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G = C48.C4  order 192 = 26·3

1st non-split extension by C48 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C48.1C4, C4.18D24, C12.36D8, C24.14Q8, C16.1Dic3, C8.13Dic6, C22.1Dic12, (C2×C16).5S3, (C2×C48).7C2, (C2×C6).7Q16, C24.71(C2×C4), (C2×C8).311D6, (C2×C4).73D12, C12.25(C4⋊C4), C6.9(C2.D8), C3⋊2(C8.4Q8), (C2×C12).393D4, C8.16(C2×Dic3), C2.5(C24⋊1C4), C24.C4.1C2, C4.10(C4⋊Dic3), (C2×C24).383C22, SmallGroup(192,65)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C48.C4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C24.C4 — C48.C4
Lower central
C3 — C6 — C12 — C24 — C48.C4
Upper central
C1 — C4 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C48.C4
 G = < a,b | a48=1, b4=a24, bab-1=a23 >

C1
C2
2C2
C3
C4
C4
C22
C6
2C6
C8
C8
C2×C4
12C8
12C8
C12
C2×C6
C12
C16
C2×C8
C16
6M4(2)
6M4(2)
C24
C24
C2×C12
4C3⋊C8
4C3⋊C8
C2×C16
3C8.C4
3C8.C4
C2×C24
C48
C48
2C4.Dic3
2C4.Dic3
3C8.4Q8
C24.C4
C24.C4
C2×C48
C48.C4

Smallest permutation representation of C48.C4
►On 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(1 55 13 91 25 79 37 67)(2 78 14 66 26 54 38 90)(3 53 15 89 27 77 39 65)(4 76 16 64 28 52 40 88)(5 51 17 87 29 75 41 63)(6 74 18 62 30 50 42 86)(7 49 19 85 31 73 43 61)(8 72 20 60 32 96 44 84)(9 95 21 83 33 71 45 59)(10 70 22 58 34 94 46 82)(11 93 23 81 35 69 47 57)(12 68 24 56 36 92 48 80)
 
G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,55,13,91,25,79,37,67)(2,78,14,66,26,54,38,90)(3,53,15,89,27,77,39,65)(4,76,16,64,28,52,40,88)(5,51,17,87,29,75,41,63)(6,74,18,62,30,50,42,86)(7,49,19,85,31,73,43,61)(8,72,20,60,32,96,44,84)(9,95,21,83,33,71,45,59)(10,70,22,58,34,94,46,82)(11,93,23,81,35,69,47,57)(12,68,24,56,36,92,48,80)>;
 
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,55,13,91,25,79,37,67)(2,78,14,66,26,54,38,90)(3,53,15,89,27,77,39,65)(4,76,16,64,28,52,40,88)(5,51,17,87,29,75,41,63)(6,74,18,62,30,50,42,86)(7,49,19,85,31,73,43,61)(8,72,20,60,32,96,44,84)(9,95,21,83,33,71,45,59)(10,70,22,58,34,94,46,82)(11,93,23,81,35,69,47,57)(12,68,24,56,36,92,48,80) );
 
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,55,13,91,25,79,37,67),(2,78,14,66,26,54,38,90),(3,53,15,89,27,77,39,65),(4,76,16,64,28,52,40,88),(5,51,17,87,29,75,41,63),(6,74,18,62,30,50,42,86),(7,49,19,85,31,73,43,61),(8,72,20,60,32,96,44,84),(9,95,21,83,33,71,45,59),(10,70,22,58,34,94,46,82),(11,93,23,81,35,69,47,57),(12,68,24,56,36,92,48,80)]])
 

54 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D16A···16H24A···24H48A···48P
order1223444666888888881212121216···1624···2448···48
size112211222222222424242422222···22···22···2

54 irreducible representations

dim11112222222222222
type++++-+-++--++-
imageC1C2C2C4S3Q8D4Dic3D6D8Q16Dic6D12D24Dic12C8.4Q8C48.C4
kernelC48.C4C24.C4C2×C48C48C2×C16C24C2×C12C16C2×C8C12C2×C6C8C2×C4C4C22C3C1
# reps121411121222244816

Matrix representation of C48.C4 ►in GL2(𝔽97) generated by

110
044
,
01
750
G:=sub<GL(2,GF(97))| [11,0,0,44],[0,75,1,0] >;
 

C48.C4 in GAP, Magma, Sage, TeX 

C_{48}.C_4
 
% in TeX
 
G:=Group("C48.C4");
 
// GroupNames label
 
G:=SmallGroup(192,65);
 
// by ID
 
G=gap.SmallGroup(192,65);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,176,184,675,192,1684,102,6278]);
 
// Polycyclic
 
G:=Group<a,b|a^48=1,b^4=a^24,b*a*b^-1=a^23>;
 
// generators/relations
 

Export

Subgroup lattice of C48.C4 in TeX

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