Copied to
clipboard

G = D248C4order 192 = 26·3

8th semidirect product of D24 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D248C4, C8.22D12, C24.38D4, Dic128C4, C12.3SD16, C8.1(C4×S3), C4.Q81S3, (C2×C6).30D8, (C2×C8).41D6, C31(D82C4), C24.21(C2×C4), C4.1(D6⋊C4), C4○D24.6C2, (C2×C12).88D4, C12.C84C2, C6.4(D4⋊C4), C12.1(C22⋊C4), (C2×C24).47C22, C22.8(D4⋊S3), C4.7(Q82S3), C2.6(C6.D8), (C3×C4.Q8)⋊1C2, (C2×C4).16(C3⋊D4), SmallGroup(192,47)

Series: Derived Chief Lower central Upper central

C1C24 — D248C4
C1C3C6C12C2×C12C2×C24C4○D24 — D248C4
C3C6C12C24 — D248C4
C1C2C2×C4C2×C8C4.Q8

Generators and relations for D248C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a15b >

Subgroups: 208 in 58 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×2], C22, C22, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8, Dic3, C12 [×2], C12, D6, C2×C6, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C4.Q8, M5(2), C4○D8, C3⋊C16, C24⋊C2, D24, Dic12, C3×C4⋊C4, C2×C24, C4○D12, D82C4, C12.C8, C3×C4.Q8, C4○D24, D248C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D6⋊C4, D4⋊S3, Q82S3, D82C4, C6.D8, D248C4

Character table of D248C4

 class 12A2B2C34A4B4C4D4E6A6B6C8A8B8C12A12B12C12D12E12F16A16B16C16D24A24B24C24D
 size 112242228824222224448888121212124444
ρ1111111111111111111111111111111    trivial
ρ2111-111111-1111111111111-1-1-1-11111    linear of order 2
ρ3111-1111-1-1-111111111-1-1-1-111111111    linear of order 2
ρ41111111-1-1111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ511-1111-1i-i-1-11-1-1-111-1ii-i-ii-ii-i1-11-1    linear of order 4
ρ611-1-111-1-ii1-11-1-1-111-1-i-iiii-ii-i1-11-1    linear of order 4
ρ711-1111-1-ii-1-11-1-1-111-1-i-iii-ii-ii1-11-1    linear of order 4
ρ811-1-111-1i-i1-11-1-1-111-1ii-i-i-ii-ii1-11-1    linear of order 4
ρ92220-122-2-20-1-1-1222-1-111110000-1-1-1-1    orthogonal lifted from D6
ρ102220-122220-1-1-1222-1-1-1-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ112220222000222-2-2-22200000000-2-2-2-2    orthogonal lifted from D4
ρ1222-2022-2000-22-222-22-200000000-22-22    orthogonal lifted from D4
ρ1322202-2-2000222000-2-20000-2-2220000    orthogonal lifted from D8
ρ1422202-2-2000222000-2-2000022-2-20000    orthogonal lifted from D8
ρ1522-20-12-20001-1122-2-113-33-300001-11-1    orthogonal lifted from D12
ρ1622-20-12-20001-1122-2-11-33-3300001-11-1    orthogonal lifted from D12
ρ1722-20-12-22i-2i01-11-2-22-11-i-iii0000-11-11    complex lifted from C4×S3
ρ1822-20-12-2-2i2i01-11-2-22-11ii-i-i0000-11-11    complex lifted from C4×S3
ρ1922-202-22000-22-2000-220000--2-2-2--20000    complex lifted from SD16
ρ2022-202-22000-22-2000-220000-2--2--2-20000    complex lifted from SD16
ρ212220-122000-1-1-1-2-2-2-1-1-3--3--3-300001111    complex lifted from C3⋊D4
ρ222220-122000-1-1-1-2-2-2-1-1--3-3-3--300001111    complex lifted from C3⋊D4
ρ234440-2-4-4000-2-2-200022000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ2444-40-2-440002-220002-2000000000000    orthogonal lifted from Q82S3
ρ254-4004000000-40-2-22-2000000000000-2-202-2    complex lifted from D82C4
ρ264-4004000000-402-2-2-20000000000002-20-2-2    complex lifted from D82C4
ρ274-400-200000-2-322-3-2-22-200000000000-6-26--2    complex faithful
ρ284-400-2000002-32-2-3-2-22-2000000000006-2-6--2    complex faithful
ρ294-400-2000002-32-2-32-2-2-200000000000-6--26-2    complex faithful
ρ304-400-200000-2-322-32-2-2-2000000000006--2-6-2    complex faithful

Smallest permutation representation of D248C4
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(2 20)(3 15)(4 10)(6 24)(7 19)(8 14)(11 23)(12 18)(16 22)(25 40 37 28)(26 35 38 47)(27 30 39 42)(29 44 41 32)(31 34 43 46)(33 48 45 36)

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

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

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

Matrix representation of D248C4 in GL6(𝔽97)

010000
96960000
00404000
00574000
00005740
00005757
,
27420000
15700000
00005740
00005757
00404000
00574000
,
7500000
0750000
001000
0009600
00004057
00005757

G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,96,0,0,0,0,0,0,40,57,0,0,0,0,40,40,0,0,0,0,0,0,57,57,0,0,0,0,40,57],[27,15,0,0,0,0,42,70,0,0,0,0,0,0,0,0,40,57,0,0,0,0,40,40,0,0,57,57,0,0,0,0,40,57,0,0],[75,0,0,0,0,0,0,75,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,40,57,0,0,0,0,57,57] >;

D248C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes_8C_4
% in TeX

G:=Group("D24:8C4");
// GroupNames label

G:=SmallGroup(192,47);
// by ID

G=gap.SmallGroup(192,47);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,675,794,192,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^15*b>;
// generators/relations

Export

Character table of D248C4 in TeX

׿
×
𝔽