C4^2.189C2^3 - GroupNames

G = C₄².₁₈₉C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₁₈₉C₂³, Q₈⋊C₈⋊₉C₂, D₄⋊C₈⋊₁₇C₂, C₄⋊C₄.₂₅D₄, (C₂×D₄).₄₆D₄, C₄.₄D₈⋊₃C₂, C₄⋊D₈.₃C₂, C₄⋊C₈.₄C₂², (C₂×Q₈).₄₄D₄, C₄⋊SD₁₆⋊₃₃C₂, C₄.₅₄(C₄○D₈), C₄.₁₀D₈⋊₉C₂, (C₄×C₈).₁₇C₂², C₄⋊Q₈.₁₁C₂², C₄.₆₀(C₈⋊C₂²), (C₄×D₄).₂₃C₂², (C₄×Q₈).₂₃C₂², C₂.₁₈(D₄⋊D₄), C₄⋊₁D₄.₁₃C₂², C₂².₁₅₅C₂²≀C₂, C₂.₁₅(D₄.₈D₄), C₂².₅₀C₂⁴⋊₁C₂, (C₂×C₄).₉₄₆(C₂×D₄), SmallGroup(128,360)

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

Derived series

C₁ — C₄² — C₄².₁₈₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₂².₅₀C₂⁴ — C₄².₁₈₉C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₁₈₉C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₁₈₉C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₁₈₉C₂³

Generators and relations for C₄².₁₈₉C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=1, d²=b², e²=a²b², ab=ba, cac=dad^-1=a^-1, eae^-1=ab², cbc=dbd^-1=ebe^-1=b^-1, dcd^-1=ac, ece^-1=bc, de=ed >

Subgroups: 272 in 108 conjugacy classes, 34 normal (32 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, D₄⋊C₄, C₄⋊C₈, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄×Q₈, C₂²⋊Q₈, C₄.₄D₄, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊Q₈, C₂×D₈, C₂×SD₁₆, D₄⋊C₈, Q₈⋊C₈, C₄.₁₀D₈, C₄⋊D₈, C₄⋊SD₁₆, C₄.₄D₈, C₂².₅₀C₂⁴, C₄².₁₈₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₄○D₈, C₈⋊C₂², D₄⋊D₄, D₄.₈D₄, C₄².₁₈₉C₂³

Character table of C₄².₁₈₉C₂³

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	8	16	2	2	2	2	4	4	4	4	4	8	8	8	4	4	4	4	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	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	-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	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	-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	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	-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	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	-1	-1	-1	linear of order 2
ρ₉	2	2	2	2	0	0	-2	-2	-2	-2	0	0	0	0	2	2	-2	0	0	0	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	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	2	2	-2	-2	-2	0	0	-2	-2	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	0	-2	-2	-2	-2	0	0	0	0	2	-2	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	0	-2	-2	2	2	0	2	2	0	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	0	-2	-2	2	2	0	-2	-2	0	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	0	0	2	-2	0	0	2i	0	0	-2i	0	0	0	0	-√-2	√-2	-√-2	√-2	0	0	√2	-√2	complex lifted from C₄○D₈
ρ₁₆	2	2	-2	-2	0	0	2	-2	0	0	-2i	0	0	2i	0	0	0	0	√-2	-√-2	√-2	-√-2	0	0	√2	-√2	complex lifted from C₄○D₈
ρ₁₇	2	-2	-2	2	0	0	0	0	-2	2	0	-2i	2i	0	0	0	0	0	√2	-√2	-√2	√2	-√-2	√-2	0	0	complex lifted from C₄○D₈
ρ₁₈	2	-2	-2	2	0	0	0	0	-2	2	0	-2i	2i	0	0	0	0	0	-√2	√2	√2	-√2	√-2	-√-2	0	0	complex lifted from C₄○D₈
ρ₁₉	2	2	-2	-2	0	0	2	-2	0	0	-2i	0	0	2i	0	0	0	0	-√-2	√-2	-√-2	√-2	0	0	-√2	√2	complex lifted from C₄○D₈
ρ₂₀	2	-2	-2	2	0	0	0	0	-2	2	0	2i	-2i	0	0	0	0	0	-√2	√2	√2	-√2	-√-2	√-2	0	0	complex lifted from C₄○D₈
ρ₂₁	2	-2	-2	2	0	0	0	0	-2	2	0	2i	-2i	0	0	0	0	0	√2	-√2	-√2	√2	√-2	-√-2	0	0	complex lifted from C₄○D₈
ρ₂₂	2	2	-2	-2	0	0	2	-2	0	0	2i	0	0	-2i	0	0	0	0	√-2	-√-2	√-2	-√-2	0	0	-√2	√2	complex lifted from C₄○D₈
ρ₂₃	4	-4	-4	4	0	0	0	0	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₄	4	4	-4	-4	0	0	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₅	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2i	2i	-2i	-2i	0	0	0	0	complex lifted from D₄.₈D₄
ρ₂₆	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2i	-2i	2i	2i	0	0	0	0	complex lifted from D₄.₈D₄

Smallest permutation representation of C₄².₁₈₉C₂³
►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 18 15 5)(2 19 16 6)(3 20 13 7)(4 17 14 8)(9 62 56 59)(10 63 53 60)(11 64 54 57)(12 61 55 58)(21 31 33 26)(22 32 34 27)(23 29 35 28)(24 30 36 25)(37 48 49 44)(38 45 50 41)(39 46 51 42)(40 47 52 43)

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

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

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

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

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

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

Matrix representation of C₄².₁₈₉C₂³ ►in GL₄(𝔽₁₇) generated by

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

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

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

5	5	0	0
5	12	0	0
0	0	11	6
0	0	14	6

3	3	0	0
3	14	0	0
0	0	13	0
0	0	0	13

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

C₄².₁₈₉C₂³ in GAP, Magma, Sage, TeX

C_4^2._{189}C_2^3

% in TeX

G:=Group("C4^2.189C2^3");

// GroupNames label

G:=SmallGroup(128,360);

// by ID

G=gap.SmallGroup(128,360);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,520,1123,570,521,136,2804,1411,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄².₁₈₉C₂³ in TeX

G = C42.189C23 order 128 = 27

50th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₁₈₉C₂³ order 128 = 2⁷

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