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G = C42.664C23order 128 = 27

79th non-split extension by C42 of C23 acting via C23/C22=C2

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

Aliases: C42.664C23, C8⋊C811C2, (C2×C8).33D4, C85D419C2, C4.7(C4○D8), C84D4.7C2, C4.4D838C2, C2.8(C83D4), C4.3(C8⋊C22), C4⋊Q8.88C22, (C4×C8).257C22, C4.SD1610C2, C41D4.48C22, C2.12(C8.12D4), C22.65(C41D4), (C2×C4).721(C2×D4), SmallGroup(128,449)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.664C23
C1C2C22C2×C4C42C4×C8C8⋊C8 — C42.664C23
C1C22C42 — C42.664C23
C1C22C42 — C42.664C23
C1C22C22C42 — C42.664C23

Generators and relations for C42.664C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=ab2, 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: 304 in 104 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, D8, SD16, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C41D4, C4⋊Q8, C2×D8, C2×SD16, C8⋊C8, C4.4D8, C4.SD16, C85D4, C84D4, C42.664C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C41D4, C4○D8, C8⋊C22, C8.12D4, C83D4, C42.664C23

Character table of C42.664C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L
 size 111116162222221616444444444444
ρ111111111111111111111111111    trivial
ρ211111-11111111-1-1-1-1-11-1-11-1-111    linear of order 2
ρ31111-11111111-11-1-1-1-11-1-11-1-111    linear of order 2
ρ41111-1-1111111-1-1111111111111    linear of order 2
ρ51111-111111111-1-1-111-1-11-1-11-1-1    linear of order 2
ρ61111-1-11111111111-1-1-11-1-11-1-1-1    linear of order 2
ρ7111111111111-1-111-1-1-11-1-11-1-1-1    linear of order 2
ρ811111-1111111-11-1-111-1-11-1-11-1-1    linear of order 2
ρ9222200-22-2-22-2000000200-200-22    orthogonal lifted from D4
ρ10222200-2-2-22-220000-2-200200200    orthogonal lifted from D4
ρ112222002-22-2-2-200-22000200-2000    orthogonal lifted from D4
ρ12222200-2-2-22-2200002200-200-200    orthogonal lifted from D4
ρ13222200-22-2-22-2000000-2002002-2    orthogonal lifted from D4
ρ142222002-22-2-2-2002-2000-2002000    orthogonal lifted from D4
ρ152-22-2000-2002000-2-2-2--202--2-2i2-22i0    complex lifted from C4○D8
ρ162-22-2000-200200022--2-20-2-2-2i-2--22i0    complex lifted from C4○D8
ρ172-22-2000-200200022-2--20-2--22i-2-2-2i0    complex lifted from C4○D8
ρ182-22-2000200-20002-2-2--2-2i2-20-2--202i    complex lifted from C4○D8
ρ192-22-2000200-2000-22-2--22i-2-202--20-2i    complex lifted from C4○D8
ρ202-22-2000200-2000-22--2-2-2i-2--202-202i    complex lifted from C4○D8
ρ212-22-2000-2002000-2-2--2-202-22i2--2-2i0    complex lifted from C4○D8
ρ222-22-2000200-20002-2--2-22i2--20-2-20-2i    complex lifted from C4○D8
ρ234-4-4400000-40400000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400-40400000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-440000040-400000000000000    orthogonal lifted from C8⋊C22
ρ2644-4-40040-400000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C42.664C23
On 64 points
Generators in S64
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(33 64 37 60)(34 57 38 61)(35 58 39 62)(36 59 40 63)
(1 61 31 40)(2 62 32 33)(3 63 25 34)(4 64 26 35)(5 57 27 36)(6 58 28 37)(7 59 29 38)(8 60 30 39)(9 21 45 56)(10 22 46 49)(11 23 47 50)(12 24 48 51)(13 17 41 52)(14 18 42 53)(15 19 43 54)(16 20 44 55)
(2 26)(3 7)(4 32)(6 30)(8 28)(9 17)(10 55)(11 23)(12 53)(13 21)(14 51)(15 19)(16 49)(18 48)(20 46)(22 44)(24 42)(25 29)(33 35)(34 59)(36 57)(37 39)(38 63)(40 61)(41 56)(43 54)(45 52)(47 50)(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 15 61 19 31 43 40 54)(2 12 62 24 32 48 33 51)(3 9 63 21 25 45 34 56)(4 14 64 18 26 42 35 53)(5 11 57 23 27 47 36 50)(6 16 58 20 28 44 37 55)(7 13 59 17 29 41 38 52)(8 10 60 22 30 46 39 49)

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

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

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

Matrix representation of C42.664C23 in GL6(𝔽17)

100000
010000
0001600
001000
0000016
000010
,
16150000
110000
000100
0016000
000001
0000160
,
100000
16160000
001000
0001600
000001
000010
,
400000
040000
005506
00125110
00601212
0006512
,
10100000
1200000
000010
000001
000100
0016000

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

C42.664C23 in GAP, Magma, Sage, TeX

C_4^2._{664}C_2^3
% in TeX

G:=Group("C4^2.664C2^3");
// GroupNames label

G:=SmallGroup(128,449);
// by ID

G=gap.SmallGroup(128,449);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,387,100,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a*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 C42.664C23 in TeX

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