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G = C42.6C22order 64 = 26

6th non-split extension by C42 of C22 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.6C22, C4⋊C812C2, C4⋊C4.5C4, C4.73(C2×D4), C4.12(C4⋊C4), (C2×C4).17Q8, C4.22(C2×Q8), C2.5(C8○D4), (C2×C4).127D4, C22⋊C4.2C4, (C22×C8).7C2, C22.8(C4⋊C4), (C2×C8).46C22, C23.17(C2×C4), (C2×C4).149C23, C42⋊C2.5C2, (C2×M4(2)).14C2, C22.44(C22×C4), (C22×C4).112C22, C2.10(C2×C4⋊C4), (C2×C4).25(C2×C4), SmallGroup(64,105)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.6C22
C1C2C4C2×C4C22×C4C42⋊C2 — C42.6C22
C1C22 — C42.6C22
C1C2×C4 — C42.6C22
C1C2C2C2×C4 — C42.6C22

Generators and relations for C42.6C22
 G = < a,b,c,d | a4=b4=1, c2=b, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b2c >

Subgroups: 73 in 57 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C23, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], C22×C4, C4⋊C8 [×4], C42⋊C2, C22×C8, C2×M4(2), C42.6C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C8○D4 [×2], C42.6C22

Character table of C42.6C22

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111221111224444222222224444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ31111-1-11111-1-111-1-1-11-1-111-11-11-11    linear of order 2
ρ41111-1-11111-1-1-1-111-11-1-111-111-11-1    linear of order 2
ρ5111111111111-1-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ61111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-11111-1-111-1-11-111-1-11-11-11-1    linear of order 2
ρ81111-1-11111-1-1-1-1111-111-1-11-1-11-11    linear of order 2
ρ9111111-1-1-1-1-1-1-11-11ii-i-i-i-iii-i-iii    linear of order 4
ρ10111111-1-1-1-1-1-1-11-11-i-iiiii-i-iii-i-i    linear of order 4
ρ11111111-1-1-1-1-1-11-11-1-i-iiiii-i-i-i-iii    linear of order 4
ρ12111111-1-1-1-1-1-11-11-1ii-i-i-i-iiiii-i-i    linear of order 4
ρ131111-1-1-1-1-1-1111-1-11i-i-i-iiii-ii-i-ii    linear of order 4
ρ141111-1-1-1-1-1-1111-1-11-iiii-i-i-ii-iii-i    linear of order 4
ρ151111-1-1-1-1-1-111-111-1-iiii-i-i-iii-i-ii    linear of order 4
ρ161111-1-1-1-1-1-111-111-1i-i-i-iiii-i-iii-i    linear of order 4
ρ172-2-22-22-22-22-220000000000000000    orthogonal lifted from D4
ρ182-2-222-2-22-222-20000000000000000    orthogonal lifted from D4
ρ192-2-22-222-22-22-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ202-2-222-22-22-2-220000000000000000    symplectic lifted from Q8, Schur index 2
ρ2122-2-2002i-2i-2i2i000000850878300800000    complex lifted from C8○D4
ρ2222-2-2002i-2i-2i2i000000808387008500000    complex lifted from C8○D4
ρ232-22-200-2i-2i2i2i000000085008387080000    complex lifted from C8○D4
ρ2422-2-200-2i2i2i-2i000000830885008700000    complex lifted from C8○D4
ρ2522-2-200-2i2i2i-2i000000870858008300000    complex lifted from C8○D4
ρ262-22-200-2i-2i2i2i000000080087830850000    complex lifted from C8○D4
ρ272-22-2002i2i-2i-2i000000083008580870000    complex lifted from C8○D4
ρ282-22-2002i2i-2i-2i000000087008850830000    complex lifted from C8○D4

Smallest permutation representation of C42.6C22
On 32 points
Generators in S32
(1 23 27 10)(2 15 28 20)(3 17 29 12)(4 9 30 22)(5 19 31 14)(6 11 32 24)(7 21 25 16)(8 13 26 18)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(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)
(1 19 31 10)(2 11 32 20)(3 21 25 12)(4 13 26 22)(5 23 27 14)(6 15 28 24)(7 17 29 16)(8 9 30 18)

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

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

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

C42.6C22 is a maximal subgroup of
C23.7M4(2)  C22⋊C4.C8  M4(2).42D4  (C2×D4).Q8  M4(2).3Q8  C4⋊C4.96D4  C4⋊C4.97D4  C4⋊C4.98D4  C22⋊C4.7D4  M4(2).12D4  M4(2).13D4  M4(2).Q8  M4(2).2Q8  C22⋊C4.Q8  C42.257C23  C42.674C23  C42.260C23  C42.261C23  C42.262C23  C42.264C23  C42.265C23  M4(2)⋊22D4  C42.286C23  C42.287C23  M4(2)⋊9Q8  C42.307C23  C42.308C23  C42.309C23  C42.310C23  C42.14C23  C42.15C23  C42.16C23  C42.17C23  C42.18C23  C42.19C23  C42.20C23  C42.21C23  C42.22C23  C42.23C23  C4.2- 1+4  C42.25C23  C42.26C23  C42.27C23  C42.28C23  C42.29C23  C42.30C23  C20⋊C8⋊C2
 C42.D2p: C42.428D4  C42.107D4  C42.9D4  C42.10D4  C42.30D6  C42.43D6  C42.30D10  C42.43D10 ...
 C4p.(C4⋊C4): C42.62Q8  C42.28Q8  Dic3⋊C8⋊C2  C12.88(C2×Q8)  C20.65(C4⋊C4)  C20.51(C4⋊C4)  C4⋊C4.9F5  Dic7⋊C8⋊C2 ...
C42.6C22 is a maximal quotient of
C23.29C42  C20⋊C8⋊C2  C4⋊C4.9F5
 C42.D2p: C42.90D4  C42.91D4  C42.Q8  C42.92D4  C42.21Q8  C42.45Q8  C42.95D4  C42.23Q8 ...
 (C2×C8).D2p: C23.21M4(2)  (C2×C8).195D4  Dic3⋊C8⋊C2  C12.88(C2×Q8)  C20.65(C4⋊C4)  C20.51(C4⋊C4)  Dic7⋊C8⋊C2  C28.439(C2×D4) ...

Matrix representation of C42.6C22 in GL4(𝔽17) generated by

0100
16000
0001
0010
,
4000
0400
0040
0004
,
2000
0200
00150
0002
,
0100
1000
0001
00160
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,15,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,16,0,0,1,0] >;

C42.6C22 in GAP, Magma, Sage, TeX

C_4^2._6C_2^2
% in TeX

G:=Group("C4^2.6C2^2");
// GroupNames label

G:=SmallGroup(64,105);
// by ID

G=gap.SmallGroup(64,105);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,332,88]);
// Polycyclic

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

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Character table of C42.6C22 in TeX

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