C4^2.353C2^3 - GroupNames

G = C₄².₃₅₃C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₅₃C₂³, (C₄×D₈)⋊₂₃C₂, C₄⋊D₈⋊₂₃C₂, C₄⋊C₄.₃₄₆D₄, C₄⋊SD₁₆⋊₇C₂, (C₄×SD₁₆)⋊₆C₂, Q₈.Q₈⋊₁₉C₂, D₄.Q₈⋊₁₉C₂, D₄⋊D₄⋊₂₀C₂, D₈⋊C₄⋊₁₀C₂, C₂.₁₈(D₄○D₈), C₄⋊C₄.₇₂C₂³, C₄⋊C₈.₅₄C₂², (C₂×C₈).₄₆C₂³, Q₈.₉(C₄○D₄), D₄.₁₀(C₄○D₄), (C₄×C₈).₁₁₀C₂², (C₂×C₄).₃₁₇C₂⁴, C₂²⋊C₄.₁₄₇D₄, (C₄×D₄).₈₀C₂², C₂³.₂₅₆(C₂×D₄), SD₁₆⋊C₄⋊₁₃C₂, (C₄×Q₈).₇₇C₂², C₈⋊C₄.₁₁C₂², C₄.Q₈.₂₀C₂², C₂.₂₇(D₄○SD₁₆), (C₂×D₈).₁₂₆C₂², (C₂×D₄).₄₀₈C₂³, C₄⋊D₄.₂₇C₂², C₂³.₄₆D₄⋊₆C₂, C₄⋊₁D₄.₆₀C₂², C₂².D₈⋊₁₇C₂, C₂²⋊C₈.₃₀C₂², (C₂×Q₈).₃₈₀C₂³, C₂.D₈.₁₇₄C₂², D₄⋊C₄.₃₅C₂², (C₂²×C₄).₂₉₀C₂³, C₄².₇C₂²⋊₂C₂, (C₂×SD₁₆).₁₈C₂², C₂².₅₇₇(C₂²×D₄), C₄².C₂.₁₂C₂², C₂².₃₄C₂⁴⋊₁C₂, Q₈⋊C₄.₁₅₅C₂², C₂³.₃₃C₂³⋊₁₂C₂, C₄²⋊C₂.₁₂₈C₂², C₂.₁₁₈(C₂².₁₉C₂⁴), C₄.₂₀₂(C₂×C₄○D₄), (C₂×C₄).₅₀₁(C₂×D₄), (C₂×C₄⋊C₄).₆₁₄C₂², (C₂×C₄○D₄).₁₄₃C₂², SmallGroup(128,1851)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₅₃C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×D₄ — C₂³.₃₃C₂³ — C₄².₃₅₃C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₅₃C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₅₃C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₅₃C₂³

Generators and relations for C₄².₃₅₃C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=e²=1, d²=b², ab=ba, ac=ca, dad^-1=ab², ae=ea, cbc=ebe=b^-1, bd=db, dcd^-1=a²c, ece=bc, de=ed >

Subgroups: 404 in 198 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, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₄⋊D₄, C₂².D₄, C₄².C₂, C₄⋊₁D₄, C₂×D₈, C₂×SD₁₆, C₂×C₄○D₄, C₄².₇C₂², C₄×D₈, C₄×SD₁₆, SD₁₆⋊C₄, D₈⋊C₄, D₄⋊D₄, C₄⋊D₈, C₄⋊SD₁₆, D₄.Q₈, Q₈.Q₈, C₂².D₈, C₂³.₄₆D₄, C₂³.₃₃C₂³, C₂².₃₄C₂⁴, C₄².₃₅₃C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○D₈, D₄○SD₁₆, C₄².₃₅₃C₂³

Smallest permutation representation of C₄².₃₅₃C₂³
►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 23 27 19)(2 24 28 20)(3 21 25 17)(4 22 26 18)(5 62 14 12)(6 63 15 9)(7 64 16 10)(8 61 13 11)(29 38 36 41)(30 39 33 42)(31 40 34 43)(32 37 35 44)(45 59 49 55)(46 60 50 56)(47 57 51 53)(48 58 52 54)

