C2^4.394C2^3 - GroupNames

G = C₂⁴.₃₉₄C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₉₄C₂³, C₂³.₅₈₇C₂⁴, C₂².₃₆₁2₊¹⁺⁴, C₂².₂₆₈2_-¹⁺⁴, C₂²⋊C₄.₁₂D₄, C₂³⋊Q₈⋊₄₃C₂, C₂³.₂₁₀(C₂×D₄), C₂.₉₂(D₄⋊₅D₄), C₂³.₇Q₈⋊₈₄C₂, C₂³.₁₇₀(C₄○D₄), C₂³.₁₁D₄⋊₇₉C₂, C₂³.₃₄D₄⋊₄₆C₂, (C₂³×C₄).₄₅₃C₂², (C₂²×C₄).₈₇₀C₂³, (C₂×C₄²).₆₄₂C₂², C₂³.₈Q₈⋊₁₀₁C₂, C₂².₃₉₆(C₂²×D₄), C₂³.₂₃D₄.₅₀C₂, C₂³.₁₀D₄.₄₀C₂, (C₂²×D₄).₂₂₆C₂², (C₂²×Q₈).₁₈₀C₂², C₂⁴.C₂²⋊₁₂₅C₂, C₂³.₈₁C₂³⋊₇₉C₂, C₂³.₈₃C₂³⋊₇₄C₂, C₂³.₆₇C₂³⋊₇₇C₂, C₂.C₄².₂₉₄C₂², C₂.₈₀(C₂².₄₆C₂⁴), C₂.₇₀(C₂².₃₆C₂⁴), C₂.₁₂(C₂².₅₆C₂⁴), C₂.₅₆(C₂³.₃₈C₂³), C₂.₅₉(C₂².₃₃C₂⁴), (C₂×C₄).₉₃(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₃₈C₂, (C₂×C₄).₄₁₉(C₄○D₄), (C₂×C₄⋊C₄).₄₀₁C₂², C₂².₄₄₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₅₄C₂², (C₂×C₂².D₄).₂₄C₂, SmallGroup(128,1419)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₉₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₃₉₄C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₉₄C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₉₄C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₉₄C₂³

Generators and relations for C₂⁴.₃₉₄C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=f²=a, e²=b, ab=ba, ac=ca, ede^-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 516 in 256 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂⁴.₃₉₄C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₈C₂³, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, D₄⋊₅D₄, C₂².₄₆C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₃₉₄C₂³

Smallest permutation representation of C₂⁴.₃₉₄C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₃₉₄C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₂²⋊Q₈

C₂×C₂².D₄

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₉₄C₂³ ►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	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	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

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

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

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

C₂⁴.₃₉₄C₂³ in GAP, Magma, Sage, TeX

C_2^4._{394}C_2^3

% in TeX

G:=Group("C2^4.394C2^3");

// GroupNames label

G:=SmallGroup(128,1419);

// by ID

G=gap.SmallGroup(128,1419);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.394C23 order 128 = 27

234th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₉₄C₂³ order 128 = 2⁷

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