C4^2.358C2^3 - GroupNames

G = C₄².₃₅₈C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₅₈C₂³, (C₄×D₈)⋊₆C₂, (C₄×Q₁₆)⋊₅C₂, C₄⋊C₄.₃₅₁D₄, C₄.Q₁₆⋊₂₅C₂, D₄⋊Q₈⋊₂₅C₂, D₄⋊D₄.₂C₂, C₂.₂₀(D₄○D₈), C₄⋊C₈.₅₇C₂², C₄⋊C₄.₇₇C₂³, (C₄×C₈).₇₁C₂², (C₂×C₈).₅₁C₂³, C₂.₁₉(Q₈○D₈), D₄.₁₄(C₄○D₄), D₄.₇D₄⋊₂₀C₂, D₄.₂D₄⋊₂₁C₂, (C₂×C₄).₃₂₂C₂⁴, Q₈.₁₂(C₄○D₄), Q₈.D₄⋊₂₁C₂, C₂²⋊C₄.₁₅₂D₄, (C₄×D₄).₈₄C₂², C₂³.₂₆₁(C₂×D₄), C₄⋊Q₈.₁₀₇C₂², SD₁₆⋊C₄⋊₁₇C₂, (C₄×Q₈).₈₀C₂², C₈⋊C₄.₁₄C₂², C₂.D₈.₉₂C₂², (C₂×D₄).₄₁₂C₂³, (C₂×D₈).₁₂₈C₂², C₄⋊D₄.₃₀C₂², C₂².D₈⋊₁₈C₂, C₂²⋊C₈.₃₅C₂², (C₂×Q₈).₃₈₂C₂³, C₂²⋊Q₈.₃₀C₂², C₂³.₄₈D₄⋊₁₈C₂, (C₂²×C₄).₂₉₅C₂³, C₄².₇C₂²⋊₇C₂, Q₈⋊C₄.₃₈C₂², (C₂×Q₁₆).₁₂₃C₂², (C₂×SD₁₆).₁₉C₂², C₄.₄D₄.₂₉C₂², C₂².₅₈₂(C₂²×D₄), D₄⋊C₄.₁₆₄C₂², C₂².₃₆C₂⁴⋊₂C₂, C₂³.₃₃C₂³⋊₁₃C₂, C₄²⋊C₂.₁₃₃C₂², C₂.₁₂₃(C₂².₁₉C₂⁴), C₄.₂₀₇(C₂×C₄○D₄), (C₂×C₄).₅₀₆(C₂×D₄), (C₂×C₄⋊C₄).₆₁₆C₂², (C₂×C₄○D₄).₁₄₅C₂², SmallGroup(128,1856)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₅₈C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×D₄ — C₂³.₃₃C₂³ — C₄².₃₅₈C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₅₈C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₅₈C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₅₈C₂³

Generators and relations for C₄².₃₅₈C₂³
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=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, C₂.D₈, 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×D₈, C₄×Q₁₆, SD₁₆⋊C₄, D₄⋊D₄, D₄.₇D₄, D₄.₂D₄, Q₈.D₄, D₄⋊Q₈, C₄.Q₁₆, C₂².D₈, C₂³.₄₈D₄, C₂³.₃₃C₂³, C₂².₃₆C₂⁴, C₄².₃₅₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○D₈, Q₈○D₈, C₄².₃₅₈C₂³

Smallest permutation representation of C₄².₃₅₈C₂³
►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)

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

(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), (5,16)(6,13)(7,14)(8,15)(9,62)(10,63)(11,64)(12,61)(17,24)(18,21)(19,22)(20,23)(29,34)(30,35)(31,36)(32,33)(37,43)(38,44)(39,41)(40,42)(45,59)(46,60)(47,57)(48,58)(49,51)(50,52)(53,55)(54,56), (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), (5,16)(6,13)(7,14)(8,15)(9,62)(10,63)(11,64)(12,61)(17,24)(18,21)(19,22)(20,23)(29,34)(30,35)(31,36)(32,33)(37,43)(38,44)(39,41)(40,42)(45,59)(46,60)(47,57)(48,58)(49,51)(50,52)(53,55)(54,56), (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)], [(5,16),(6,13),(7,14),(8,15),(9,62),(10,63),(11,64),(12,61),(17,24),(18,21),(19,22),(20,23),(29,34),(30,35),(31,36),(32,33),(37,43),(38,44),(39,41),(40,42),(45,59),(46,60),(47,57),(48,58),(49,51),(50,52),(53,55),(54,56)], [(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○D₈

Q₈○D₈

kernel

C₄².₃₅₈C₂³

C₄².₇C₂²

C₄×D₈

C₄×Q₁₆

SD₁₆⋊C₄

D₄⋊D₄

D₄.₇D₄

D₄.₂D₄

Q₈.D₄

D₄⋊Q₈

C₄.Q₁₆

C₂².D₈

C₂³.₄₈D₄

C₂³.₃₃C₂³

C₂².₃₆C₂⁴

C₂²⋊C₄

C₄⋊C₄

D₄

Q₈

C₂

# reps

Matrix representation of C₄².₃₅₈C₂³ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	13	0	0	0	0
0	0	6	7	15	0
0	0	7	6	0	15
0	0	9	8	11	10
0	0	8	9	10	11

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	10	0	0	1
0	0	0	7	16	0

1	0	0	0	0	0
1	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	10	1	0
0	0	7	0	0	16

1	15	0	0	0	0
0	16	0	0	0	0
0	0	4	10	2	0
0	0	10	4	0	2
0	0	1	11	13	7
0	0	11	1	7	13

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

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,6,7,9,8,0,0,7,6,8,9,0,0,15,0,11,10,0,0,0,15,10,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,10,0,0,0,1,0,0,7,0,0,0,0,0,16,0,0,0,0,1,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,7,0,0,0,16,10,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,4,10,1,11,0,0,10,4,11,1,0,0,2,0,13,7,0,0,0,2,7,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,4,0,0,3,14,13,0,0,0,0,0,3,3,0,0,0,0,3,14] >;

C₄².₃₅₈C₂³ in GAP, Magma, Sage, TeX

C_4^2._{358}C_2^3

% in TeX

G:=Group("C4^2.358C2^3");

// GroupNames label

G:=SmallGroup(128,1856);

// by ID

G=gap.SmallGroup(128,1856);

# by ID

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

// 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*b^2*c,e*c*e=b*c,d*e=e*d>;

// generators/relations

G = C42.358C23 order 128 = 27

219th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₅₈C₂³ order 128 = 2⁷

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