C4^2.191C2^3 - GroupNames

G = C₄².₁₉₁C₂³ order 128 = 2⁷

52^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₁₉₁C₂³, Q₈⋊C₈⋊₁₆C₂, C₄⋊C₄.₂₉₁D₄, C₄.Q₁₆⋊₁C₂, (C₂×D₄).₂₄D₄, C₈⋊₅D₄.₉C₂, C₄⋊C₈.₅C₂², (C₂×Q₈).₄₅D₄, C₄.₅₅(C₄○D₈), C₄.D₈.₁C₂, C₄⋊Q₈.₁₂C₂², (C₄×C₈).₂₄₃C₂², (C₄×Q₈).₂₄C₂², C₂.₁₉(D₄⋊₄D₄), C₄⋊₁D₄.₁₅C₂², C₄.₅₉(C₈.C₂²), C₂².₁₅₇C₂²≀C₂, C₂.₁₂(D₄.₇D₄), C₂².₅₃C₂⁴.₁C₂, (C₂×C₄).₉₄₈(C₂×D₄), SmallGroup(128,362)

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

Derived series

C₁ — C₄² — C₄².₁₉₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₂².₅₃C₂⁴ — C₄².₁₉₁C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₁₉₁C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₁₉₁C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₁₉₁C₂³

Generators and relations for C₄².₁₉₁C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=1, c²=e²=a², ab=ba, cac^-1=dad=a^-1, eae^-1=ab², cbc^-1=dbd=ebe^-1=b^-1, dcd=ac, ece^-1=bc, de=ed >

Subgroups: 264 in 109 conjugacy classes, 34 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₂.D₈, C₄×D₄, C₄×Q₈, C₂².D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂×SD₁₆, Q₈⋊C₈, C₄.D₈, C₄.Q₁₆, C₈⋊₅D₄, C₂².₅₃C₂⁴, C₄².₁₉₁C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₄○D₈, C₈.C₂², D₄.₇D₄, D₄⋊₄D₄, C₄².₁₉₁C₂³

Character table of C₄².₁₉₁C₂³

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	8	2	2	2	2	4	4	4	4	4	8	8	16	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	-2	2	-2	-2	-2	-2	0	0	0	0	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	-2	-2	2	2	0	-2	0	-2	-2	0	2	0	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	-2	0	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	-2	-2	-2	-2	0	0	0	0	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	0	2	0	2	-2	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	0	0	2	2	-2	-2	2	0	2	0	-2	-2	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	2i	0	0	0	0	0	√-2	-√-2	√-2	-√-2	0	0	-√2	√2	complex lifted from C₄○D₈
ρ₁₆	2	-2	-2	2	0	0	0	0	-2	2	0	-2i	0	2i	0	0	0	0	-√-2	√-2	√-2	-√-2	√2	-√2	0	0	complex lifted from C₄○D₈
ρ₁₇	2	-2	-2	2	0	0	0	0	-2	2	0	2i	0	-2i	0	0	0	0	√-2	-√-2	-√-2	√-2	√2	-√2	0	0	complex lifted from C₄○D₈
ρ₁₈	2	2	-2	-2	0	0	2	-2	0	0	2i	0	-2i	0	0	0	0	0	√-2	-√-2	√-2	-√-2	0	0	√2	-√2	complex lifted from C₄○D₈
ρ₁₉	2	-2	-2	2	0	0	0	0	-2	2	0	2i	0	-2i	0	0	0	0	-√-2	√-2	√-2	-√-2	-√2	√2	0	0	complex lifted from C₄○D₈
ρ₂₀	2	-2	-2	2	0	0	0	0	-2	2	0	-2i	0	2i	0	0	0	0	√-2	-√-2	-√-2	√-2	-√2	√2	0	0	complex lifted from C₄○D₈
ρ₂₁	2	2	-2	-2	0	0	2	-2	0	0	2i	0	-2i	0	0	0	0	0	-√-2	√-2	-√-2	√-2	0	0	-√2	√2	complex lifted from C₄○D₈
ρ₂₂	2	2	-2	-2	0	0	2	-2	0	0	-2i	0	2i	0	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	0	0	0	0	0	0	0	0	0	0	2	2	-2	-2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
ρ₂₄	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	-2	2	2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
ρ₂₅	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	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₆	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	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of C₄².₁₉₁C₂³
►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 44 45 50)(38 41 46 51)(39 42 47 52)(40 43 48 49)

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

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

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

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

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

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

Matrix representation of C₄².₁₉₁C₂³ ►in GL₄(𝔽₁₇) generated by

0	1	0	0
16	0	0	0
0	0	1	2
0	0	16	16

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

12	5	0	0
5	5	0	0
0	0	0	7
0	0	12	0

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

13	0	0	0
0	4	0	0
0	0	13	0
0	0	0	13

G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,16,0,0,2,16],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,5,0,0,5,5,0,0,0,0,0,12,0,0,7,0],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[13,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13] >;

C₄².₁₉₁C₂³ in GAP, Magma, Sage, TeX

C_4^2._{191}C_2^3

% in TeX

G:=Group("C4^2.191C2^3");

// GroupNames label

G:=SmallGroup(128,362);

// by ID

G=gap.SmallGroup(128,362);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,184,1123,570,521,136,2804,1411,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄².₁₉₁C₂³ in TeX

G = C42.191C23 order 128 = 27

52nd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₁₉₁C₂³ order 128 = 2⁷

52^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²