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G = C42.10D4order 128 = 27

10th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.10D4, (C2×C8).8D4, (C2×D4).17Q8, (C2×Q8).13Q8, C4.9C426C2, C4.60(C41D4), M4(2).C49C2, C23.14(C4○D4), C4.102(C22⋊Q8), (C22×C8).89C22, (C22×C4).743C23, C22.39(C22⋊Q8), C42⋊C2.73C22, C42⋊C22.10C2, C42.6C2225C2, C2.11(C23.4Q8), C4.121(C22.D4), (C2×M4(2)).239C22, C22.36(C22.D4), (C2×C4).24(C2×Q8), (C2×C4).86(C4○D4), (C2×C4).1385(C2×D4), (C22×C8)⋊C2.2C2, (C2×C4○D4).70C22, SmallGroup(128,830)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C42.10D4
C1C2C4C2×C4C22×C4C2×C4○D4C42⋊C22 — C42.10D4
C1C2C22×C4 — C42.10D4
C1C4C22×C4 — C42.10D4
C1C2C2C22×C4 — C42.10D4

Generators and relations for C42.10D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b-1, ab=ba, cac-1=a-1b-1, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b2c3 >

Subgroups: 208 in 105 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×4], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C22⋊C8 [×2], C4≀C2 [×4], C4⋊C8 [×4], C8.C4 [×2], C42⋊C2 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×C4○D4, C4.9C42, (C22×C8)⋊C2, C42⋊C22 [×2], C42.6C22 [×2], M4(2).C4, C42.10D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C23.4Q8, C42.10D4

Character table of C42.10D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11222811222888884444888888
ρ111111111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ311111-1111111-11-1-111111-11-1-1-1    linear of order 2
ρ411111-1111111-1111-1-1-1-1-11-11-1-1    linear of order 2
ρ511111111111-11-111-1-1-1-11-11-1-1-1    linear of order 2
ρ611111111111-11-1-1-11111-11-11-1-1    linear of order 2
ρ711111-111111-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ811111-111111-1-1-1111111-1-1-1-111    linear of order 2
ρ922-2-22022-22-2000-220000000000    orthogonal lifted from D4
ρ10222-2-20222-2-220-2000000000000    orthogonal lifted from D4
ρ1122-22-2022-2-22000000000020-200    orthogonal lifted from D4
ρ1222-22-2022-2-220000000000-20200    orthogonal lifted from D4
ρ13222-2-20222-2-2-202000000000000    orthogonal lifted from D4
ρ1422-2-22022-22-20002-20000000000    orthogonal lifted from D4
ρ1522-22-2-2-2-222-2020000000000000    symplectic lifted from Q8, Schur index 2
ρ1622-22-22-2-222-20-20000000000000    symplectic lifted from Q8, Schur index 2
ρ1722-2-220-2-22-220000000000000-2i2i    complex lifted from C4○D4
ρ18222220-2-2-2-2-2000002i2i-2i-2i000000    complex lifted from C4○D4
ρ1922-2-220-2-22-2200000000000002i-2i    complex lifted from C4○D4
ρ20222-2-20-2-2-222000000000-2i02i000    complex lifted from C4○D4
ρ21222220-2-2-2-2-200000-2i-2i2i2i000000    complex lifted from C4○D4
ρ22222-2-20-2-2-2220000000002i0-2i000    complex lifted from C4○D4
ρ234-40000-4i4i000000008858387000000    complex faithful
ρ244-400004i-4i000000008783858000000    complex faithful
ρ254-40000-4i4i000000008588783000000    complex faithful
ρ264-400004i-4i000000008387885000000    complex faithful

Smallest permutation representation of C42.10D4
On 32 points
Generators in S32
(1 7)(2 27 6 31)(3 5)(4 25 8 29)(9 15)(10 23 14 19)(11 13)(12 21 16 17)(18 20)(22 24)(26 28)(30 32)
(1 32 5 28)(2 29 6 25)(3 26 7 30)(4 31 8 27)(9 20 13 24)(10 17 14 21)(11 22 15 18)(12 19 16 23)
(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 20 28 9 5 24 32 13)(2 19 29 16 6 23 25 12)(3 18 30 15 7 22 26 11)(4 17 31 14 8 21 27 10)

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

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

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

Matrix representation of C42.10D4 in GL4(𝔽17) generated by

10107
216161
00013
00130
,
40134
0400
00130
00013
,
152143
0008
0200
415152
,
8998
0900
0008
0080
G:=sub<GL(4,GF(17))| [1,2,0,0,0,16,0,0,10,16,0,13,7,1,13,0],[4,0,0,0,0,4,0,0,13,0,13,0,4,0,0,13],[15,0,0,4,2,0,2,15,14,0,0,15,3,8,0,2],[8,0,0,0,9,9,0,0,9,0,0,8,8,0,8,0] >;

C42.10D4 in GAP, Magma, Sage, TeX

C_4^2._{10}D_4
% in TeX

G:=Group("C4^2.10D4");
// GroupNames label

G:=SmallGroup(128,830);
// by ID

G=gap.SmallGroup(128,830);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,1018,248,1411,4037,2028,1027,124]);
// Polycyclic

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

Export

Character table of C42.10D4 in TeX

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