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G = C42.199D4order 128 = 27

181st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.199D4, C23.720C24, C24.103C23, C22.3762- 1+4, C22.4932+ 1+4, C428C473C2, C23.Q893C2, C23.4Q866C2, (C2×C42).732C22, (C22×C4).231C23, C22.452(C22×D4), C23.11D4129C2, C23.10D4.72C2, (C22×D4).297C22, (C22×Q8).235C22, C23.78C2364C2, C2.72(C22.29C24), C2.C42.423C22, C2.59(C22.31C24), C2.52(C22.56C24), C2.59(C22.57C24), C2.69(C23.38C23), (C2×C4).437(C2×D4), (C2×C42.C2)⋊28C2, (C2×C4⋊C4).529C22, (C2×C4.4D4).35C2, (C2×C22⋊C4).339C22, SmallGroup(128,1552)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.199D4
C1C2C22C23C22×C4C22×D4C23.10D4 — C42.199D4
C1C23 — C42.199D4
C1C23 — C42.199D4
C1C23 — C42.199D4

Generators and relations for C42.199D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 484 in 232 conjugacy classes, 92 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×16], C22 [×3], C22 [×4], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], Q8 [×4], C23, C23 [×14], C42 [×4], C22⋊C4 [×16], C4⋊C4 [×18], C22×C4 [×3], C22×C4 [×10], C2×D4 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C4.4D4 [×4], C42.C2 [×4], C22×D4, C22×Q8, C428C4, C23.10D4 [×2], C23.78C23 [×2], C23.Q8 [×4], C23.11D4 [×2], C23.4Q8 [×2], C2×C4.4D4, C2×C42.C2, C42.199D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×3], 2- 1+4 [×3], C22.29C24, C23.38C23, C22.31C24, C22.56C24 [×2], C22.57C24 [×2], C42.199D4

Character table of C42.199D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111884444448888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1-111-1-1-1-11111-1-111    linear of order 2
ρ311111111-1-1111111-1-111-1-111-1-1    linear of order 2
ρ41111111111-1-111-1-11111-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-111-1-111-1-11111-1-1    linear of order 2
ρ61111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ71111111111-1-111-1-1-1-1-1-1-1-11111    linear of order 2
ρ811111111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ911111111-1111-1-1-1-1-11-11-111-11-1    linear of order 2
ρ10111111111-1-1-1-1-1111-1-11-11-111-1    linear of order 2
ρ11111111111-111-1-1-1-11-1-111-11-1-11    linear of order 2
ρ1211111111-11-1-1-1-111-11-111-1-11-11    linear of order 2
ρ13111111111-1-1-1-1-111-111-1-111-1-11    linear of order 2
ρ1411111111-1111-1-1-1-11-11-1-11-11-11    linear of order 2
ρ1511111111-11-1-1-1-1111-11-11-11-11-1    linear of order 2
ρ16111111111-111-1-1-1-1-111-11-1-111-1    linear of order 2
ρ172-22-22-22-200-222-2-220000000000    orthogonal lifted from D4
ρ182-22-22-22-200-22-222-20000000000    orthogonal lifted from D4
ρ192-22-22-22-2002-2-22-220000000000    orthogonal lifted from D4
ρ202-22-22-22-2002-22-22-20000000000    orthogonal lifted from D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ244-444-4-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ25444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C42.199D4
On 64 points
Generators in S64
(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 44 55 13)(2 41 56 14)(3 42 53 15)(4 43 54 16)(5 47 38 20)(6 48 39 17)(7 45 40 18)(8 46 37 19)(9 59 24 32)(10 60 21 29)(11 57 22 30)(12 58 23 31)(25 36 49 62)(26 33 50 63)(27 34 51 64)(28 35 52 61)
(1 52 22 39)(2 51 23 38)(3 50 24 37)(4 49 21 40)(5 56 27 12)(6 55 28 11)(7 54 25 10)(8 53 26 9)(13 63 57 19)(14 62 58 18)(15 61 59 17)(16 64 60 20)(29 47 43 34)(30 46 44 33)(31 45 41 36)(32 48 42 35)
(1 63 55 33)(2 34 56 64)(3 61 53 35)(4 36 54 62)(5 29 38 60)(6 57 39 30)(7 31 40 58)(8 59 37 32)(9 48 24 17)(10 18 21 45)(11 46 22 19)(12 20 23 47)(13 52 44 28)(14 25 41 49)(15 50 42 26)(16 27 43 51)

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

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

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

Matrix representation of C42.199D4 in GL10(ℤ)

1000000000
0100000000
0000100000
0000010000
0010000000
0001000000
000000000-1
0000000010
0000000-100
0000001000
,
-1000000000
0-100000000
000-1000000
0010000000
00000-10000
0000100000
0000000100
0000001000
000000000-1
00000000-10
,
0100000000
-1000000000
0000010000
0000100000
0001000000
0010000000
0000000001
0000000010
0000000100
0000001000
,
0100000000
1000000000
0000100000
0000010000
00-10000000
000-1000000
0000000010
0000000001
0000001000
0000000100

G:=sub<GL(10,Integers())| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0],[0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

C42.199D4 in GAP, Magma, Sage, TeX

C_4^2._{199}D_4
% in TeX

G:=Group("C4^2.199D4");
// GroupNames label

G:=SmallGroup(128,1552);
// by ID

G=gap.SmallGroup(128,1552);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,794,185,80]);
// Polycyclic

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

Export

Character table of C42.199D4 in TeX

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