C4^2.287C2^3 - GroupNames

G = C₄².₂₈₇C₂³ order 128 = 2⁷

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

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₄².₂₈₇C₂³, (C₂×C₈)⋊₉Q₈, C₄⋊Q₈.₃₀C₄, C₄.₂₂(C₄×Q₈), C₈.₂₈(C₂×Q₈), C₈⋊₄Q₈⋊₃₂C₂, C₂².₇(C₄×Q₈), C₂²⋊Q₈.₁₉C₄, C₄.₆₅(C₂²×Q₈), C₄⋊C₈.₂₃₃C₂², (C₄×C₈).₃₃₀C₂², (C₂×C₈).₄₂₄C₂³, C₄².₂₁₂(C₂×C₄), (C₂×C₄).₆₅₈C₂⁴, C₄².C₂.₁₅C₄, (C₄×Q₈).₅₇C₂², C₈⋊C₄.₁₅₉C₂², C₈○₂M₄(2).₂₃C₂, C₂.₁₉(Q₈○M₄(2)), C₄⋊M₄(2).₃₇C₂, C₂³.₁₄₃(C₂²×C₄), (C₂×C₄²).₇₇₁C₂², C₂².₁₈₄(C₂³×C₄), (C₂²×C₈).₄₄₄C₂², (C₂²×C₄).₁₅₂₀C₂³, C₄²⋊C₂.₃₀₄C₂², (C₂×M₄(2)).₃₆₁C₂², C₄².₆C₂².₁₃C₂, C₂³.₃₇C₂³.₂₂C₂, C₂.₂₅(C₂×C₄×Q₈), C₄⋊C₄.₁₁₇(C₂×C₄), C₄.₃₀₉(C₂×C₄○D₄), (C₂×C₄).₂₄₃(C₂×Q₈), (C₂×C₈⋊C₄).₃₉C₂, C₂²⋊C₄.₃₉(C₂×C₄), (C₂×C₄).₇₅(C₂²×C₄), (C₂×Q₈).₁₁₈(C₂×C₄), (C₂×C₄).₆₉₆(C₄○D₄), (C₂²×C₄).₃₄₅(C₂×C₄), SmallGroup(128,1693)

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

Derived series

C₁ — C₂² — C₄².₂₈₇C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₂×C₈⋊C₄ — C₄².₂₈₇C₂³

Lower central

C₁ — C₂² — C₄².₂₈₇C₂³

Upper central

C₁ — C₂×C₄ — C₄².₂₈₇C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₈₇C₂³

Generators and relations for C₄².₂₈₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=b, d²=a²b, e²=a², ab=ba, cac^-1=a^-1b², dad^-1=ab², ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=a²c, ede^-1=b²d >

Subgroups: 220 in 174 conjugacy classes, 140 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×Q₈, C₄×C₈, C₈⋊C₄, C₄⋊C₈, C₂×C₄², C₄²⋊C₂, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×C₈⋊C₄, C₈○₂M₄(2), C₄⋊M₄(2), C₄².₆C₂², C₈⋊₄Q₈, C₂³.₃₇C₂³, C₄².₂₈₇C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, Q₈, C₂³, C₂²×C₄, C₂×Q₈, C₄○D₄, C₂⁴, C₄×Q₈, C₂³×C₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄×Q₈, Q₈○M₄(2), C₄².₂₈₇C₂³

Smallest permutation representation of C₄².₂₈₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	···	4R	8A	···	8H	8I	···	8T
order	1	2	2	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	1	1	2	2	1	1	1	1	2	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+				-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	Q₈	C₄○D₄	Q₈○M₄(2)
kernel	C₄².₂₈₇C₂³	C₂×C₈⋊C₄	C₈○₂M₄(2)	C₄⋊M₄(2)	C₄².₆C₂²	C₈⋊₄Q₈	C₂³.₃₇C₂³	C₂²⋊Q₈	C₄².C₂	C₄⋊Q₈	C₂×C₈	C₂×C₄	C₂
# reps	1	1	2	1	2	8	1	8	4	4	4	4	4

Matrix representation of C₄².₂₈₇C₂³ ►in GL₆(𝔽₁₇)

14	15	0	0	0	0
5	3	0	0	0	0
0	0	11	15	0	0
0	0	9	6	0	0
0	0	0	0	6	2
0	0	0	0	8	11

16	0	0	0	0	0
0	16	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	13	0
0	0	0	0	0	13

14	7	0	0	0	0
1	3	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	13	0	0	0
0	0	0	13	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	13	0	0	0
0	0	0	0	0	1
0	0	0	0	13	0

3	2	0	0	0	0
12	14	0	0	0	0
0	0	11	15	0	0
0	0	9	6	0	0
0	0	0	0	11	15
0	0	0	0	9	6

G:=sub<GL(6,GF(17))| [14,5,0,0,0,0,15,3,0,0,0,0,0,0,11,9,0,0,0,0,15,6,0,0,0,0,0,0,6,8,0,0,0,0,2,11],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[14,1,0,0,0,0,7,3,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0],[3,12,0,0,0,0,2,14,0,0,0,0,0,0,11,9,0,0,0,0,15,6,0,0,0,0,0,0,11,9,0,0,0,0,15,6] >;

C₄².₂₈₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{287}C_2^3

% in TeX

G:=Group("C4^2.287C2^3");

// GroupNames label

G:=SmallGroup(128,1693);

// by ID

G=gap.SmallGroup(128,1693);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,268,2019,124]);

// Polycyclic

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

// generators/relations

G = C42.287C23 order 128 = 27

148th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₈₇C₂³ order 128 = 2⁷

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