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G = C24.19D4order 192 = 26·3

19th non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.19D4, C8.21D12, D12.21D4, Dic6.21D4, M4(2).10D6, C8○D126C2, (C2×C8).71D6, (C2×D24)⋊21C2, C8⋊D610C2, C8.C46S3, C4.136(S3×D4), C4.57(C2×D12), C12.137(C2×D4), C32(D4.4D4), C12.46D43C2, C6.50(C4⋊D4), C2.23(C12⋊D4), (C2×C24).103C22, (C2×C12).313C23, C4○D12.40C22, (C2×D12).88C22, C22.7(Q83S3), (C3×M4(2)).7C22, C4.Dic3.38C22, (C3×C8.C4)⋊7C2, (C2×C6).4(C4○D4), (C2×C4).114(C22×S3), SmallGroup(192,456)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.19D4
C1C3C6C12C2×C12C4○D12C8○D12 — C24.19D4
C3C6C2×C12 — C24.19D4
C1C2C2×C4C8.C4

Generators and relations for C24.19D4
 G = < a,b,c | a24=c2=1, b4=a12, bab-1=a7, cac=a-1, cbc=a12b3 >

Subgroups: 416 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C22×S3, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, S3×C8, C8⋊S3, C24⋊C2, D24, C4.Dic3, C2×C24, C3×M4(2), C2×D12, C4○D12, D4.4D4, C12.46D4, C3×C8.C4, C8○D12, C2×D24, C8⋊D6, C24.19D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q83S3, D4.4D4, C12⋊D4, C24.19D4

Character table of C24.19D4

 class 12A2B2C2D2E34A4B4C6A6B8A8B8C8D8E8F8G12A12B12C24A24B24C24D24E24F24G24H
 size 112122424222122422488121222444448888
ρ1111111111111111111111111111111    trivial
ρ2111-1-1-1111-11111111-1-111111111111    linear of order 2
ρ3111-111111-111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ41111-1-1111111111-1-1111111111-1-1-1-1    linear of order 2
ρ511111-1111111-1-1-11-1-1-1111-1-1-1-111-1-1    linear of order 2
ρ6111-1-11111-111-1-1-11-111111-1-1-1-111-1-1    linear of order 2
ρ7111-11-1111-111-1-1-1-1111111-1-1-1-1-1-111    linear of order 2
ρ81111-11111111-1-1-1-11-1-1111-1-1-1-1-1-111    linear of order 2
ρ9222000-1220-1-1-2-2-2-2200-1-1-1111111-1-1    orthogonal lifted from D6
ρ10222000-1220-1-12222200-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122-2-2002-2222-2000000022-200000000    orthogonal lifted from D4
ρ1222-200022-202-2-2-220000-2-22-22-220000    orthogonal lifted from D4
ρ1322-22002-22-22-2000000022-200000000    orthogonal lifted from D4
ρ14222000-1220-1-1222-2-200-1-1-1-1-1-1-11111    orthogonal lifted from D6
ρ15222000-1220-1-1-2-2-22-200-1-1-11111-1-111    orthogonal lifted from D6
ρ1622-200022-202-222-20000-2-222-22-20000    orthogonal lifted from D4
ρ1722-2000-12-20-1122-2000011-1-11-11-33-33    orthogonal lifted from D12
ρ1822-2000-12-20-1122-2000011-1-11-113-33-3    orthogonal lifted from D12
ρ1922-2000-12-20-11-2-22000011-11-11-1-333-3    orthogonal lifted from D12
ρ2022-2000-12-20-11-2-22000011-11-11-13-3-33    orthogonal lifted from D12
ρ212220002-2-202200000-2i2i-2-2-200000000    complex lifted from C4○D4
ρ222220002-2-2022000002i-2i-2-2-200000000    complex lifted from C4○D4
ρ2344-4000-2-440-220000000-2-2200000000    orthogonal lifted from S3×D4
ρ24444000-2-4-40-2-2000000022200000000    orthogonal lifted from Q83S3, Schur index 2
ρ254-400004000-40-222200000000220-2200000    orthogonal lifted from D4.4D4
ρ264-400004000-4022-2200000000-2202200000    orthogonal lifted from D4.4D4
ρ274-40000-20002022-220000023-23026-2-60000    orthogonal faithful
ρ284-40000-200020-22220000023-230-2-6260000    orthogonal faithful
ρ294-40000-20002022-2200000-232302-6-260000    orthogonal faithful
ρ304-40000-200020-222200000-23230-262-60000    orthogonal faithful

Smallest permutation representation of C24.19D4
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 44 19 26 13 32 7 38)(2 27 20 33 14 39 8 45)(3 34 21 40 15 46 9 28)(4 41 22 47 16 29 10 35)(5 48 23 30 17 36 11 42)(6 31 24 37 18 43 12 25)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 24)(18 23)(19 22)(20 21)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(43 48)(44 47)(45 46)

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

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

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

Matrix representation of C24.19D4 in GL6(𝔽73)

9450000
0650000
00132500
00702800
000182850
004533513
,
27270000
0460000
000010
00046071
0021300
002510027
,
9450000
55640000
00132500
00406000
000182850
0044531545

G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,45,65,0,0,0,0,0,0,13,70,0,4,0,0,25,28,18,53,0,0,0,0,28,35,0,0,0,0,50,13],[27,0,0,0,0,0,27,46,0,0,0,0,0,0,0,0,21,25,0,0,0,46,3,10,0,0,1,0,0,0,0,0,0,71,0,27],[9,55,0,0,0,0,45,64,0,0,0,0,0,0,13,40,0,44,0,0,25,60,18,53,0,0,0,0,28,15,0,0,0,0,50,45] >;

C24.19D4 in GAP, Magma, Sage, TeX

C_{24}._{19}D_4
% in TeX

G:=Group("C24.19D4");
// GroupNames label

G:=SmallGroup(192,456);
// by ID

G=gap.SmallGroup(192,456);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,226,1123,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=a^12*b^3>;
// generators/relations

Export

Character table of C24.19D4 in TeX

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