C4^2.385C2^3 - GroupNames

G = C₄².₃₈₅C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₈₅C₂³, C₈⋊₃Q₈⋊₆C₂, C₈⋊₈D₄⋊₄₅C₂, C₄⋊C₄.₂₄₅D₄, C₈.₈(C₄○D₄), C₈.D₄⋊₆C₂, C₈⋊₂D₄.₄C₂, D₈⋊C₄⋊₁₃C₂, (C₄×SD₁₆)⋊₅₃C₂, C₂²⋊C₄.₈₅D₄, C₂³.₈₂(C₂×D₄), Q₁₆⋊C₄⋊₁₃C₂, C₈.₁₂D₄⋊₁₈C₂, C₄⋊C₄.₁₁₂C₂³, (C₄×C₈).₂₉₀C₂², (C₂×C₈).₆₀₀C₂³, (C₂×C₄).₃₇₁C₂⁴, (C₄×D₄).₉₂C₂², (C₂×D₈).₆₃C₂², C₄⋊Q₈.₁₁₄C₂², (C₄×Q₈).₈₉C₂², C₈○₂M₄(2)⋊₁₅C₂, C₂.₃₇(D₄○SD₁₆), (C₂×D₄).₁₂₆C₂³, C₄⋊D₄.₃₃C₂², (C₂×Q₈).₁₁₄C₂³, (C₂×Q₁₆).₆₄C₂², C₄.Q₈.₁₆₃C₂², C₈⋊C₄.₁₂₈C₂², C₂²⋊Q₈.₃₃C₂², (C₂²×C₈).₃₅₉C₂², C₄.₄D₄.₃₄C₂², C₂².₆₃₁(C₂²×D₄), D₄⋊C₄.₂₀₄C₂², (C₂²×C₄).₁₀₅₁C₂³, C₂².₃₆C₂⁴⋊₄C₂, Q₈⋊C₄.₂₀₆C₂², (C₂×SD₁₆).₁₅₁C₂², C₄².₂₈C₂²⋊₃₃C₂, C₄²⋊C₂.₃₂₈C₂², (C₂×M₄(2)).₂₈₁C₂², C₂.₆₈(C₂².₂₆C₂⁴), C₄.₅₆(C₂×C₄○D₄), (C₂×C₄).₁₄₃(C₂×D₄), SmallGroup(128,1905)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₈₅C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×C₈ — C₈○₂M₄(2) — C₄².₃₈₅C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₈₅C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₈₅C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₈₅C₂³

Generators and relations for C₄².₃₈₅C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=d²=1, e²=b, ab=ba, ac=ca, dad=ab², ae=ea, cbc=b^-1, bd=db, be=eb, dcd=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), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₄.Q₈, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₈○₂M₄(2), C₄×SD₁₆, Q₁₆⋊C₄, D₈⋊C₄, C₈⋊₈D₄, C₈⋊₂D₄, C₈.D₄, C₄².₂₈C₂², C₈.₁₂D₄, C₈⋊₃Q₈, C₂².₃₆C₂⁴, C₄².₃₈₅C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₂₆C₂⁴, D₄○SD₁₆, C₄².₃₈₅C₂³

Smallest permutation representation of C₄².₃₈₅C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

C₄².₃₈₅C₂³

C₈○₂M₄(2)

C₄×SD₁₆

Q₁₆⋊C₄

D₈⋊C₄

C₈⋊₈D₄

C₈⋊₂D₄

C₈.D₄

C₄².₂₈C₂²

C₈.₁₂D₄

C₈⋊₃Q₈

C₂².₃₆C₂⁴

C₂²⋊C₄

C₄⋊C₄

C₈

C₂

# reps

Matrix representation of C₄².₃₈₅C₂³ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	13	0	0	0	0
0	0	0	0	1	0
0	0	4	13	1	15
0	0	16	0	0	0
0	0	0	0	0	4

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	4	13	1	15
0	0	4	0	1	16

1	0	0	0	0	0
0	16	0	0	0	0
0	0	0	16	0	4
0	0	0	0	13	4
0	0	13	4	16	1
0	0	13	0	16	1

0	4	0	0	0	0
13	0	0	0	0	0
0	0	11	3	12	10
0	0	0	14	5	0
0	0	0	12	3	0
0	0	5	7	14	6

0	1	0	0	0	0
16	0	0	0	0	0
0	0	12	5	0	0
0	0	12	12	0	0
0	0	3	14	0	7
0	0	3	0	5	7

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,16,0,0,0,0,13,0,0,0,0,1,1,0,0,0,0,0,15,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,4,4,0,0,1,0,13,0,0,0,0,0,1,1,0,0,0,0,15,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,13,0,0,16,0,4,0,0,0,0,13,16,16,0,0,4,4,1,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,11,0,0,5,0,0,3,14,12,7,0,0,12,5,3,14,0,0,10,0,0,6],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,3,3,0,0,5,12,14,0,0,0,0,0,0,5,0,0,0,0,7,7] >;

C₄².₃₈₅C₂³ in GAP, Magma, Sage, TeX

C_4^2._{385}C_2^3

% in TeX

G:=Group("C4^2.385C2^3");

// GroupNames label

G:=SmallGroup(128,1905);

// by ID

G=gap.SmallGroup(128,1905);

# by ID

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

// Polycyclic

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

// generators/relations

G = C42.385C23 order 128 = 27

246th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₈₅C₂³ order 128 = 2⁷

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