₈C₂³ order 128 = 2⁷

288^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²