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G = C42.666C23order 128 = 27

81st non-split extension by C42 of C23 acting via C23/C22=C2

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

Aliases: C42.666C23, C82Q87C2, C8⋊C812C2, (C2×C8).35D4, C4.9(C4○D8), C85D4.13C2, C2.9(C83D4), C4.4(C8⋊C22), C4⋊Q8.90C22, (C4×C8).259C22, C4.SD1638C2, C4.4D8.14C2, C2.9(C8.2D4), C4.4(C8.C22), C41D4.49C22, C2.14(C8.12D4), C22.67(C41D4), (C2×C4).723(C2×D4), SmallGroup(128,451)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.666C23
C1C2C22C2×C4C42C4×C8C8⋊C8 — C42.666C23
C1C22C42 — C42.666C23
C1C22C42 — C42.666C23
C1C22C22C42 — C42.666C23

Generators and relations for C42.666C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=a-1b2, e2=b, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a-1c, ece-1=b-1c, ede-1=a2d >

Subgroups: 240 in 93 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×6], C2×C4 [×3], C2×C4 [×3], D4 [×6], Q8 [×6], C23, C42, C4⋊C4 [×6], C2×C8 [×6], SD16 [×4], C2×D4 [×3], C2×Q8 [×3], C4×C8 [×3], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×2], C41D4, C4⋊Q8 [×3], C2×SD16 [×2], C8⋊C8, C4.4D8 [×2], C4.SD16 [×2], C85D4, C82Q8, C42.666C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, C4○D8 [×2], C8⋊C22 [×2], C8.C22 [×2], C8.12D4, C83D4, C8.2D4, C42.666C23

Character table of C42.666C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 111116222222161616444444444444
ρ111111111111111111111111111    trivial
ρ21111-1111111-111-1-111-1-11-1-11-1-1    linear of order 2
ρ311111111111-11-1-1-1-1-11-1-11-1-111    linear of order 2
ρ41111-111111111-111-1-1-11-1-11-1-1-1    linear of order 2
ρ51111-11111111-11-1-1-1-11-1-11-1-111    linear of order 2
ρ611111111111-1-1111-1-1-11-1-11-1-1-1    linear of order 2
ρ71111-1111111-1-1-1111111111111    linear of order 2
ρ8111111111111-1-1-1-111-1-11-1-11-1-1    linear of order 2
ρ922220-22-2-22-20000000200-200-22    orthogonal lifted from D4
ρ10222202-22-2-2-20002-2000-2002000    orthogonal lifted from D4
ρ1122220-2-2-22-2200000-2-200200200    orthogonal lifted from D4
ρ1222220-2-2-22-22000002200-200-200    orthogonal lifted from D4
ρ1322220-22-2-22-20000000-2002002-2    orthogonal lifted from D4
ρ14222202-22-2-2-2000-22000200-2000    orthogonal lifted from D4
ρ152-22-200-2002000022--2-20-2-2-2i-2--22i0    complex lifted from C4○D8
ρ162-22-200-20020000-2-2-2--202--2-2i2-22i0    complex lifted from C4○D8
ρ172-22-200-20020000-2-2--2-202-22i2--2-2i0    complex lifted from C4○D8
ρ182-22-200200-200002-2-2--2-2i2-20-2--202i    complex lifted from C4○D8
ρ192-22-200200-20000-22-2--22i-2-202--20-2i    complex lifted from C4○D8
ρ202-22-200200-20000-22--2-2-2i-2--202-202i    complex lifted from C4○D8
ρ212-22-200-2002000022-2--20-2--22i-2-2-2i0    complex lifted from C4○D8
ρ222-22-200200-200002-2--2-22i2--20-2-20-2i    complex lifted from C4○D8
ρ2344-4-4000040-4000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-40000-404000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-44040-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-440-404000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.666C23
On 64 points
Generators in S64
(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(25 56 29 52)(26 49 30 53)(27 50 31 54)(28 51 32 55)(33 62 37 58)(34 63 38 59)(35 64 39 60)(36 57 40 61)
(1 61 22 38)(2 62 23 39)(3 63 24 40)(4 64 17 33)(5 57 18 34)(6 58 19 35)(7 59 20 36)(8 60 21 37)(9 27 47 52)(10 28 48 53)(11 29 41 54)(12 30 42 55)(13 31 43 56)(14 32 44 49)(15 25 45 50)(16 26 46 51)
(2 21)(3 7)(4 19)(6 17)(8 23)(9 31)(10 55)(11 29)(12 53)(13 27)(14 51)(15 25)(16 49)(20 24)(26 44)(28 42)(30 48)(32 46)(33 35)(34 57)(36 63)(37 39)(38 61)(40 59)(41 54)(43 52)(45 50)(47 56)(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 15 61 25 22 45 38 50)(2 12 62 30 23 42 39 55)(3 9 63 27 24 47 40 52)(4 14 64 32 17 44 33 49)(5 11 57 29 18 41 34 54)(6 16 58 26 19 46 35 51)(7 13 59 31 20 43 36 56)(8 10 60 28 21 48 37 53)

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

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

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

Matrix representation of C42.666C23 in GL6(𝔽17)

1600000
0160000
0001600
001000
0000016
000010
,
1150000
1160000
000100
0016000
000001
0000160
,
100000
1160000
001000
0001600
000001
000010
,
490000
4130000
005500
0012500
00001212
0000512
,
0100000
12100000
000010
000001
000100
0016000

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

C42.666C23 in GAP, Magma, Sage, TeX

C_4^2._{666}C_2^3
% in TeX

G:=Group("C4^2.666C2^3");
// GroupNames label

G:=SmallGroup(128,451);
// by ID

G=gap.SmallGroup(128,451);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,387,100,1123,136,2804,172]);
// Polycyclic

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

Export

Character table of C42.666C23 in TeX

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