C4^2.386C2^3 - GroupNames

G = C₄².₃₈₆C₂³ order 128 = 2⁷

247^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄².₃₈₆C₂³, C₈⋊D₄⋊₇C₂, C₈⋊₅D₄⋊₆C₂, C₈⋊₈D₄⋊₄₆C₂, C₄⋊C₄.₂₄₆D₄, C₈.₉(C₄○D₄), (C₄×SD₁₆)⋊₅₄C₂, C₂²⋊C₄.₈₆D₄, C₈.₅Q₈⋊₁₇C₂, C₂³.₈₃(C₂×D₄), C₄⋊C₄.₁₁₃C₂³, (C₂×C₈).₆₀₁C₂³, (C₄×C₈).₂₉₁C₂², (C₂×C₄).₃₇₂C₂⁴, (C₄×D₄).₉₃C₂², C₄⋊Q₈.₁₁₅C₂², SD₁₆⋊C₄⋊₁₉C₂, (C₄×Q₈).₉₀C₂², C₈○₂M₄(2)⋊₁₆C₂, C₂.D₈.₉₆C₂², C₂.₃₈(D₄○SD₁₆), (C₂×D₄).₁₂₇C₂³, C₄⋊D₄.₃₄C₂², C₄⋊₁D₄.₆₄C₂², (C₂×Q₈).₁₁₅C₂³, C₈⋊C₄.₁₂₉C₂², C₄.Q₈.₁₆₄C₂², C₂²⋊Q₈.₃₄C₂², (C₂²×C₈).₃₆₀C₂², (C₂×SD₁₆).₂₂C₂², C₂².₆₃₂(C₂²×D₄), C₄².C₂.₁₉C₂², D₄⋊C₄.₂₀₅C₂², C₂².₃₅C₂⁴⋊₃C₂, (C₂²×C₄).₁₀₅₂C₂³, Q₈⋊C₄.₁₂₈C₂², C₄²⋊C₂.₃₂₉C₂², C₄².₃₀C₂²⋊₂₀C₂, C₄².₂₉C₂²⋊₂₀C₂, (C₂×M₄(2)).₂₈₂C₂², C₂².₃₄C₂⁴.₂C₂, C₂.₆₉(C₂².₂₆C₂⁴), C₄.₅₇(C₂×C₄○D₄), (C₂×C₄).₁₄₄(C₂×D₄), SmallGroup(128,1906)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₈₆C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×C₈ — C₈○₂M₄(2) — C₄².₃₈₆C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₈₆C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₈₆C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₈₆C₂³

Generators and relations for C₄².₃₈₆C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=1, d²=b², e²=b, ab=ba, ac=ca, dad^-1=ab², ae=ea, cbc=b^-1, bd=db, be=eb, dcd^-1=a²c, ece^-1=b^-1c, de=ed >

Subgroups: 348 in 181 conjugacy classes, 88 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄².C₂, C₄².C₂, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×SD₁₆, C₈○₂M₄(2), C₄×SD₁₆, SD₁₆⋊C₄, C₈⋊₈D₄, C₈⋊D₄, C₄².₂₉C₂², C₄².₃₀C₂², C₈⋊₅D₄, C₈.₅Q₈, C₂².₃₄C₂⁴, C₂².₃₅C₂⁴, C₄².₃₈₆C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₂₆C₂⁴, D₄○SD₁₆, C₄².₃₈₆C₂³

Smallest permutation representation of C₄².₃₈₆C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	···	4F	4G	4H	4I	4J	···	4O	8A	8B	8C	8D	8E	···	8J
order	1	2	2	2	2	2	2	4	···	4	4	4	4	4	···	4	8	8	8	8	8	···	8
size	1	1	1	1	4	8	8	2	···	2	4	4	4	8	···	8	2	2	2	2	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

kernel

C₄².₃₈₆C₂³

C₈○₂M₄(2)

C₄×SD₁₆

SD₁₆⋊C₄

C₈⋊₈D₄

C₈⋊D₄

C₄².₂₉C₂²

C₄².₃₀C₂²

C₈⋊₅D₄

C₈.₅Q₈

C₂².₃₄C₂⁴

C₂².₃₅C₂⁴

C₂²⋊C₄

C₄⋊C₄

C₈

C₂

# reps

Matrix representation of C₄².₃₈₆C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	16	0	0	0
0	0	0	16	0	0

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0

1	15	0	0	0	0
0	16	0	0	0	0
0	0	0	14	5	0
0	0	14	0	0	12
0	0	12	0	0	14
0	0	0	5	14	0

4	9	0	0	0	0
4	13	0	0	0	0
0	0	12	0	0	14
0	0	0	12	3	0
0	0	0	14	5	0
0	0	3	0	0	5

1	15	0	0	0	0
1	16	0	0	0	0
0	0	5	12	0	0
0	0	5	5	0	0
0	0	0	0	5	12
0	0	0	0	5	5

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,14,12,0,0,0,14,0,0,5,0,0,5,0,0,14,0,0,0,12,14,0],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,12,0,0,3,0,0,0,12,14,0,0,0,0,3,5,0,0,0,14,0,0,5],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5] >;

C₄².₃₈₆C₂³ in GAP, Magma, Sage, TeX

C_4^2._{386}C_2^3

% in TeX

G:=Group("C4^2.386C2^3");

// GroupNames label

G:=SmallGroup(128,1906);

// by ID

G=gap.SmallGroup(128,1906);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,184,1018,80,4037,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=b^2,e^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,e*c*e^-1=b^-1*c,d*e=e*d>;

// generators/relations

G = C42.386C23 order 128 = 27

247th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₈₆C₂³ order 128 = 2⁷

247^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²