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G = D12.38D4order 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⋊C223S3, Q8○D122C2, C4○D4.22D6, (C2×D4).81D6, (C3×D4).13D4, C4.105(S3×D4), (C3×Q8).13D4, D12⋊C46C2, C6.64C22≀C2, D126C226C2, C12.197(C2×D4), C34(D4.8D4), (C2×Dic3).5D4, C22.36(S3×D4), Q83Dic37C2, C23.12D67C2, C12.47D46C2, D4.10(C3⋊D4), (C2×C12).16C23, Q8.17(C3⋊D4), C2.32(C232D6), 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

C1C2×C12 — D12.38D4
C1C3C6C12C2×C12C4○D12Q8○D12 — D12.38D4
C3C6C2×C12 — D12.38D4
C1C2C2×C4C8⋊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 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×5], S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, Dic3 [×4], C12 [×2], C12, D6, C2×C6, C2×C6 [×4], C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6 [×4], C4×S3 [×3], D12, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×3], C2×C12, C2×C12, C3×D4, C3×D4 [×3], C3×Q8, C22×C6, C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22, C8⋊C22, 2- 1+4, C4.Dic3, C4×Dic3, D4⋊S3, D4.S3, C6.D4 [×2], C3×M4(2), C3×D8, C3×SD16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3 [×3], S3×Q8, C6×D4, C3×C4○D4, D4.8D4, C12.47D4, D12⋊C4, Q83Dic3, D126C22, C23.12D6, C3×C8⋊C22, Q8○D12, D12.38D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, D4.8D4, C232D6, 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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 37 10 40 7 43 4 46)(2 44 11 47 8 38 5 41)(3 39 12 42 9 45 6 48)(13 25 22 28 19 31 16 34)(14 32 23 35 20 26 17 29)(15 27 24 30 21 33 18 36)
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 48)

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

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

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

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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