C2^4.309C2^3 - GroupNames

G = C₂⁴.₃₀₉C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₀₉C₂³, C₂³.₄₁₅C₂⁴, C₂².₂₀₉2₊¹⁺⁴, C₂².₁₅₉2_-¹⁺⁴, C₄²⋊₅C₄⋊₁₃C₂, C₄²⋊₈C₄⋊₃₅C₂, (C₂²×C₄).₈₃C₂³, C₂³.Q₈.₈C₂, C₂³.₁₄₆(C₄○D₄), (C₂×C₄²).₅₃₀C₂², (C₂³×C₄).₁₀₄C₂², C₂³.₈Q₈.₂₃C₂, C₂³.₃₄D₄.₁₇C₂, C₂³.₁₁D₄.₁₁C₂, C₂³.₈₃C₂³⋊₂₉C₂, C₂³.₆₃C₂³⋊₆₉C₂, C₂³.₈₁C₂³⋊₂₈C₂, C₂⁴.C₂².₂₃C₂, C₂.₃₂(C₂².₄₅C₂⁴), C₂.C₄².₁₆₃C₂², C₂.₄₈(C₂².₄₆C₂⁴), C₂.₃₆(C₂².₃₆C₂⁴), C₂.₄₃(C₂².₄₇C₂⁴), C₂.₅₈(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₇₇C₂, (C₄×C₂²⋊C₄).₅₅C₂, (C₂×C₄).₁₃₆(C₄○D₄), (C₂×C₄⋊C₄).₂₇₉C₂², C₂².₂₉₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₆₃C₂², SmallGroup(128,1247)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₀₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₀₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₀₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₀₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₀₉C₂³

Generators and relations for C₂⁴.₃₀₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=c, e²=a, g²=b, ab=ba, ac=ca, ede^-1=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: 372 in 206 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₃₄D₄, C₄²⋊₈C₄, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₃₀₉C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂⁴.₃₀₉C₂³

Smallest permutation representation of C₂⁴.₃₀₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
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

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₃₀₉C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₃₄D₄

C₄²⋊₈C₄

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₀₉C₂³ ►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

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	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

0	1	0	0	0	0
1	0	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	0	3

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

1	0	0	0	0	0
0	4	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	0	4

2	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	1	0
0	0	0	0	0	1

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

C₂⁴.₃₀₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{309}C_2^3

% in TeX

G:=Group("C2^4.309C2^3");

// GroupNames label

G:=SmallGroup(128,1247);

// by ID

G=gap.SmallGroup(128,1247);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=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 = C24.309C23 order 128 = 27

149th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₀₉C₂³ order 128 = 2⁷

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