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G = C24.3Dic3  order 192 = 26·3

1st non-split extension by C24 of Dic3 acting via Dic3/C3=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.3Dic3, C23⋊2(C3⋊C8), C3⋊2(C23⋊C8), (C22×C6)⋊2C8, (C23×C6).3C4, (C2×C12).227D4, (C22×C12).1C4, (C22×C4).19D6, C6.13(C22⋊C8), C6.16(C23⋊C4), C6.6(C4.D4), (C2×C6).22M4(2), C12.55D4⋊21C2, (C22×C4).4Dic3, C2.1(C12.D4), C23.25(C2×Dic3), C2.3(C12.55D4), C2.1(C23.7D6), C22.4(C4.Dic3), (C22×C12).324C22, C22.23(C6.D4), C22.2(C2×C3⋊C8), (C2×C6).29(C2×C8), (C2×C22⋊C4).1S3, (C6×C22⋊C4).21C2, (C2×C4).159(C3⋊D4), (C2×C6).85(C22⋊C4), (C22×C6).123(C2×C4), SmallGroup(192,84)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.3Dic3
Chief series
C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C12.55D4 — C24.3Dic3
Lower central
C3 — C6 — C2×C6 — C24.3Dic3
Upper central
C1 — C22 — C22×C4 — C2×C22⋊C4

Generators and relations for C24.3Dic3
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=ce3, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 248 in 98 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C22⋊C4, C2×C8, C22×C4, C24, C3⋊C8, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C22⋊C8, C2×C22⋊C4, C2×C3⋊C8, C3×C22⋊C4, C22×C12, C23×C6, C23⋊C8, C12.55D4, C6×C22⋊C4, C24.3Dic3
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C23⋊C4, C4.D4, C2×C3⋊C8, C4.Dic3, C6.D4, C23⋊C8, C12.55D4, C12.D4, C23.7D6, C24.3Dic3

Smallest permutation representation of C24.3Dic3
►On 48 points
Generators in S48
(1 7)(2 48)(3 9)(4 38)(5 11)(6 40)(8 42)(10 44)(12 46)(13 19)(14 27)(15 21)(16 29)(17 23)(18 31)(20 33)(22 35)(24 25)(26 32)(28 34)(30 36)(37 43)(39 45)(41 47)

(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F6A···6G6H6I6J6K8A···8H12A···12H
order1222222234444446···666668···812···12
size1111224422222442···2444412···124···4

42 irreducible representations

dim1111112222222224444
type+++++-+-++
imageC1C2C2C4C4C8S3D4Dic3D6Dic3M4(2)C3⋊D4C3⋊C8C4.Dic3C23⋊C4C4.D4C12.D4C23.7D6
kernelC24.3Dic3C12.55D4C6×C22⋊C4C22×C12C23×C6C22×C6C2×C22⋊C4C2×C12C22×C4C22×C4C24C2×C6C2×C4C23C22C6C6C2C2
# reps1212281211124441122

Matrix representation of C24.3Dic3 ►in GL8(𝔽73)

7272000000
01000000
00110000
000720000
000072000
00000100
000000720
00000001
,
720000000
072000000
007200000
000720000
00001000
00000100
000000720
000000072
,
10000000
01000000
007200000
000720000
000072000
000007200
000000720
000000072
,
10000000
01000000
00100000
00010000
000072000
000007200
000000720
000000072
,
6428000000
08000000
004600000
000460000
00000100
000072000
00000001
000000720
,
2321000000
2750000000
005100000
0044220000
00000010
00000001
00000100
000072000

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[64,0,0,0,0,0,0,0,28,8,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0],[23,27,0,0,0,0,0,0,21,50,0,0,0,0,0,0,0,0,51,44,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

C24.3Dic3 in GAP, Magma, Sage, TeX 

C_2^4._3{\rm Dic}_3
 
% in TeX
 
G:=Group("C2^4.3Dic3");
 
// GroupNames label
 
G:=SmallGroup(192,84);
 
// by ID
 
G=gap.SmallGroup(192,84);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,100,1123,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=c*e^3,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
 
// generators/relations
 

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