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

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

(1 36)(2 33)(3 34)(4 35)(5 60)(6 57)(7 58)(8 59)(9 51)(10 52)(11 49)(12 50)(13 55)(14 56)(15 53)(16 54)(17 43)(18 44)(19 41)(20 42)(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), (1,23,27,19)(2,24,28,20)(3,21,25,17)(4,22,26,18)(5,62,14,12)(6,63,15,9)(7,64,16,10)(8,61,13,11)(29,38,36,41)(30,39,33,42)(31,40,34,43)(32,37,35,44)(45,59,49,55)(46,60,50,56)(47,57,51,53)(48,58,52,54), (5,10)(6,11)(7,12)(8,9)(13,63)(14,64)(15,61)(16,62)(17,21)(18,22)(19,23)(20,24)(29,38)(30,39)(31,40)(32,37)(33,42)(34,43)(35,44)(36,41)(45,47)(46,48)(49,51)(50,52)(53,59)(54,60)(55,57)(56,58), (1,52,27,48)(2,45,28,49)(3,50,25,46)(4,47,26,51)(5,40,14,43)(6,44,15,37)(7,38,16,41)(8,42,13,39)(9,35,63,32)(10,29,64,36)(11,33,61,30)(12,31,62,34)(17,60,21,56)(18,53,22,57)(19,58,23,54)(20,55,24,59), (1,36)(2,33)(3,34)(4,35)(5,60)(6,57)(7,58)(8,59)(9,51)(10,52)(11,49)(12,50)(13,55)(14,56)(15,53)(16,54)(17,43)(18,44)(19,41)(20,42)(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), (1,23,27,19)(2,24,28,20)(3,21,25,17)(4,22,26,18)(5,62,14,12)(6,63,15,9)(7,64,16,10)(8,61,13,11)(29,38,36,41)(30,39,33,42)(31,40,34,43)(32,37,35,44)(45,59,49,55)(46,60,50,56)(47,57,51,53)(48,58,52,54), (5,10)(6,11)(7,12)(8,9)(13,63)(14,64)(15,61)(16,62)(17,21)(18,22)(19,23)(20,24)(29,38)(30,39)(31,40)(32,37)(33,42)(34,43)(35,44)(36,41)(45,47)(46,48)(49,51)(50,52)(53,59)(54,60)(55,57)(56,58), (1,52,27,48)(2,45,28,49)(3,50,25,46)(4,47,26,51)(5,40,14,43)(6,44,15,37)(7,38,16,41)(8,42,13,39)(9,35,63,32)(10,29,64,36)(11,33,61,30)(12,31,62,34)(17,60,21,56)(18,53,22,57)(19,58,23,54)(20,55,24,59), (1,36)(2,33)(3,34)(4,35)(5,60)(6,57)(7,58)(8,59)(9,51)(10,52)(11,49)(12,50)(13,55)(14,56)(15,53)(16,54)(17,43)(18,44)(19,41)(20,42)(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)], [(1,23,27,19),(2,24,28,20),(3,21,25,17),(4,22,26,18),(5,62,14,12),(6,63,15,9),(7,64,16,10),(8,61,13,11),(29,38,36,41),(30,39,33,42),(31,40,34,43),(32,37,35,44),(45,59,49,55),(46,60,50,56),(47,57,51,53),(48,58,52,54)], [(5,10),(6,11),(7,12),(8,9),(13,63),(14,64),(15,61),(16,62),(17,21),(18,22),(19,23),(20,24),(29,38),(30,39),(31,40),(32,37),(33,42),(34,43),(35,44),(36,41),(45,47),(46,48),(49,51),(50,52),(53,59),(54,60),(55,57),(56,58)], [(1,52,27,48),(2,45,28,49),(3,50,25,46),(4,47,26,51),(5,40,14,43),(6,44,15,37),(7,38,16,41),(8,42,13,39),(9,35,63,32),(10,29,64,36),(11,33,61,30),(12,31,62,34),(17,60,21,56),(18,53,22,57),(19,58,23,54),(20,55,24,59)], [(1,36),(2,33),(3,34),(4,35),(5,60),(6,57),(7,58),(8,59),(9,51),(10,52),(11,49),(12,50),(13,55),(14,56),(15,53),(16,54),(17,43),(18,44),(19,41),(20,42),(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	2H	4A	···	4F	4G	···	4O	4P	4Q	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	2	2	2	2	4	···	4	4	···	4	4	4	8	8	8	8	8	8
size	1	1	1	1	4	4	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○D₈

D₄○SD₁₆

kernel

C₄².₃₅₃C₂³

C₄².₇C₂²

C₄×D₈

C₄×SD₁₆

SD₁₆⋊C₄

D₈⋊C₄

D₄⋊D₄

C₄⋊D₈

C₄⋊SD₁₆

D₄.Q₈

Q₈.Q₈

C₂².D₈

C₂³.₄₆D₄

C₂³.₃₃C₂³

C₂².₃₄C₂⁴

C₂²⋊C₄

C₄⋊C₄

D₄

Q₈

C₂

# reps

Matrix representation of C₄².₃₅₃C₂³ ►in GL₆(𝔽₁₇)

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

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	13	0	1
0	0	4	13	16	0

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

16	2	0	0	0	0
0	1	0	0	0	0
0	0	8	11	0	6
0	0	8	14	3	3
0	0	15	2	0	9
0	0	0	5	3	12

1	15	0	0	0	0
0	16	0	0	0	0
0	0	11	0	10	10
0	0	14	0	0	10
0	0	0	5	3	3
0	0	5	12	3	3

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,15,16,13,13,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,4,0,0,15,16,13,13,0,0,0,0,0,16,0,0,0,0,1,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,4,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,8,8,15,0,0,0,11,14,2,5,0,0,0,3,0,3,0,0,6,3,9,12],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,11,14,0,5,0,0,0,0,5,12,0,0,10,0,3,3,0,0,10,10,3,3] >;

C₄².₃₅₃C₂³ in GAP, Magma, Sage, TeX

C_4^2._{353}C_2^3

% in TeX

G:=Group("C4^2.353C2^3");

// GroupNames label

G:=SmallGroup(128,1851);

// by ID

G=gap.SmallGroup(128,1851);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,1018,80,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.353C23 order 128 = 27

214th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₅₃C₂³ order 128 = 2⁷

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