C4^2.360C2^3 - GroupNames

G = C₄².₃₆₀C₂³ order 128 = 2⁷

221^st non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₆₀C₂³, C₄⋊C₄.₃₅₃D₄, Q₈⋊D₄⋊₆C₂, C₄⋊SD₁₆⋊₈C₂, Q₈.Q₈⋊₂₁C₂, (C₄×SD₁₆)⋊₃₂C₂, C₄⋊C₈.₅₉C₂², C₄⋊C₄.₇₉C₂³, (C₂×C₈).₅₃C₂³, (C₂×C₄).₃₂₄C₂⁴, (C₄×C₈).₂₆₃C₂², Q₈.₁₄(C₄○D₄), C₂²⋊C₄.₁₅₄D₄, (C₄×D₄).₈₆C₂², (C₂×D₄).₉₅C₂³, C₂³.₂₆₃(C₂×D₄), SD₁₆⋊C₄⋊₁₈C₂, (C₄×Q₈).₈₂C₂², C₈⋊C₄.₁₆C₂², C₂.D₈.₉₃C₂², C₂.₃₂(D₄○SD₁₆), C₄⋊D₄.₃₂C₂², C₄⋊₁D₄.₆₁C₂², C₂²⋊C₈.₃₇C₂², (C₂×Q₈).₃₈₃C₂³, C₄.Q₈.₁₅₆C₂², D₄⋊C₄.₃₉C₂², C₂³.₁₉D₄⋊₂₁C₂, C₄².₇C₂²⋊₉C₂, (C₂²×C₄).₂₉₇C₂³, Q₈⋊C₄.₃₉C₂², C₂².₅₈₄(C₂²×D₄), C₄².C₂.₁₄C₂², (C₂×SD₁₆).₁₄₇C₂², (C₂²×Q₈).₂₉₅C₂², C₂³.₃₂C₂³⋊₁₀C₂, C₄²⋊C₂.₁₃₅C₂², C₂².₃₄C₂⁴.₁C₂, C₂.₁₂₅(C₂².₁₉C₂⁴), C₄.₂₀₉(C₂×C₄○D₄), (C₂×C₄).₅₀₈(C₂×D₄), SmallGroup(128,1858)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₆₀C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×Q₈ — C₂³.₃₂C₂³ — C₄².₃₆₀C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₆₀C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₆₀C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₆₀C₂³

Generators and relations for C₄².₃₆₀C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=d²=1, e²=b², ab=ba, ac=ca, dad=ab², ae=ea, cbc=ebe^-1=b^-1, bd=db, dcd=a²b²c, ece^-1=bc, de=ed >

Subgroups: 364 in 191 conjugacy classes, 88 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄×Q₈, C₄⋊D₄, C₄⋊D₄, C₂².D₄, C₄².C₂, C₄⋊₁D₄, C₂×SD₁₆, C₂²×Q₈, C₄².₇C₂², C₄×SD₁₆, SD₁₆⋊C₄, Q₈⋊D₄, C₄⋊SD₁₆, Q₈.Q₈, C₂³.₁₉D₄, C₂³.₃₂C₂³, C₂².₃₄C₂⁴, C₄².₃₆₀C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○SD₁₆, C₄².₃₆₀C₂³

Smallest permutation representation of C₄².₃₆₀C₂³
►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)(2 45)(3 46)(4 47)(5 35)(6 36)(7 33)(8 34)(9 32)(10 29)(11 30)(12 31)(13 41)(14 42)(15 43)(16 44)(17 54)(18 55)(19 56)(20 53)(21 50)(22 51)(23 52)(24 49)(25 59)(26 60)(27 57)(28 58)(37 63)(38 64)(39 61)(40 62)

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

kernel

C₄².₃₆₀C₂³

C₄².₇C₂²

C₄×SD₁₆

SD₁₆⋊C₄

Q₈⋊D₄

C₄⋊SD₁₆

Q₈.Q₈

C₂³.₁₉D₄

C₂³.₃₂C₂³

C₂².₃₄C₂⁴

C₂²⋊C₄

C₄⋊C₄

Q₈

C₂

# reps

Matrix representation of C₄².₃₆₀C₂³ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	13	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

1	0	0	0	0	0
0	1	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

6	16	0	0	0	0
1	11	0	0	0	0
0	0	0	13	6	0
0	0	13	0	0	11
0	0	6	0	0	13
0	0	0	11	13	0

7	13	0	0	0	0
12	10	0	0	0	0
0	0	11	0	0	4
0	0	0	11	13	0
0	0	0	13	6	0
0	0	4	0	0	6

1	0	0	0	0	0
0	1	0	0	0	0
0	0	13	13	3	14
0	0	13	4	14	14
0	0	3	14	13	13
0	0	14	14	13	4

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

C₄².₃₆₀C₂³ in GAP, Magma, Sage, TeX

C_4^2._{360}C_2^3

% in TeX

G:=Group("C4^2.360C2^3");

// GroupNames label

G:=SmallGroup(128,1858);

// by ID

G=gap.SmallGroup(128,1858);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,1018,304,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.360C23 order 128 = 27

221st non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₆₀C₂³ order 128 = 2⁷

221^st non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²