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G = C83D8order 128 = 27

3rd semidirect product of C8 and D8 acting via D8/C4=C22

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

Aliases: C83D8, C42.668C23, C4.5(C2×D8), C85D42C2, C84D45C2, C8⋊C814C2, (C2×C8).37D4, C4.4D810C2, C2.8(C84D4), C4.6(C8⋊C22), C4⋊Q8.92C22, C2.11(C83D4), (C4×C8).154C22, C41D4.51C22, C22.69(C41D4), (C2×C4).725(C2×D4), SmallGroup(128,453)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C83D8
C1C2C22C2×C4C42C4×C8C8⋊C8 — C83D8
C1C22C42 — C83D8
C1C22C42 — C83D8
C1C22C22C42 — C83D8

Generators and relations for C83D8
 G = < a,b,c | a8=b8=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 384 in 121 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×6], C4, C22, C22 [×9], C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×18], Q8 [×2], C23 [×3], C42, C4⋊C4 [×2], C2×C8 [×6], D8 [×12], SD16 [×4], C2×D4 [×9], C2×Q8, C4×C8, C4×C8 [×2], D4⋊C4 [×4], C41D4, C41D4 [×2], C4⋊Q8, C2×D8 [×6], C2×SD16 [×2], C8⋊C8, C4.4D8 [×2], C85D4, C84D4, C84D4 [×2], C83D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C8⋊C22 [×4], C84D4, C83D4 [×2], C83D8

Character table of C83D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J8K8L
 size 111116161622222216444444444444
ρ111111111111111111111111111    trivial
ρ21111-1-111111111-1-111-1-11-1-11-1-1    linear of order 2
ρ31111-1-1-1111111-1111111111111    linear of order 2
ρ4111111-1111111-1-1-111-1-11-1-11-1-1    linear of order 2
ρ51111-111111111-1-1-1-1-11-1-11-1-111    linear of order 2
ρ611111-11111111-111-1-1-11-1-11-1-1-1    linear of order 2
ρ711111-1-11111111-1-1-1-11-1-11-1-111    linear of order 2
ρ81111-11-1111111111-1-1-11-1-11-1-1-1    linear of order 2
ρ92222000-22-2-22-200000-2002002-2    orthogonal lifted from D4
ρ102222000-2-2-22-220002200-200-200    orthogonal lifted from D4
ρ1122220002-22-2-2-202-2000-2002000    orthogonal lifted from D4
ρ122222000-2-2-22-22000-2-200200200    orthogonal lifted from D4
ρ1322220002-22-2-2-20-22000200-2000    orthogonal lifted from D4
ρ142222000-22-2-22-200000200-200-22    orthogonal lifted from D4
ρ152-22-20000-20020022-220-222-2-2-20    orthogonal lifted from D8
ρ162-22-20000-200200222-20-2-2-2-2220    orthogonal lifted from D8
ρ172-22-20000200-200-222-2-2-2202-202    orthogonal lifted from D8
ρ182-22-20000200-2002-22-22220-2-20-2    orthogonal lifted from D8
ρ192-22-20000-200200-2-2-22022-22-220    orthogonal lifted from D8
ρ202-22-20000-200200-2-22-202-2222-20    orthogonal lifted from D8
ρ212-22-20000200-200-22-222-2-20220-2    orthogonal lifted from D8
ρ222-22-20000200-2002-2-22-22-20-2202    orthogonal lifted from D8
ρ234-4-44000000-4040000000000000    orthogonal lifted from C8⋊C22
ρ244-4-4400000040-40000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-4000-4040000000000000000    orthogonal lifted from C8⋊C22
ρ2644-4-400040-40000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C83D8
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 25 53 46 11 61 23 37)(2 30 54 43 12 58 24 34)(3 27 55 48 13 63 17 39)(4 32 56 45 14 60 18 36)(5 29 49 42 15 57 19 33)(6 26 50 47 16 62 20 38)(7 31 51 44 9 59 21 35)(8 28 52 41 10 64 22 40)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 56)(25 61)(26 60)(27 59)(28 58)(29 57)(30 64)(31 63)(32 62)(34 40)(35 39)(36 38)(41 43)(44 48)(45 47)

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

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

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

Matrix representation of C83D8 in GL6(𝔽17)

1600000
0160000
001616314
0011633
0031411
0033161
,
3140000
330000
000010
000001
0016000
0001600
,
010000
100000
0016000
000100
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,3,3,0,0,16,16,14,3,0,0,3,3,1,16,0,0,14,3,1,1],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C83D8 in GAP, Magma, Sage, TeX

C_8\rtimes_3D_8
% in TeX

G:=Group("C8:3D8");
// GroupNames label

G:=SmallGroup(128,453);
// by ID

G=gap.SmallGroup(128,453);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,436,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Character table of C83D8 in TeX

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