C2^4.276C2^3 - GroupNames

G = C₂⁴.₂₇₆C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₇₆C₂³, C₂³.₃₅₅C₂⁴, C₂².₁₆₂2₊¹⁺⁴, C₂².₁₂₀2_-¹⁺⁴, C₂.₂₁D₄², C₂²⋊C₄⋊₃₇D₄, C₂³⋊₂D₄.₃C₂, C₂³.₁₆₈(C₂×D₄), C₂.₂₈(D₄⋊₆D₄), C₂³.₄Q₈⋊₉C₂, C₂³.₂₆(C₄○D₄), (C₂³×C₄).₈₁C₂², C₂³.₈Q₈⋊₄₅C₂, C₂³.₁₀D₄⋊₂₆C₂, (C₂²×C₄).₈₁₀C₂³, (C₂×C₄²).₄₉₈C₂², C₂².₂₃₅(C₂²×D₄), C₂⁴.C₂²⋊₄₀C₂, (C₂²×D₄).₅₁₄C₂², C₂³.₆₅C₂³⋊₅₈C₂, C₂.₂₇(C₂².₁₉C₂⁴), C₂.C₄².₁₁₂C₂², C₂.₉(C₂².₅₃C₂⁴), C₂.₁₄(C₂².₃₃C₂⁴), (C₂×C₄×D₄)⋊₃₃C₂, (C₂×C₄).₃₃₄(C₂×D₄), (C₂×C₄).₁₁₀(C₄○D₄), (C₂×C₄⋊C₄).₂₃₇C₂², C₂².₂₃₂(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₁₂C₂, (C₂×C₂²⋊C₄).₁₃₂C₂², SmallGroup(128,1187)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₇₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₂⁴.₂₇₆C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₇₆C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₇₆C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₇₆C₂³

Generators and relations for C₂⁴.₂₇₆C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=1, g²=b, ab=ba, ac=ca, ede=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 644 in 330 conjugacy classes, 108 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₈Q₈, C₂³.₈Q₈, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₄Q₈, C₂×C₄×D₄, C₂×C₂².D₄, C₂⁴.₂₇₆C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂².₃₃C₂⁴, D₄², D₄⋊₆D₄, C₂².₅₃C₂⁴, C₂⁴.₂₇₆C₂³

Smallest permutation representation of C₂⁴.₂₇₆C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(9 57)(10 58)(11 59)(12 60)(21 29)(22 30)(23 31)(24 32)(25 38)(26 39)(27 40)(28 37)(33 35)(34 36)(41 43)(42 44)(45 47)(46 48)(49 64)(50 61)(51 62)(52 63)(53 55)(54 56)

(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)

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	···	2M	4A	···	4H	4I	···	4T	4U	4V	4W	4X
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

2_-¹⁺⁴

kernel

C₂⁴.₂₇₆C₂³

C₂³.₈Q₈

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₄Q₈

C₂×C₄×D₄

C₂×C₂².D₄

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₇₆C₂³ ►in 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	4	0	0	0
0	0	0	4	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

0	4	0	0	0	0
4	0	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

0	1	0	0	0	0
1	0	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

1	0	0	0	0	0
0	4	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

3	0	0	0	0	0
0	3	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	2	0
0	0	0	0	0	3

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

C₂⁴.₂₇₆C₂³ in GAP, Magma, Sage, TeX

C_2^4._{276}C_2^3

% in TeX

G:=Group("C2^4.276C2^3");

// GroupNames label

G:=SmallGroup(128,1187);

// by ID

G=gap.SmallGroup(128,1187);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24.276C23 order 128 = 27

116th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₇₆C₂³ order 128 = 2⁷

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