C4^2.361C2^3 - GroupNames

G = C₄².₃₆₁C₂³ order 128 = 2⁷

222^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₆₁C₂³, (C₄×Q₁₆)⋊₆C₂, C₄⋊C₄.₃₅₄D₄, Q₈.Q₈⋊₂₂C₂, C₄⋊₂Q₁₆⋊₂₅C₂, C₄⋊C₈.₆₀C₂², C₄⋊C₄.₈₀C₂³, (C₄×C₈).₇₂C₂², (C₂×C₈).₅₄C₂³, C₂.₂₀(Q₈○D₈), Q₁₆⋊C₄⋊₁₂C₂, (C₂×C₄).₃₂₅C₂⁴, Q₈.₁₅(C₄○D₄), C₂²⋊C₄.₁₅₅D₄, C₂³.₂₆₄(C₂×D₄), C₄⋊Q₈.₁₀₉C₂², C₂²⋊Q₁₆.₄C₂, (C₄×Q₈).₈₃C₂², C₈⋊C₄.₁₇C₂², C₄.Q₈.₂₃C₂², C₂²⋊C₈.₃₈C₂², (C₂×Q₈).₃₈₄C₂³, C₂.D₈.₁₇₆C₂², C₂²⋊Q₈.₃₂C₂², (C₂²×C₄).₂₉₈C₂³, Q₈⋊C₄.₄₀C₂², (C₂×Q₁₆).₁₂₄C₂², C₂³.₂₀D₄.₂C₂, C₂².₅₈₅(C₂²×D₄), C₄².C₂.₁₅C₂², C₄².₇C₂².₁C₂, (C₂²×Q₈).₂₉₆C₂², C₄²⋊C₂.₁₃₆C₂², C₂³.₃₂C₂³.₆C₂, C₂².₃₅C₂⁴.₁C₂, C₂.₁₂₆(C₂².₁₉C₂⁴), C₄.₂₁₀(C₂×C₄○D₄), (C₂×C₄).₅₀₉(C₂×D₄), SmallGroup(128,1859)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₆₁C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×Q₈ — C₂³.₃₂C₂³ — C₄².₃₆₁C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₆₁C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₆₁C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₆₁C₂³

Generators and relations for C₄².₃₆₁C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=1, c²=e²=b², ab=ba, ac=ca, dad=ab², ae=ea, cbc^-1=ebe^-1=b^-1, bd=db, dcd=a²b²c, ece^-1=bc, de=ed >

Subgroups: 284 in 179 conjugacy classes, 88 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄²⋊C₂, C₄×Q₈, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂×Q₁₆, C₂²×Q₈, C₄².₇C₂², C₄×Q₁₆, Q₁₆⋊C₄, C₂²⋊Q₁₆, C₄⋊₂Q₁₆, Q₈.Q₈, C₂³.₂₀D₄, C₂³.₃₂C₂³, C₂².₃₅C₂⁴, C₄².₃₆₁C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, Q₈○D₈, C₄².₃₆₁C₂³

Smallest permutation representation of C₄².₃₆₁C₂³
►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 12 62 15)(6 9 63 16)(7 10 64 13)(8 11 61 14)(29 33 41 40)(30 34 42 37)(31 35 43 38)(32 36 44 39)(45 49 57 56)(46 50 58 53)(47 51 59 54)(48 52 60 55)

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

(2 27)(4 25)(5 64)(6 8)(7 62)(9 11)(10 15)(12 13)(14 16)(17 24)(19 22)(29 41)(31 43)(33 40)(35 38)(45 47)(46 60)(48 58)(49 51)(50 55)(52 53)(54 56)(57 59)(61 63)

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

Q₈○D₈

kernel

C₄².₃₆₁C₂³

C₄².₇C₂²

C₄×Q₁₆

Q₁₆⋊C₄

C₂²⋊Q₁₆

C₄⋊₂Q₁₆

Q₈.Q₈

C₂³.₂₀D₄

C₂³.₃₂C₂³

C₂².₃₅C₂⁴

C₂²⋊C₄

C₄⋊C₄

Q₈

C₂

# reps

Matrix representation of C₄².₃₆₁C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	16	0	0	0
0	0	0	16	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	4	0	0	0	0
4	11	0	0	0	0
0	0	0	4	0	0
0	0	4	0	0	0
0	0	0	0	0	4
0	0	0	0	4	0

1	0	0	0	0	0
14	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	0	0	0	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	5	5	0	0
0	0	5	12	0	0
0	0	0	0	5	5
0	0	0	0	5	12

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

C₄².₃₆₁C₂³ in GAP, Magma, Sage, TeX

C_4^2._{361}C_2^3

% in TeX

G:=Group("C4^2.361C2^3");

// GroupNames label

G:=SmallGroup(128,1859);

// by ID

G=gap.SmallGroup(128,1859);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,1018,304,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=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^-1=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 = C42.361C23 order 128 = 27

222nd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₆₁C₂³ order 128 = 2⁷

222^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²