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G = (C2×C8).55D4order 128 = 27

23rd non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).55D4, (C2×D4).119D4, (C2×Q8).110D4, C4.C4213C2, C4.57(C4.4D4), C22.C4224C2, C2.22(D4.4D4), C2.22(D4.5D4), C23.282(C4○D4), (C22×C4).737C23, (C22×C8).117C22, C22.254(C4⋊D4), C23.36D4.12C2, C22.16(C422C2), C4.113(C22.D4), C2.13(C23.11D4), (C2×M4(2)).235C22, (C2×C2.D8)⋊12C2, (C2×C4).84(C4○D4), (C2×C4).1376(C2×D4), (C2×C4⋊C4).152C22, (C22×C8)⋊C2.6C2, (C2×C4○D4).67C22, SmallGroup(128,810)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).55D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.36D4 — (C2×C8).55D4
C1C2C22×C4 — (C2×C8).55D4
C1C22C22×C4 — (C2×C8).55D4
C1C2C2C22×C4 — (C2×C8).55D4

Generators and relations for (C2×C8).55D4
 G = < a,b,c,d | a2=b8=d2=1, c4=b4, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=ab4c3 >

Subgroups: 216 in 99 conjugacy classes, 38 normal (22 characteristic)
C1, C2 [×3], 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], C2.D8 [×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×C2.D8, (C2×C8).55D4
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, D4.4D4, D4.5D4, (C2×C8).55D4

Character table of (C2×C8).55D4

 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
ρ234-44-40000000000000-22220000000    orthogonal lifted from D4.4D4
ρ244-44-4000000000000022-220000000    orthogonal lifted from D4.4D4
ρ2544-4-40000000000002200-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ2644-4-4000000000000-220022000000    symplectic lifted from D4.5D4, Schur index 2

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

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

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

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

Matrix representation of (C2×C8).55D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
5160000
9120000
003636
0014141414
00141136
00331414
,
400000
040000
000700
0051000
0000010
0000127
,
100000
10160000
00161500
000100
00001615
000001

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,9,0,0,0,0,16,12,0,0,0,0,0,0,3,14,14,3,0,0,6,14,11,3,0,0,3,14,3,14,0,0,6,14,6,14],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,0,0,0,0,7,10,0,0,0,0,0,0,0,12,0,0,0,0,10,7],[1,10,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1] >;

(C2×C8).55D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{55}D_4
% in TeX

G:=Group("(C2xC8).55D4");
// GroupNames label

G:=SmallGroup(128,810);
// by ID

G=gap.SmallGroup(128,810);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,718,172,2028,1027,124]);
// 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^-1,d*c*d=a*b^4*c^3>;
// generators/relations

Export

Character table of (C2×C8).55D4 in TeX

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