C4^2.359C2^3 - GroupNames

G = C₄².₃₅₉C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₅₉C₂³, C₄⋊C₄.₃₅₂D₄, D₄⋊₂Q₈⋊₈C₂, Q₈⋊Q₈⋊₈C₂, D₈⋊C₄⋊₁₂C₂, (C₄×SD₁₆)⋊₃₁C₂, D₄⋊D₄.₃C₂, C₄⋊C₄.₇₈C₂³, C₄⋊C₈.₅₈C₂², (C₂×C₈).₅₂C₂³, Q₁₆⋊C₄⋊₁₁C₂, D₄.₁₅(C₄○D₄), D₄.₂D₄⋊₂₂C₂, D₄.₇D₄⋊₂₁C₂, (C₂×C₄).₃₂₃C₂⁴, (C₄×C₈).₂₆₂C₂², Q₈.₁₃(C₄○D₄), Q₈.D₄⋊₂₂C₂, C₂²⋊C₄.₁₅₃D₄, (C₄×D₄).₈₅C₂², (C₂×D₈).₆₁C₂², C₂³.₂₆₂(C₂×D₄), C₄⋊Q₈.₁₀₈C₂², (C₂×Q₈).₈₃C₂³, (C₄×Q₈).₈₁C₂², C₄.Q₈.₂₂C₂², C₈⋊C₄.₁₅C₂², C₂.₃₁(D₄○SD₁₆), (C₂×D₄).₄₁₃C₂³, C₂³.₄₇D₄⋊₇C₂, C₂³.₄₆D₄⋊₇C₂, C₄⋊D₄.₃₁C₂², C₂²⋊C₈.₃₆C₂², (C₂×Q₁₆).₆₂C₂², C₂²⋊Q₈.₃₁C₂², D₄⋊C₄.₃₈C₂², C₄².₇C₂²⋊₈C₂, (C₂²×C₄).₂₉₆C₂³, C₄.₄D₄.₃₀C₂², C₂².₅₈₃(C₂²×D₄), C₂².₃₆C₂⁴⋊₃C₂, Q₈⋊C₄.₁₇₆C₂², (C₂×SD₁₆).₁₄₆C₂², C₄²⋊C₂.₁₃₄C₂², C₂³.₃₃C₂³⋊₁₄C₂, C₂.₁₂₄(C₂².₁₉C₂⁴), C₄.₂₀₈(C₂×C₄○D₄), (C₂×C₄).₅₀₇(C₂×D₄), (C₂×C₄⋊C₄).₆₁₇C₂², (C₂×C₄○D₄).₁₄₆C₂², SmallGroup(128,1857)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₅₉C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×D₄ — C₂³.₃₃C₂³ — C₄².₃₅₉C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₅₉C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₅₉C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₅₉C₂³

Generators and relations for C₄².₃₅₉C₂³
G = < a,b,c,d,e | a⁴=b⁴=e²=1, c²=d²=b², ab=ba, ac=ca, dad^-1=ab², ae=ea, cbc^-1=ebe=b^-1, bd=db, dcd^-1=a²b²c, ece=bc, de=ed >

Subgroups: 364 in 192 conjugacy classes, 88 normal (84 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄⋊Q₈, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₂×C₄○D₄, C₄².₇C₂², C₄×SD₁₆, Q₁₆⋊C₄, D₈⋊C₄, D₄⋊D₄, D₄.₇D₄, D₄.₂D₄, Q₈.D₄, Q₈⋊Q₈, D₄⋊₂Q₈, C₂³.₄₆D₄, C₂³.₄₇D₄, C₂³.₃₃C₂³, C₂².₃₆C₂⁴, C₄².₃₅₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○SD₁₆, C₄².₃₅₉C₂³

Smallest permutation representation of C₄².₃₅₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4F	4G	···	4O	4P	4Q	4R	8A	8B	8C	8D	8E	8F
order	1	2	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	4	4	8	2	···	2	4	···	4	8	8	8	4	4	4	4	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

kernel

C₄².₃₅₉C₂³

C₄².₇C₂²

C₄×SD₁₆

Q₁₆⋊C₄

D₈⋊C₄

D₄⋊D₄

D₄.₇D₄

D₄.₂D₄

Q₈.D₄

Q₈⋊Q₈

D₄⋊₂Q₈

C₂³.₄₆D₄

C₂³.₄₇D₄

C₂³.₃₃C₂³

C₂².₃₆C₂⁴

C₂²⋊C₄

C₄⋊C₄

D₄

Q₈

C₂

# reps

Matrix representation of C₄².₃₅₉C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	16	2	0	0
0	0	16	1	0	0
0	0	0	0	16	2
0	0	0	0	16	1

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	15	0	0
0	0	1	16	0	0
0	0	0	0	16	2
0	0	0	0	16	1

0	13	0	0	0	0
4	0	0	0	0	0
0	0	0	0	10	7
0	0	0	0	5	0
0	0	0	10	0	0
0	0	12	10	0	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	0	7
0	0	0	0	12	0
0	0	0	7	0	0
0	0	12	0	0	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,10,5,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,0,12,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₄².₃₅₉C₂³ in GAP, Magma, Sage, TeX

C_4^2._{359}C_2^3

% in TeX

G:=Group("C4^2.359C2^3");

// GroupNames label

G:=SmallGroup(128,1857);

// by ID

G=gap.SmallGroup(128,1857);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,1018,304,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.359C23 order 128 = 27

220th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₅₉C₂³ order 128 = 2⁷

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