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G = D12.38D4  order 192 = 26·3

8th non-split extension by D12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.38D4, Dic6.38D4, M4(2).14D6, C8⋊C22⋊3S3, Q8○D12⋊2C2, C4○D4.22D6, (C2×D4).81D6, (C3×D4).13D4, C4.105(S3×D4), (C3×Q8).13D4, D12⋊C4⋊6C2, C6.64C22≀C2, D12⋊6C22⋊6C2, C12.197(C2×D4), C3⋊4(D4.8D4), (C2×Dic3).5D4, C22.36(S3×D4), Q8⋊3Dic3⋊7C2, C23.12D6⋊7C2, C12.47D4⋊6C2, D4.10(C3⋊D4), (C2×C12).16C23, Q8.17(C3⋊D4), C2.32(C23⋊2D6), C4○D12.24C22, (C6×D4).106C22, (C4×Dic3).58C22, C4.Dic3.26C22, (C2×Dic6).135C22, (C3×M4(2)).11C22, (C3×C8⋊C22)⋊7C2, (C2×C6).35(C2×D4), C4.53(C2×C3⋊D4), (C2×C4).16(C22×S3), (C3×C4○D4).14C22, SmallGroup(192,760)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — D12.38D4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8○D12 — D12.38D4
Lower central
C3 — C6 — C2×C12 — D12.38D4
Upper central
C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D12.38D4
 G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c3 >

Subgroups: 432 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, C8⋊C22, 2- 1+4, C4.Dic3, C4×Dic3, D4⋊S3, D4.S3, C6.D4, C3×M4(2), C3×D8, C3×SD16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D4⋊2S3, S3×Q8, C6×D4, C3×C4○D4, D4.8D4, C12.47D4, D12⋊C4, Q8⋊3Dic3, D12⋊6C22, C23.12D6, C3×C8⋊C22, Q8○D12, D12.38D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.8D4, C23⋊2D6, D12.38D4

Character table of D12.38D4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C6D6E8A8B12A12B12C24A24B
 size 1124812222412121212122488882444888
ρ1111111111111111111111111111    trivial
ρ21111-1-111111-1-1-11111-1-1-11111-1-1    linear of order 2
ρ3111-1-11111-111-1-1111-1-1-11-111-111    linear of order 2
ρ4111-11-1111-11-111111-111-1-111-1-1-1    linear of order 2
ρ51111-111111-1111-1111-1-1-1-1111-1-1    linear of order 2
ρ611111-11111-1-1-1-1-1111111-111111    linear of order 2
ρ7111-1-1-1111-1-1-111-111-1-1-11111-111    linear of order 2
ρ8111-111111-1-11-1-1-111-111-1111-1-1-1    linear of order 2
ρ92222-20-122200000-1-1-111-20-1-1-111    orthogonal lifted from D6
ρ10222-2-20-122-200000-1-111120-1-11-1-1    orthogonal lifted from D6
ρ1122-200222-200-20002-200000-22000    orthogonal lifted from D4
ρ1222-200-222-20020002-200000-22000    orthogonal lifted from D4
ρ13222220-122200000-1-1-1-1-120-1-1-1-1-1    orthogonal lifted from S3
ρ1422-2-2002-222000002-2-200002-2200    orthogonal lifted from D4
ρ152220002-2-2000-2202200000-2-2000    orthogonal lifted from D4
ρ162220002-2-20002-202200000-2-2000    orthogonal lifted from D4
ρ1722-22002-22-2000002-2200002-2-200    orthogonal lifted from D4
ρ18222-220-122-200000-1-11-1-1-20-1-1111    orthogonal lifted from D6
ρ1922-2200-1-22-200000-11-1√-3-√-300-111-√-3√-3    complex lifted from C3⋊D4
ρ2022-2200-1-22-200000-11-1-√-3√-300-111√-3-√-3    complex lifted from C3⋊D4
ρ2122-2-200-1-22200000-111√-3-√-300-11-1√-3-√-3    complex lifted from C3⋊D4
ρ2222-2-200-1-22200000-111-√-3√-300-11-1-√-3√-3    complex lifted from C3⋊D4
ρ23444000-2-4-4000000-2-20000022000    orthogonal lifted from S3×D4
ρ2444-4000-24-4000000-22000002-2000    orthogonal lifted from S3×D4
ρ254-400004000-2i0002i-400000000000    complex lifted from D4.8D4
ρ264-4000040002i000-2i-400000000000    complex lifted from D4.8D4
ρ278-80000-400000000400000000000    symplectic faithful, Schur index 2

Smallest permutation representation of D12.38D4
►On 48 points
Generators in S48
(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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

Matrix representation of D12.38D4 ►in GL8(𝔽73)

01000000
721000000
10110000
0727200000
0000275800
000004600
000000271
000000046
,
67675800000
360150000
606670000
0673670000
000014525829
000065591972
0000314146
000057451972
,
91215150000
360150000
06167610000
70070640000
00001470932
000065594759
000033310
000057451972
,
7012330000
646030000
06167610000
90930000
0000102511
000001072
00007040720
000002072

G:=sub<GL(8,GF(73))| [0,72,1,0,0,0,0,0,1,1,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,58,46,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,1,46],[67,3,6,0,0,0,0,0,67,6,0,67,0,0,0,0,58,0,6,3,0,0,0,0,0,15,67,67,0,0,0,0,0,0,0,0,14,65,3,57,0,0,0,0,52,59,14,45,0,0,0,0,58,19,1,19,0,0,0,0,29,72,46,72],[9,3,0,70,0,0,0,0,12,6,61,0,0,0,0,0,15,0,67,70,0,0,0,0,15,15,61,64,0,0,0,0,0,0,0,0,14,65,3,57,0,0,0,0,70,59,33,45,0,0,0,0,9,47,1,19,0,0,0,0,32,59,0,72],[70,64,0,9,0,0,0,0,12,6,61,0,0,0,0,0,3,0,67,9,0,0,0,0,3,3,61,3,0,0,0,0,0,0,0,0,1,0,70,0,0,0,0,0,0,1,40,2,0,0,0,0,25,0,72,0,0,0,0,0,11,72,0,72] >;
 

D12.38D4 in GAP, Magma, Sage, TeX 

D_{12}._{38}D_4
 
% in TeX
 
G:=Group("D12.38D4");
 
// GroupNames label
 
G:=SmallGroup(192,760);
 
// by ID
 
G=gap.SmallGroup(192,760);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,1123,297,136,851,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^4=d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Character table of D12.38D4 in TeX

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