C4^2.252C2^3 - GroupNames

G = C₄².₂₅₂C₂³ order 128 = 2⁷

113^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₂₅₂C₂³, C₄⋊C₄.₇₄D₄, C₈⋊₂C₈⋊₁₉C₂, C₈⋊₆D₄⋊₂₇C₂, (C₂×D₄).₆₅D₄, C₄⋊D₈.₉C₂, (C₂×C₈).₁₉₀D₄, C₄.₄D₈⋊₃₃C₂, C₄.D₈⋊₁₉C₂, C₄⋊Q₈.₇₃C₂², D₄.D₄⋊₃₆C₂, C₄.₁₀₇(C₄○D₈), C₄.₁₀D₈⋊₁₇C₂, C₂.₁₀(C₈⋊₂D₄), C₄⋊C₈.₁₉₁C₂², C₄.₇₃(C₈⋊C₂²), (C₄×C₈).₂₈₅C₂², (C₄×D₄).₅₁C₂², C₄⋊₁D₄.₃₈C₂², C₂.₁₄(D₄.₂D₄), C₂.₂₀(D₄.₃D₄), C₂².₂₁₃(C₄⋊D₄), (C₂×C₄).₃₇(C₄○D₄), (C₂×C₄).₁₂₈₇(C₂×D₄), SmallGroup(128,433)

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

Derived series

C₁ — C₄² — C₄².₂₅₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈⋊₆D₄ — C₄².₂₅₂C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₂₅₂C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₂₅₂C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₂₅₂C₂³

Generators and relations for C₄².₂₅₂C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=e²=1, d²=a^-1b², ab=ba, cac=a^-1, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, dcd^-1=a^-1c, ece=bc, ede=a²d >

Subgroups: 240 in 88 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, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄×D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₄.D₈, C₄.₁₀D₈, C₈⋊₂C₈, C₈⋊₆D₄, C₄⋊D₈, D₄.D₄, C₄.₄D₈, C₄².₂₅₂C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₄⋊D₄, C₄○D₈, C₈⋊C₂², D₄.₂D₄, C₈⋊₂D₄, D₄.₃D₄, C₄².₂₅₂C₂³

Character table of C₄².₂₅₂C₂³

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	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
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	complex lifted from D₄.₃D₄
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	complex lifted from D₄.₃D₄

Smallest permutation representation of C₄².₂₅₂C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

Matrix representation of C₄².₂₅₂C₂³ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	15	0
0	0	1	0	0	15
0	0	1	0	0	16
0	0	0	1	16	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	16	0	0	1
0	0	0	1	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	16	0
0	0	0	0	0	1

13	0	0	0	0	0
0	13	0	0	0	0
0	0	5	5	0	7
0	0	12	12	10	0
0	0	12	0	5	5
0	0	0	5	12	12

3	3	0	0	0	0
3	14	0	0	0	0
0	0	10	6	4	13
0	0	10	11	13	13
0	0	0	11	6	6
0	0	10	0	10	7

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

C₄².₂₅₂C₂³ in GAP, Magma, Sage, TeX

C_4^2._{252}C_2^3

% in TeX

G:=Group("C4^2.252C2^3");

// GroupNames label

G:=SmallGroup(128,433);

// by ID

G=gap.SmallGroup(128,433);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,387,1123,136,2804,718,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^-1*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 C₄².₂₅₂C₂³ in TeX

G = C42.252C23 order 128 = 27

113rd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₅₂C₂³ order 128 = 2⁷

113^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²