C4^2.250C2^3 - GroupNames

G = C₄².₂₅₀C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₂₅₀C₂³, C₄⋊C₄.₇₂D₄, C₈⋊₁C₈⋊₂₄C₂, (C₂×D₄).₆₄D₄, C₈⋊₆D₄.₈C₂, (C₂×C₈).₁₈₈D₄, D₄⋊Q₈.₈C₂, C₄⋊Q₈.₇₁C₂², C₄.₁₀₅(C₄○D₈), C₄.₁₀D₈⋊₁₆C₂, C₂.₁₄(C₈⋊D₄), C₄⋊C₈.₁₉₀C₂², C₄.₄₅(C₈⋊C₂²), (C₄×C₈).₂₁₆C₂², C₄.SD₁₆⋊₁₉C₂, (C₄×D₄).₅₀C₂², D₄.D₄.₁₁C₂, C₄.₇₆(C₈.C₂²), C₂.₁₃(D₄.₂D₄), C₂.₁₆(D₄.₅D₄), C₂².₂₁₁(C₄⋊D₄), (C₂×C₄).₃₅(C₄○D₄), (C₂×C₄).₁₂₈₅(C₂×D₄), SmallGroup(128,431)

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

Derived series

C₁ — C₄² — C₄².₂₅₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈⋊₆D₄ — C₄².₂₅₀C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₂₅₀C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₂₅₀C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₂₅₀C₂³

Generators and relations for C₄².₂₅₀C₂³
G = < a,b,c,d,e | a⁴=b⁴=e²=1, c²=a², d²=ab², ab=ba, cac^-1=a^-1, ad=da, ae=ea, cbc^-1=ebe=b^-1, bd=db, dcd^-1=a^-1c, ece=bc, ede=a²d >

Subgroups: 184 in 80 conjugacy classes, 32 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₂.D₈, C₄×D₄, C₄⋊Q₈, C₂×M₄(2), C₂×SD₁₆, C₄.₁₀D₈, C₈⋊₁C₈, C₈⋊₆D₄, D₄.D₄, D₄⋊Q₈, C₄.SD₁₆, C₄².₂₅₀C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₄⋊D₄, C₄○D₈, C₈⋊C₂², C₈.C₂², D₄.₂D₄, C₈⋊D₄, D₄.₅D₄, C₄².₂₅₀C₂³

Character table of C₄².₂₅₀C₂³

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	8	2	2	2	2	4	8	16	16	4	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	-1	1	1	1	1	1	-1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	-1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	linear of order 2
ρ₅	1	1	1	1	-1	1	1	1	1	1	-1	-1	-1	1	1	1	1	1	1	1	-1	-1	1	linear of order 2
ρ₆	1	1	1	1	-1	1	1	1	1	1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	-1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	1	1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	-1	-1	1	linear of order 2
ρ₉	2	2	2	2	0	2	-2	2	-2	-2	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	-2	2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	2	-2	2	-2	-2	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	2i	-2i	0	complex lifted from C₄○D₄
ρ₁₄	2	2	2	2	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	-2i	2i	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	2i	-2i	0	√-2	-√2	-√-2	0	0	√2	complex lifted from C₄○D₈
ρ₁₆	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	-2i	2i	0	-√-2	-√2	√-2	0	0	√2	complex lifted from C₄○D₈
ρ₁₇	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	2i	-2i	0	-√-2	√2	√-2	0	0	-√2	complex lifted from C₄○D₈
ρ₁₈	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	-2i	2i	0	√-2	√2	-√-2	0	0	-√2	complex lifted from C₄○D₈
ρ₁₉	4	-4	4	-4	0	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₀	4	-4	-4	4	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₁	4	-4	4	-4	0	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₂	4	4	-4	-4	0	0	0	0	0	0	0	0	0	2√2	0	0	-2√2	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	0	0	0	-2√2	0	0	2√2	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2

Smallest permutation representation of C₄².₂₅₀C₂³
►On 64 points

Generators in S₆₄

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

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

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

(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)

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

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

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

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

Matrix representation of C₄².₂₅₀C₂³ ►in GL₈(𝔽₁₇)

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

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

3	14	0	0	0	0	0	0
14	14	0	0	0	0	0	0
5	13	14	3	0	0	0	0
13	14	3	3	0	0	0	0
0	0	0	0	0	0	14	5
0	0	0	0	0	0	12	3
0	0	0	0	3	12	0	0
0	0	0	0	5	14	0	0

7	11	15	0	0	0	0	0
11	7	0	15	0	0	0	0
8	9	10	6	0	0	0	0
9	8	6	10	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	13	15	0	0
0	0	0	0	0	4	0	0

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

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4],[0,16,6,0,0,0,0,0,1,0,0,11,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[3,14,5,13,0,0,0,0,14,14,13,14,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,3,5,0,0,0,0,0,0,12,14,0,0,0,0,14,12,0,0,0,0,0,0,5,3,0,0],[7,11,8,9,0,0,0,0,11,7,9,8,0,0,0,0,15,0,10,6,0,0,0,0,0,15,6,10,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,11,0,0,0,0,0,16,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;

C₄².₂₅₀C₂³ in GAP, Magma, Sage, TeX

C_4^2._{250}C_2^3

% in TeX

G:=Group("C4^2.250C2^3");

// GroupNames label

G:=SmallGroup(128,431);

// by ID

G=gap.SmallGroup(128,431);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,387,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄².₂₅₀C₂³ in TeX

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

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

3	14	0	0	0	0	0	0
14	14	0	0	0	0	0	0
5	13	14	3	0	0	0	0
13	14	3	3	0	0	0	0
0	0	0	0	0	0	14	5
0	0	0	0	0	0	12	3
0	0	0	0	3	12	0	0
0	0	0	0	5	14	0	0

7	11	15	0	0	0	0	0
11	7	0	15	0	0	0	0
8	9	10	6	0	0	0	0
9	8	6	10	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	13	15	0	0
0	0	0	0	0	4	0	0

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

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

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

3	14	0	0	0	0	0	0
14	14	0	0	0	0	0	0
5	13	14	3	0	0	0	0
13	14	3	3	0	0	0	0
0	0	0	0	0	0	14	5
0	0	0	0	0	0	12	3
0	0	0	0	3	12	0	0
0	0	0	0	5	14	0	0

7	11	15	0	0	0	0	0
11	7	0	15	0	0	0	0
8	9	10	6	0	0	0	0
9	8	6	10	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	13	15	0	0
0	0	0	0	0	4	0	0

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

G = C42.250C23 order 128 = 27

111st non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₅₀C₂³ order 128 = 2⁷

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

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

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

3	14	0	0	0	0	0	0
14	14	0	0	0	0	0	0
5	13	14	3	0	0	0	0
13	14	3	3	0	0	0	0
0	0	0	0	0	0	14	5
0	0	0	0	0	0	12	3
0	0	0	0	3	12	0	0
0	0	0	0	5	14	0	0

7	11	15	0	0	0	0	0
11	7	0	15	0	0	0	0
8	9	10	6	0	0	0	0
9	8	6	10	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	13	15	0	0
0	0	0	0	0	4	0	0

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