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G = (C2×C8).165D4order 128 = 27

133rd non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).165D4, (C2×D4).120D4, (C2×Q8).111D4, C4.C4215C2, C4.58(C4.4D4), C22.C4225C2, C2.30(D4.3D4), C23.283(C4○D4), (C22×C4).738C23, (C22×C8).326C22, C22.255(C4⋊D4), C23.36D4.13C2, C22.17(C422C2), C4.114(C22.D4), C2.14(C23.11D4), (C2×M4(2)).236C22, (C2×C4.Q8)⋊24C2, (C2×C4).85(C4○D4), (C2×C4).1377(C2×D4), (C2×C4⋊C4).153C22, (C22×C8)⋊C2.7C2, (C2×C4○D4).68C22, SmallGroup(128,811)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).165D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.36D4 — (C2×C8).165D4
C1C2C22×C4 — (C2×C8).165D4
C1C22C22×C4 — (C2×C8).165D4
C1C2C2C22×C4 — (C2×C8).165D4

Generators and relations for (C2×C8).165D4
 G = < a,b,c,d | a2=b8=d2=1, c4=b4, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab3, dcd=ab4c3 >

Subgroups: 216 in 99 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×5], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×4], Q8 [×2], C23, C23, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×C4○D4, C4.C42, C22.C42 [×2], (C22×C8)⋊C2, C23.36D4 [×2], C2×C4.Q8, (C2×C8).165D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C23.11D4, D4.3D4 [×2], (C2×C8).165D4

Character table of (C2×C8).165D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J
 size 11112282222888884444888888
ρ111111111111111111111111111    trivial
ρ211111111111-1111-1-1-1-1-1-1-111-1-1    linear of order 2
ρ3111111-111111-1-1-11-1-1-1-11-111-11    linear of order 2
ρ4111111-11111-1-1-1-1-11111-11111-1    linear of order 2
ρ5111111111111-1-111-1-1-1-1-11-1-11-1    linear of order 2
ρ611111111111-1-1-11-111111-1-1-1-11    linear of order 2
ρ7111111-11111111-111111-1-1-1-1-1-1    linear of order 2
ρ8111111-11111-111-1-1-1-1-1-111-1-111    linear of order 2
ρ92222-2-2022-2-200000-222-2000000    orthogonal lifted from D4
ρ102222-2-2022-2-2000002-2-22000000    orthogonal lifted from D4
ρ112222-2-2-2-2-222000200000000000    orthogonal lifted from D4
ρ122222-2-22-2-222000-200000000000    orthogonal lifted from D4
ρ132-2-222-20-22-220-2i2i000000000000    complex lifted from C4○D4
ρ142-2-22-2202-2-2200000000000-2i2i00    complex lifted from C4○D4
ρ152-2-22-2202-2-22000000000002i-2i00    complex lifted from C4○D4
ρ162-2-222-202-22-200000000002i00-2i0    complex lifted from C4○D4
ρ172222220-2-2-2-2000000000-2i00002i    complex lifted from C4○D4
ρ182-2-222-202-22-20000000000-2i002i0    complex lifted from C4○D4
ρ192-2-22-220-222-22i000-2i0000000000    complex lifted from C4○D4
ρ202-2-222-20-22-2202i-2i000000000000    complex lifted from C4○D4
ρ212222220-2-2-2-20000000002i0000-2i    complex lifted from C4○D4
ρ222-2-22-220-222-2-2i0002i0000000000    complex lifted from C4○D4
ρ2344-4-40000000000002-200-2-2000000    complex lifted from D4.3D4
ρ244-44-400000000000002-2-2-20000000    complex lifted from D4.3D4
ρ2544-4-4000000000000-2-2002-2000000    complex lifted from D4.3D4
ρ264-44-40000000000000-2-22-20000000    complex lifted from D4.3D4

Smallest permutation representation of (C2×C8).165D4
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(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 45 27 59 5 41 31 63)(2 15 28 35 6 11 32 39)(3 43 29 57 7 47 25 61)(4 13 30 33 8 9 26 37)(10 53 38 23 14 49 34 19)(12 51 40 21 16 55 36 17)(18 48 52 62 22 44 56 58)(20 46 54 60 24 42 50 64)
(2 18)(4 20)(6 22)(8 24)(9 33)(10 61)(11 35)(12 63)(13 37)(14 57)(15 39)(16 59)(25 29)(26 54)(27 31)(28 56)(30 50)(32 52)(34 43)(36 45)(38 47)(40 41)(42 60)(44 62)(46 64)(48 58)(49 53)(51 55)

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

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

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

Matrix representation of (C2×C8).165D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
1620000
1610000
009000
000900
0000150
0000015
,
1300000
0130000
000001
000010
001000
0001600
,
100000
1160000
001000
0001600
000001
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,1,0,0,0],[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] >;

(C2×C8).165D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{165}D_4
% in TeX

G:=Group("(C2xC8).165D4");
// GroupNames label

G:=SmallGroup(128,811);
// by ID

G=gap.SmallGroup(128,811);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,422,387,58,718,172,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*c*d=a*b^4*c^3>;
// generators/relations

Export

Character table of (C2×C8).165D4 in TeX

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