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G = C12.15C42  order 192 = 26·3

8th non-split extension by C12 of C42 acting via C42/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.15C42, C42.1Dic3, C12.24M4(2), C3⋊C16⋊3C4, C8.32(C4×S3), (C4×C12).1C4, (C2×C24).1C4, C3⋊1(C16⋊C4), C8⋊C4.4S3, C24.42(C2×C4), (C2×C8).150D6, (C2×C8).1Dic3, C6.2(C8⋊C4), C4.22(C4×Dic3), C12.C8.6C2, (C2×C6).20M4(2), C4.6(C4.Dic3), (C2×C24).260C22, C2.3(C42.S3), C22.3(C4.Dic3), (C3×C8⋊C4).5C2, (C2×C12).302(C2×C4), (C2×C4).70(C2×Dic3), SmallGroup(192,25)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C12.15C42
Chief series
C1 — C3 — C6 — C12 — C24 — C2×C24 — C12.C8 — C12.15C42
Lower central
C3 — C12 — C12.15C42
Upper central
C1 — C4 — C8⋊C4

Generators and relations for C12.15C42
 G = < a,b,c | a12=c4=1, b4=a9, bab-1=a5, ac=ca, cbc-1=a9b >

C1
C2
2C2
C3
C22
C4
C4
4C4
C6
2C6
C8
C2×C4
C8
2C8
2C2×C4
C12
C2×C6
C12
4C12
C2×C8
C2×C8
C42
3C16
3C16
3C16
3C16
C2×C12
C24
C24
2C2×C12
2C24
C8⋊C4
3M5(2)
3M5(2)
C2×C24
C3⋊C16
C3⋊C16
C2×C24
C4×C12
C3⋊C16
C3⋊C16
3C16⋊C4
C3×C8⋊C4
C12.C8
C12.C8
C12.15C42

Smallest permutation representation of C12.15C42
►On 48 points
Generators in S48
(1 29 36 13 25 48 9 21 44 5 17 40)(2 33 18 14 45 30 10 41 26 6 37 22)(3 31 38 15 27 34 11 23 46 7 19 42)(4 35 20 16 47 32 12 43 28 8 39 24)

(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)

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

42 conjugacy classes

class 1 2A2B 3 4A4B4C4D4E6A6B6C8A8B8C8D8E8F12A12B12C12D12E12F12G12H16A···16H24A···24H
order122344444666888888121212121212121216···1624···24
size1122112442222222442222444412···124···4

42 irreducible representations

dim11111122222222244
type++++--+
imageC1C2C2C4C4C4S3Dic3Dic3D6M4(2)M4(2)C4×S3C4.Dic3C4.Dic3C16⋊C4C12.15C42
kernelC12.15C42C12.C8C3×C8⋊C4C3⋊C16C4×C12C2×C24C8⋊C4C42C2×C8C2×C8C12C2×C6C8C4C22C3C1
# reps12182211112244424

Matrix representation of C12.15C42 ►in GL4(𝔽97) generated by

9103646
091091
00810
00081
,
87904095
90503363
59239683
53515558
,
155235
096065
002259
00075
G:=sub<GL(4,GF(97))| [91,0,0,0,0,91,0,0,36,0,81,0,46,91,0,81],[87,90,59,53,90,50,23,51,40,33,96,55,95,63,83,58],[1,0,0,0,55,96,0,0,2,0,22,0,35,65,59,75] >;
 

C12.15C42 in GAP, Magma, Sage, TeX 

C_{12}._{15}C_4^2
 
% in TeX
 
G:=Group("C12.15C4^2");
 
// GroupNames label
 
G:=SmallGroup(192,25);
 
// by ID
 
G=gap.SmallGroup(192,25);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,100,1123,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=c^4=1,b^4=a^9,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^9*b>;
 
// generators/relations
 

Export

Subgroup lattice of C12.15C42 in TeX

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