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G = D242C4order 192 = 26·3

2nd semidirect product of D24 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D242C4, C24.86D4, Dic122C4, M5(2)⋊6S3, C12.7SD16, C22.3D24, C8.6(C4×S3), (C2×C6).2D8, C24.3(C2×C4), C8⋊Dic31C2, (C2×C8).48D6, C32(D82C4), C4○D24.7C2, (C2×C4).11D12, C4.20(D6⋊C4), (C2×C12).101D4, C8.43(C3⋊D4), C4.12(C24⋊C2), (C3×M5(2))⋊10C2, (C2×C24).52C22, C6.19(D4⋊C4), C12.44(C22⋊C4), C2.11(C2.D24), SmallGroup(192,77)

Series: Derived Chief Lower central Upper central

C1C24 — D242C4
C1C3C6C12C24C2×C24C4○D24 — D242C4
C3C6C12C24 — D242C4
C1C2C2×C4C2×C8M5(2)

Generators and relations for D242C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a7b >

Subgroups: 232 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 [×2], C12 [×2], D6, C2×C6, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C24 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C4.Q8, M5(2), C4○D8, C48, C24⋊C2, D24, Dic12, C4⋊Dic3, C2×C24, C4○D12, D82C4, C8⋊Dic3, C3×M5(2), C4○D24, D242C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, C24⋊C2, D24, D6⋊C4, D82C4, C2.D24, D242C4

Smallest permutation representation of D242C4
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 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 48)(23 47)(24 46)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 46 37 34)(26 33 38 45)(27 44 39 32)(28 31 40 43)(29 42 41 30)(35 36 47 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)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,48)(23,47)(24,46), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,46,37,34)(26,33,38,45)(27,44,39,32)(28,31,40,43)(29,42,41,30)(35,36,47,48)>;

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

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

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E6A6B8A8B8C12A12B12C16A16B16C16D24A24B24C24D24E24F48A···48H
order1222344444668881212121616161624242424242448···48
size112242222424242422422444442222444···4

36 irreducible representations

dim1111112222222222244
type+++++++++++
imageC1C2C2C2C4C4S3D4D4D6SD16D8C4×S3C3⋊D4D12C24⋊C2D24D82C4D242C4
kernelD242C4C8⋊Dic3C3×M5(2)C4○D24D24Dic12M5(2)C24C2×C12C2×C8C12C2×C6C8C8C2×C4C4C22C3C1
# reps1111221111222224424

Matrix representation of D242C4 in GL6(𝔽97)

010000
96960000
00177400
0059000
0081854040
0089855740
,
0960000
9600000
007302323
0080017
0055571212
0017401212
,
2200000
75750000
001000
00649600
007605757
006805740

G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,96,0,0,0,0,0,0,17,59,81,89,0,0,74,0,85,85,0,0,0,0,40,57,0,0,0,0,40,40],[0,96,0,0,0,0,96,0,0,0,0,0,0,0,73,8,55,17,0,0,0,0,57,40,0,0,23,0,12,12,0,0,23,17,12,12],[22,75,0,0,0,0,0,75,0,0,0,0,0,0,1,64,76,68,0,0,0,96,0,0,0,0,0,0,57,57,0,0,0,0,57,40] >;

D242C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes_2C_4
% in TeX

G:=Group("D24:2C4");
// GroupNames label

G:=SmallGroup(192,77);
// by ID

G=gap.SmallGroup(192,77);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,387,268,570,136,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^7*b>;
// generators/relations

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