C4^2.248C2^3 - GroupNames

G = C₄².₂₄₈C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₂₄₈C₂³, C₄⋊C₄.₇₀D₄, C₈⋊₁C₈⋊₂₂C₂, C₈⋊₆D₄⋊₂₆C₂, (C₂×D₄).₆₃D₄, C₄⋊D₈.₈C₂, (C₂×C₈).₁₈₆D₄, D₄⋊₂Q₈⋊₃₄C₂, C₄.₄D₈⋊₂₀C₂, C₄.D₈⋊₁₈C₂, C₄.₆Q₁₆⋊₉C₂, C₄⋊Q₈.₆₉C₂², C₄.₁₀₃(C₄○D₈), C₂.₁₂(C₈⋊D₄), C₄⋊C₈.₁₈₉C₂², C₄.₉₃(C₈⋊C₂²), (C₄×C₈).₂₁₄C₂², (C₄×D₄).₄₉C₂², C₄⋊₁D₄.₃₆C₂², C₄.₇₄(C₈.C₂²), C₂.₁₆(D₄.₄D₄), C₂.₁₂(D₄.₂D₄), C₂².₂₀₉(C₄⋊D₄), (C₂×C₄).₃₃(C₄○D₄), (C₂×C₄).₁₂₈₃(C₂×D₄), SmallGroup(128,429)

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

Derived series

C₁ — C₄² — C₄².₂₄₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈⋊₆D₄ — C₄².₂₄₈C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₂₄₈C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₂₄₈C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₂₄₈C₂³

Generators and relations for C₄².₂₄₈C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=e²=1, d²=ab², ab=ba, cac=a^-1, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, dcd^-1=a^-1c, ece=bc, ede=a²d >

Subgroups: 232 in 85 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), D₈, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₂²⋊C₈, D₄⋊C₄, C₄⋊C₈, C₄.Q₈, C₄×D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂×M₄(2), C₂×D₈, C₄.D₈, C₄.₆Q₁₆, C₈⋊₁C₈, C₈⋊₆D₄, C₄⋊D₈, D₄⋊₂Q₈, C₄.₄D₈, C₄².₂₄₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₄⋊D₄, C₄○D₈, C₈⋊C₂², C₈.C₂², D₄.₂D₄, C₈⋊D₄, D₄.₄D₄, C₄².₂₄₈C₂³

Character table of C₄².₂₄₈C₂³

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	8	16	2	2	2	2	4	8	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	0	2	-2	2	-2	-2	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	2	-2	2	-2	-2	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	0	-2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	0	-2	2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	-2i	2i	complex lifted from C₄○D₄
ρ₁₄	2	2	2	2	0	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	2i	-2i	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	2i	-2i	0	√-2	√2	-√-2	-√2	0	0	complex lifted from C₄○D₈
ρ₁₆	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	-2i	2i	0	-√-2	√2	√-2	-√2	0	0	complex lifted from C₄○D₈
ρ₁₇	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	2i	-2i	0	-√-2	-√2	√-2	√2	0	0	complex lifted from C₄○D₈
ρ₁₈	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	-2i	2i	0	√-2	-√2	-√-2	√2	0	0	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	0	0	4	0	-4	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	0	0	0	0	0	0	0	2√2	0	0	-2√2	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
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	orthogonal lifted from D₄.₄D₄
ρ₂₃	4	-4	4	-4	0	0	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of C₄².₂₄₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

Matrix representation of C₄².₂₄₈C₂³ ►in GL₆(𝔽₁₇)

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	14	3	0	16
0	0	3	3	1	0

0	1	0	0	0	0
16	0	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	0	0	0	16
0	0	0	0	1	0

1	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	0	1
0	0	0	0	1	0

4	0	0	0	0	0
0	4	0	0	0	0
0	0	14	14	0	0
0	0	3	14	0	0
0	0	9	9	3	3
0	0	9	8	14	3

14	14	0	0	0	0
14	3	0	0	0	0
0	0	14	14	15	0
0	0	14	3	0	15
0	0	0	0	3	3
0	0	0	0	3	14

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

C₄².₂₄₈C₂³ in GAP, Magma, Sage, TeX

C_4^2._{248}C_2^3

% in TeX

G:=Group("C4^2.248C2^3");

// GroupNames label

G:=SmallGroup(128,429);

// by ID

G=gap.SmallGroup(128,429);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a*b^2,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=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 C₄².₂₄₈C₂³ in TeX

G = C42.248C23 order 128 = 27

109th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₄₈C₂³ order 128 = 2⁷

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