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G = C22.D8order 64 = 26

3rd non-split extension by C22 of D8 acting via D8/D4=C2

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

Aliases: C22.4D8, C23.45D4, C2.D86C2, C2.8(C2×D8), C22⋊C84C2, (C2×C4).37D4, D4⋊C47C2, C4⋊D4.5C2, (C2×C8).7C22, C4.29(C4○D4), C4⋊C4.63C22, (C2×C4).105C23, (C2×D4).21C22, C22.101(C2×D4), C2.17(C8.C22), (C22×C4).51C22, C2.11(C22.D4), (C2×C4⋊C4)⋊11C2, SmallGroup(64,161)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C22.D8
C1C2C4C2×C4C4⋊C4C2×C4⋊C4 — C22.D8
C1C2C2×C4 — C22.D8
C1C22C22×C4 — C22.D8
C1C2C2C2×C4 — C22.D8

Generators and relations for C22.D8
 G = < a,b,c,d | a2=b2=c8=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=bc-1 >

Subgroups: 113 in 57 conjugacy classes, 27 normal (15 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C22.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, C4○D4 [×2], C22.D4, C2×D8, C8.C22, C22.D8

Character table of C22.D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111228224444484444
ρ11111111111111111111    trivial
ρ2111111-11111111-1-1-1-1-1    linear of order 2
ρ3111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ4111111-111-1-1-1-11-11111    linear of order 2
ρ51111-1-111111-1-1-1-1-1-111    linear of order 2
ρ61111-1-1-11111-1-1-1111-1-1    linear of order 2
ρ71111-1-1111-1-111-1-111-1-1    linear of order 2
ρ81111-1-1-111-1-111-11-1-111    linear of order 2
ρ92222-2-20-2-20000200000    orthogonal lifted from D4
ρ102222220-2-20000-200000    orthogonal lifted from D4
ρ112-2-22-220000000002-22-2    orthogonal lifted from D8
ρ122-2-222-20000000002-2-22    orthogonal lifted from D8
ρ132-2-22-22000000000-22-22    orthogonal lifted from D8
ρ142-2-222-2000000000-222-2    orthogonal lifted from D8
ρ152-22-2000-22-2i2i00000000    complex lifted from C4○D4
ρ162-22-20002-200-2i2i000000    complex lifted from C4○D4
ρ172-22-20002-2002i-2i000000    complex lifted from C4○D4
ρ182-22-2000-222i-2i00000000    complex lifted from C4○D4
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C22.D8
On 32 points
Generators in S32
(1 14)(2 27)(3 16)(4 29)(5 10)(6 31)(7 12)(8 25)(9 18)(11 20)(13 22)(15 24)(17 28)(19 30)(21 32)(23 26)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(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)
(2 22)(3 7)(4 20)(6 18)(8 24)(9 11)(10 30)(12 28)(13 15)(14 26)(16 32)(17 21)(25 27)(29 31)

G:=sub<Sym(32)| (1,14)(2,27)(3,16)(4,29)(5,10)(6,31)(7,12)(8,25)(9,18)(11,20)(13,22)(15,24)(17,28)(19,30)(21,32)(23,26), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,30)(12,28)(13,15)(14,26)(16,32)(17,21)(25,27)(29,31)>;

G:=Group( (1,14)(2,27)(3,16)(4,29)(5,10)(6,31)(7,12)(8,25)(9,18)(11,20)(13,22)(15,24)(17,28)(19,30)(21,32)(23,26), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,30)(12,28)(13,15)(14,26)(16,32)(17,21)(25,27)(29,31) );

G=PermutationGroup([(1,14),(2,27),(3,16),(4,29),(5,10),(6,31),(7,12),(8,25),(9,18),(11,20),(13,22),(15,24),(17,28),(19,30),(21,32),(23,26)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28)], [(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)], [(2,22),(3,7),(4,20),(6,18),(8,24),(9,11),(10,30),(12,28),(13,15),(14,26),(16,32),(17,21),(25,27),(29,31)])

C22.D8 is a maximal subgroup of
C24.115D4  C24.183D4  C24.117D4  (C2×D4).301D4  (C2×D4).303D4  C42.221D4  C42.226D4  C42.230D4  C42.232D4  C233D8  C24.123D4  C24.126D4  C24.127D4  C4.2+ 1+4  C4.182+ 1+4  C42.284D4  C42.286D4  C42.290D4
 (C2×C2p).D8: (C2×C4).5D8  C42.278D4  C22.D24  (C2×C6).40D8  (C2×C6).D8  C22.D40  (C2×C10).40D8  (C2×C10).D8 ...
 C4⋊C4.D2p: C23.5D8  C4⋊C4.12D4  C24.15D4  C42.353C23  C42.358C23  C42.423C23  C42.425C23  D44D8 ...
C22.D8 is a maximal quotient of
C23.36D8  C23.37D8  C23.38D8  C24.83D4
 (C2×C2p).D8: C2.D84C4  D4⋊C4⋊C4  (C2×C4).24D8  (C2×C8).1Q8  (C2×C4).27D8  (C2×C4).28D8  C22.D16  C23.49D8 ...
 (C2×C8).D2p: C23.12D8  D6.D8  D6.5D8  D10.12D8  D10.13D8  D14.D8  D14.5D8 ...

Matrix representation of C22.D8 in GL4(𝔽17) generated by

1000
0100
0001
0010
,
1000
0100
00160
00016
,
31400
3300
0040
00013
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[3,3,0,0,14,3,0,0,0,0,4,0,0,0,0,13],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C22.D8 in GAP, Magma, Sage, TeX

C_2^2.D_8
% in TeX

G:=Group("C2^2.D8");
// GroupNames label

G:=SmallGroup(64,161);
// by ID

G=gap.SmallGroup(64,161);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,50,1444,376,88]);
// Polycyclic

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

Export

Character table of C22.D8 in TeX

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