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G = (C2×C12).D4order 192 = 26·3

16th non-split extension by C2×C12 of D4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C4).4D12, (C2×C12).16D4, C4.10D4.S3, (C2×Q8).26D6, (C4×Dic3).2C4, (C2×Dic6).3C4, (C6×Q8).2C22, C6.14(C23⋊C4), C31(C42.3C4), C22.15(D6⋊C4), Dic3⋊Q8.1C2, C12.10D4.1C2, C2.15(C23.6D6), (C2×C4).4(C4×S3), (C2×C12).4(C2×C4), (C2×C4).4(C3⋊D4), (C2×C6).8(C22⋊C4), (C3×C4.10D4).1C2, SmallGroup(192,37)

Series: Derived Chief Lower central Upper central

C1C2×C12 — (C2×C12).D4
C1C3C6C2×C6C2×C12C6×Q8Dic3⋊Q8 — (C2×C12).D4
C3C6C2×C6C2×C12 — (C2×C12).D4
C1C2C22C2×Q8C4.10D4

Generators and relations for (C2×C12).D4
 G = < a,b,c,d | a2=b12=1, c4=d2=b6, ab=ba, cac-1=ab6, ad=da, cbc-1=dbd-1=ab5, dcd-1=b9c3 >

Subgroups: 176 in 60 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, M4(2), C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C3×Q8, C4.10D4, C4.10D4, C4⋊Q8, C4.Dic3, C4×Dic3, Dic3⋊C4, C3×M4(2), C2×Dic6, C6×Q8, C42.3C4, C12.10D4, C3×C4.10D4, Dic3⋊Q8, (C2×C12).D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C42.3C4, C23.6D6, (C2×C12).D4

Character table of (C2×C12).D4

 class 12A2B34A4B4C4D4E4F6A6B8A8B8C8D12A12B12C12D24A24B24C24D
 size 11224441212242488242444888888
ρ1111111111111111111111111    trivial
ρ2111111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ31111111-1-1-111-1-1111111-1-1-1-1    linear of order 2
ρ41111111-1-1-11111-1-111111111    linear of order 2
ρ51111-11-1-1-1111i-ii-i-1-1-11ii-i-i    linear of order 4
ρ61111-11-111-111i-i-ii-1-1-11ii-i-i    linear of order 4
ρ71111-11-111-111-iii-i-1-1-11-i-iii    linear of order 4
ρ81111-11-1-1-1111-ii-ii-1-1-11-i-iii    linear of order 4
ρ9222-1222000-1-1-2-200-1-1-1-11111    orthogonal lifted from D6
ρ102222-2-22000220000-2-22-20000    orthogonal lifted from D4
ρ1122222-2-200022000022-2-20000    orthogonal lifted from D4
ρ12222-1222000-1-12200-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ13222-1-2-22000-1-1000011-113-33-3    orthogonal lifted from D12
ρ14222-1-2-22000-1-1000011-11-33-33    orthogonal lifted from D12
ρ15222-1-22-2000-1-1-2i2i00111-1ii-i-i    complex lifted from C4×S3
ρ16222-1-22-2000-1-12i-2i00111-1-i-iii    complex lifted from C4×S3
ρ17222-12-2-2000-1-10000-1-111--3-3-3--3    complex lifted from C3⋊D4
ρ18222-12-2-2000-1-10000-1-111-3--3--3-3    complex lifted from C3⋊D4
ρ1944-440000004-4000000000000    orthogonal lifted from C23⋊C4
ρ204-4040002-20-40000000000000    symplectic lifted from C42.3C4, Schur index 2
ρ214-404000-220-40000000000000    symplectic lifted from C42.3C4, Schur index 2
ρ2244-4-2000000-2200002-3-2-3000000    complex lifted from C23.6D6
ρ2344-4-2000000-220000-2-32-3000000    complex lifted from C23.6D6
ρ248-80-400000040000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C2×C12).D4
On 48 points
Generators in S48
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(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)
(1 19 38 26 7 13 44 32)(2 24 45 25 8 18 39 31)(3 17 40 36 9 23 46 30)(4 22 47 35 10 16 41 29)(5 15 42 34 11 21 48 28)(6 20 37 33 12 14 43 27)
(1 10 7 4)(2 3 8 9)(5 6 11 12)(13 26 19 32)(14 25 20 31)(15 36 21 30)(16 35 22 29)(17 34 23 28)(18 33 24 27)(37 42 43 48)(38 47 44 41)(39 40 45 46)

G:=sub<Sym(48)| (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (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), (1,19,38,26,7,13,44,32)(2,24,45,25,8,18,39,31)(3,17,40,36,9,23,46,30)(4,22,47,35,10,16,41,29)(5,15,42,34,11,21,48,28)(6,20,37,33,12,14,43,27), (1,10,7,4)(2,3,8,9)(5,6,11,12)(13,26,19,32)(14,25,20,31)(15,36,21,30)(16,35,22,29)(17,34,23,28)(18,33,24,27)(37,42,43,48)(38,47,44,41)(39,40,45,46)>;

G:=Group( (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (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), (1,19,38,26,7,13,44,32)(2,24,45,25,8,18,39,31)(3,17,40,36,9,23,46,30)(4,22,47,35,10,16,41,29)(5,15,42,34,11,21,48,28)(6,20,37,33,12,14,43,27), (1,10,7,4)(2,3,8,9)(5,6,11,12)(13,26,19,32)(14,25,20,31)(15,36,21,30)(16,35,22,29)(17,34,23,28)(18,33,24,27)(37,42,43,48)(38,47,44,41)(39,40,45,46) );

G=PermutationGroup([[(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(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)], [(1,19,38,26,7,13,44,32),(2,24,45,25,8,18,39,31),(3,17,40,36,9,23,46,30),(4,22,47,35,10,16,41,29),(5,15,42,34,11,21,48,28),(6,20,37,33,12,14,43,27)], [(1,10,7,4),(2,3,8,9),(5,6,11,12),(13,26,19,32),(14,25,20,31),(15,36,21,30),(16,35,22,29),(17,34,23,28),(18,33,24,27),(37,42,43,48),(38,47,44,41),(39,40,45,46)]])

Matrix representation of (C2×C12).D4 in GL6(𝔽73)

100000
010000
001000
000100
0000720
005858072
,
1720000
100000
00505700
00242300
0021215032
0019476123
,
5970000
66140000
0069693753
00444491
0072000
0065292033
,
0720000
7200000
00505700
00242300
0069693753
007353236

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,58,0,0,0,1,0,58,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,50,24,21,19,0,0,57,23,21,47,0,0,0,0,50,61,0,0,0,0,32,23],[59,66,0,0,0,0,7,14,0,0,0,0,0,0,69,44,72,65,0,0,69,44,0,29,0,0,37,9,0,20,0,0,53,1,0,33],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,50,24,69,7,0,0,57,23,69,35,0,0,0,0,37,32,0,0,0,0,53,36] >;

(C2×C12).D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12}).D_4
% in TeX

G:=Group("(C2xC12).D4");
// GroupNames label

G:=SmallGroup(192,37);
// by ID

G=gap.SmallGroup(192,37);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,184,1123,794,297,136,851,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^4=d^2=b^6,a*b=b*a,c*a*c^-1=a*b^6,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^5,d*c*d^-1=b^9*c^3>;
// generators/relations

Export

Character table of (C2×C12).D4 in TeX

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