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G = C48.C4order 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), C32(C8.4Q8), (C2×C12).393D4, C8.16(C2×Dic3), C2.5(C241C4), C24.C4.1C2, C4.10(C4⋊Dic3), (C2×C24).383C22, SmallGroup(192,65)

Series: Derived Chief Lower central Upper central

C1C24 — C48.C4
C1C3C6C12C2×C12C2×C24C24.C4 — C48.C4
C3C6C12C24 — C48.C4
C1C4C2×C4C2×C8C2×C16

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

2C2
2C6
12C8
12C8
6M4(2)
6M4(2)
4C3⋊C8
4C3⋊C8
3C8.C4
3C8.C4
2C4.Dic3
2C4.Dic3
3C8.4Q8

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

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

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

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

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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