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G = D2411C4order 192 = 26·3

The semidirect product of D24 and C4 acting through Inn(D24)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2411C4, C8.14D12, C24.74D4, Dic1211C4, C42.267D6, (C4×C8)⋊10S3, C31(C8○D8), C6.9(C4×D4), C24⋊C27C4, (C4×C24)⋊15C2, C8.22(C4×S3), C8○D1211C2, C24.52(C2×C4), (C2×C8).324D6, C4.76(C2×D12), C2.12(C4×D12), C4○D24.10C2, D12.13(C2×C4), C12.296(C2×D4), C24.C417C2, C424S315C2, Dic6.13(C2×C4), (C2×C24).426C22, C12.104(C22×C4), (C4×C12).328C22, (C2×C12).791C23, C4○D12.35C22, C22.20(C4○D12), C4.Dic3.33C22, C4.62(S3×C2×C4), (C2×C6).62(C4○D4), (C2×C4).672(C22×S3), SmallGroup(192,259)

Series: Derived Chief Lower central Upper central

C1C12 — D2411C4
C1C3C6C12C2×C12C4○D12C4○D24 — D2411C4
C3C6C12 — D2411C4
C1C8C2×C8C4×C8

Generators and relations for D2411C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, ac=ca, cbc-1=a18b >

Subgroups: 264 in 106 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], S3 [×2], C6, C6, C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], Dic3 [×2], C12 [×2], C12 [×2], 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 [×4], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C2×C12, C4×C8, 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○D8, C424S3 [×2], C24.C4, C4×C24, C8○D12 [×2], C4○D24, D2411C4
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○D8, C4×D12, D2411C4

Smallest permutation representation of D2411C4
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 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)
(1 7 13 19)(2 8 14 20)(3 9 15 21)(4 10 16 22)(5 11 17 23)(6 12 18 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40)], [(1,7,13,19),(2,8,14,20),(3,9,15,21),(4,10,16,22),(5,11,17,23),(6,12,18,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)])

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C···4G4H4I6A6B6C8A8B8C8D8E···8J8K8L8M8N12A···12L24A···24P
order122223444···44466688888···8888812···1224···24
size11212122112···2121222211112···2121212122···22···2

60 irreducible representations

dim1111111112222222222
type+++++++++++
imageC1C2C2C2C2C2C4C4C4S3D4D6D6C4○D4C4×S3D12C4○D12C8○D8D2411C4
kernelD2411C4C424S3C24.C4C4×C24C8○D12C4○D24C24⋊C2D24Dic12C4×C8C24C42C2×C8C2×C6C8C8C22C3C1
# reps12112142212122444816

Matrix representation of D2411C4 in GL2(𝔽73) generated by

170
043
,
043
170
,
460
072
G:=sub<GL(2,GF(73))| [17,0,0,43],[0,17,43,0],[46,0,0,72] >;

D2411C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes_{11}C_4
% in TeX

G:=Group("D24:11C4");
// GroupNames label

G:=SmallGroup(192,259);
// by ID

G=gap.SmallGroup(192,259);
# by ID

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

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^18*b>;
// generators/relations

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