C2^4.308C2^3 - GroupNames

G = C₂⁴.₃₀₈C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₀₈C₂³, C₂³.₃₉₉C₂⁴, C₂².₁₅₀2_-¹⁺⁴, C₄²⋊₉C₄⋊₂₁C₂, C₄²⋊₈C₄⋊₂₉C₂, C₂³.₁₈₃(C₂×D₄), (C₂²×C₄).₃₈₆D₄, (C₂²×C₄).₇₈C₂³, (C₂×C₄²).₄₈C₂², (C₂³×C₄).₃₈₃C₂², C₂².₂₇₅(C₂²×D₄), C₂³.₁₁D₄.₁₀C₂, C₄.₅₀(C₂².D₄), (C₂²×Q₈).₁₁₈C₂², C₂³.₆₃C₂³⋊₆₆C₂, C₂³.₆₇C₂³⋊₅₁C₂, C₂.C₄².₁₅₁C₂², C₂.₂₄(C₂².₂₆C₂⁴), C₂.₁₇(C₂³.₃₈C₂³), C₂.₄₂(C₂².₄₆C₂⁴), C₂.₂₄(C₂².₅₀C₂⁴), (C₂×C₄).₈₃₀(C₂×D₄), (C₄×C₂²⋊C₄).₅₁C₂, (C₂×C₂²⋊Q₈).₃₁C₂, (C₂×C₄).₁₂₆(C₄○D₄), (C₂×C₄⋊C₄).₂₆₈C₂², C₂².₂₇₆(C₂×C₄○D₄), (C₂×C₄²⋊C₂).₄₂C₂, C₂.₃₄(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₅₀₀C₂², SmallGroup(128,1231)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₀₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₃₀₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₀₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₀₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₀₈C₂³

Generators and relations for C₂⁴.₃₀₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=c, f²=b, g²=a, ab=ba, ac=ca, ede=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 420 in 242 conjugacy classes, 104 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₄²⋊₈C₄, C₄²⋊₉C₄, C₂³.₆₃C₂³, C₂³.₆₇C₂³, C₂³.₁₁D₄, C₂×C₄²⋊C₂, C₂×C₂²⋊Q₈, C₂⁴.₃₀₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₂².D₄, C₂².₂₆C₂⁴, C₂³.₃₈C₂³, C₂².₄₆C₂⁴, C₂².₅₀C₂⁴, C₂⁴.₃₀₈C₂³

Smallest permutation representation of C₂⁴.₃₀₈C₂³
►On 64 points

Generators in S₆₄

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

(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)

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

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

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

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

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

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

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

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂⁴.₃₀₈C₂³

C₄×C₂²⋊C₄

C₄²⋊₈C₄

C₄²⋊₉C₄

C₂³.₆₃C₂³

C₂³.₆₇C₂³

C₂³.₁₁D₄

C₂×C₄²⋊C₂

C₂×C₂²⋊Q₈

C₂²×C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₃₀₈C₂³ ►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	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	1	0
0	0	0	0	0	1

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

0	1	0	0	0	0
4	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	2

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	3	0
0	0	0	0	0	2

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

C₂⁴.₃₀₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{308}C_2^3

% in TeX

G:=Group("C2^4.308C2^3");

// GroupNames label

G:=SmallGroup(128,1231);

// by ID

G=gap.SmallGroup(128,1231);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,184,675,80]);

// Polycyclic

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

// generators/relations

G = C24.308C23 order 128 = 27

148th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₀₈C₂³ order 128 = 2⁷

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