C2^4.249C2^3 - GroupNames

G = C₂⁴.₂₄₉C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₄₉C₂³, C₂³.₃₁₄C₂⁴, C₂².₁₂₉2₊¹⁺⁴, C₄⋊C₄⋊₃₃D₄, (C₂²×C₄)⋊₂₃D₄, C₂³⋊₂D₄⋊₈C₂, C₂.₇(Q₈⋊₆D₄), C₂³.₁₅₃(C₂×D₄), C₂.₁₉(D₄⋊₅D₄), C₄.₁₆₂(C₄⋊D₄), C₂³.₂₀(C₄○D₄), (C₂²×C₄).₄₉C₂³, C₂³.₇Q₈⋊₃₄C₂, C₂³.₁₀D₄⋊₁₀C₂, (C₂×C₄²).₄₆₃C₂², (C₂³×C₄).₃₃₂C₂², C₂².₁₉₄(C₂²×D₄), C₂⁴.C₂²⋊₃₁C₂, C₂⁴.₃C₂²⋊₂₈C₂, (C₂²×D₄).₅₀₂C₂², C₂³.₆₅C₂³⋊₃₄C₂, C₂.₂₁(C₂².₁₉C₂⁴), C₂.₁₁(C₂².₂₉C₂⁴), C₂.C₄².₇₈C₂², C₂.₆(C₂².₄₉C₂⁴), C₂.₁₂(C₂².₄₇C₂⁴), (C₂×C₄×D₄)⋊₂₁C₂, (C₂×C₄⋊D₄)⋊₅C₂, (C₂×C₄).₆₇₉(C₂×D₄), C₂.₁₈(C₂×C₄⋊D₄), (C₂×C₄).₉₄(C₄○D₄), (C₂×C₄²⋊C₂)⋊₂₀C₂, (C₂×C₄⋊C₄).₂₀₅C₂², C₂².₁₉₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₀₉C₂², SmallGroup(128,1146)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₄₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₂⁴.₂₄₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₄₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₄₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₄₉C₂³

Generators and relations for C₂⁴.₂₄₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=f²=1, d²=b, g²=a, 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: 740 in 352 conjugacy classes, 112 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊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, C₄²⋊C₂, C₄×D₄, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₇Q₈, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂×C₄²⋊C₂, C₂×C₄×D₄, C₂×C₄⋊D₄, C₂⁴.₂₄₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₄⋊D₄, C₂².₁₉C₂⁴, C₂².₂₉C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂⁴.₂₄₉C₂³

Smallest permutation representation of C₂⁴.₂₄₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	···	4H	4I	···	4V	4W	4X
order	1	2	···	2	2	2	2	2	2	2	4	···	4	4	···	4	4	4
size	1	1	···	1	4	4	4	4	8	8	2	···	2	4	···	4	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₂₄₉C₂³

C₂³.₇Q₈

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂×C₄²⋊C₂

C₂×C₄×D₄

C₂×C₄⋊D₄

C₄⋊C₄

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₄₉C₂³ ►in 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

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

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

0	2	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	1
0	0	0	0	1	0

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

0	1	0	0	0	0
4	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	4

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

C₂⁴.₂₄₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{249}C_2^3

% in TeX

G:=Group("C2^4.249C2^3");

// GroupNames label

G:=SmallGroup(128,1146);

// by ID

G=gap.SmallGroup(128,1146);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b,g^2=a,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 = C24.249C23 order 128 = 27

89th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₄₉C₂³ order 128 = 2⁷

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