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G = D244C4order 192 = 26·3

4th semidirect product of D24 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D244C4, C8.27D12, C24.53D4, Dic124C4, C42.23D6, C8.8(C4×S3), C24⋊C22C4, C8⋊C45S3, C24.5(C2×C4), C6.14(C4×D4), C8○D1212C2, C31(C8.26D4), C4○D24.8C2, C4.78(C2×D12), C2.17(C4×D12), (C2×C8).161D6, C424S31C2, D12.15(C2×C4), C12.298(C2×D4), C24.C412C2, (C4×C12).17C22, Dic6.15(C2×C4), C12.108(C22×C4), (C2×C12).792C23, (C2×C24).272C22, C4○D12.36C22, C22.21(C4○D12), C4.Dic3.34C22, C4.66(S3×C2×C4), (C3×C8⋊C4)⋊1C2, (C2×C6).63(C4○D4), (C2×C4).682(C22×S3), SmallGroup(192,276)

Series: Derived Chief Lower central Upper central

C1C12 — D244C4
C1C3C6C12C2×C12C4○D12C4○D24 — D244C4
C3C6C12 — D244C4
C1C4C2×C8C8⋊C4

Generators and relations for D244C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a6b >

Subgroups: 264 in 104 conjugacy classes, 47 normal (29 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], Dic3 [×2], C12 [×2], C12, D6 [×2], C2×C6, C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], C3⋊C8 [×2], C24 [×2], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C2×C12, C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24, Dic12, C4.Dic3 [×2], C4×C12, C2×C24 [×2], C4○D12 [×2], C8.26D4, C424S3 [×2], C24.C4, C3×C8⋊C4, C8○D12 [×2], C4○D24, D244C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], D12 [×2], C22×S3, C4×D4, S3×C2×C4, C2×D12, C4○D12, C8.26D4, C4×D12, D244C4

Smallest permutation representation of D244C4
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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)
(1 7 13 19)(2 20 14 8)(3 9 15 21)(4 22 16 10)(5 11 17 23)(6 24 18 12)(26 38)(28 40)(30 42)(32 44)(34 46)(36 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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34), (1,7,13,19)(2,20,14,8)(3,9,15,21)(4,22,16,10)(5,11,17,23)(6,24,18,12)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48)>;

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

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

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G6A6B6C8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E12F12G12H24A···24H
order12222344444446668888888888121212121212121224···24
size1121212211244121222222224412121212222244444···4

42 irreducible representations

dim1111111112222222244
type+++++++++++
imageC1C2C2C2C2C2C4C4C4S3D4D6D6C4○D4C4×S3D12C4○D12C8.26D4D244C4
kernelD244C4C424S3C24.C4C3×C8⋊C4C8○D12C4○D24C24⋊C2D24Dic12C8⋊C4C24C42C2×C8C2×C6C8C8C22C3C1
# reps1211214221212244424

Matrix representation of D244C4 in GL4(𝔽73) generated by

704200
67300
006453
00559
,
006453
00559
704200
67300
,
462700
02700
00721
0001
G:=sub<GL(4,GF(73))| [70,67,0,0,42,3,0,0,0,0,64,55,0,0,53,9],[0,0,70,67,0,0,42,3,64,55,0,0,53,9,0,0],[46,0,0,0,27,27,0,0,0,0,72,0,0,0,1,1] >;

D244C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes_4C_4
% in TeX

G:=Group("D24:4C4");
// GroupNames label

G:=SmallGroup(192,276);
// by ID

G=gap.SmallGroup(192,276);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,387,58,136,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^6*b>;
// generators/relations

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