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G = C4.5D24order 192 = 26·3

5th non-split extension by C4 of D24 acting via D24/C24=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.5D24, C12.30D8, C12.26SD16, C42.261D6, (C4×C8)⋊5S3, (C4×C24)⋊5C2, C6.3(C2×D8), C2.5(C2×D24), C122Q82C2, (C2×C8).287D6, C2.D241C2, (C2×C4).80D12, C31(C4.4D8), C6.5(C2×SD16), C4.5(C24⋊C2), (C2×C12).377D4, C4⋊D12.2C2, C6.5(C4.4D4), (C2×D12).2C22, C22.90(C2×D12), C4⋊Dic3.5C22, C4.102(C4○D12), C12.218(C4○D4), (C2×C24).347C22, (C2×C12).723C23, (C4×C12).307C22, C2.10(C427S3), C2.8(C2×C24⋊C2), (C2×C6).106(C2×D4), (C2×C4).666(C22×S3), SmallGroup(192,253)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C4.5D24
C1C3C6C12C2×C12C2×D12C4⋊D12 — C4.5D24
C3C6C2×C12 — C4.5D24
C1C22C42C4×C8

Generators and relations for C4.5D24
 G = < a,b,c | a4=b24=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 472 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, D4⋊C4, C41D4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C4.4D8, C2.D24, C4×C24, C122Q8, C4⋊D12, C4.5D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, SD16, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C2×D8, C2×SD16, C24⋊C2, D24, C2×D12, C4○D12, C4.4D8, C427S3, C2×C24⋊C2, C2×D24, C4.5D24

Smallest permutation representation of C4.5D24
On 96 points
Generators in S96
(1 95 25 59)(2 96 26 60)(3 73 27 61)(4 74 28 62)(5 75 29 63)(6 76 30 64)(7 77 31 65)(8 78 32 66)(9 79 33 67)(10 80 34 68)(11 81 35 69)(12 82 36 70)(13 83 37 71)(14 84 38 72)(15 85 39 49)(16 86 40 50)(17 87 41 51)(18 88 42 52)(19 89 43 53)(20 90 44 54)(21 91 45 55)(22 92 46 56)(23 93 47 57)(24 94 48 58)
(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 48 25 24)(2 23 26 47)(3 46 27 22)(4 21 28 45)(5 44 29 20)(6 19 30 43)(7 42 31 18)(8 17 32 41)(9 40 33 16)(10 15 34 39)(11 38 35 14)(12 13 36 37)(49 68 85 80)(50 79 86 67)(51 66 87 78)(52 77 88 65)(53 64 89 76)(54 75 90 63)(55 62 91 74)(56 73 92 61)(57 60 93 96)(58 95 94 59)(69 72 81 84)(70 83 82 71)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A···4F4G4H6A6B6C8A···8H12A···12L24A···24P
order12222234···4446668···812···1224···24
size1111242422···224242222···22···22···2

54 irreducible representations

dim1111122222222222
type++++++++++++
imageC1C2C2C2C2S3D4D6D6D8SD16C4○D4D12C24⋊C2D24C4○D12
kernelC4.5D24C2.D24C4×C24C122Q8C4⋊D12C4×C8C2×C12C42C2×C8C12C12C12C2×C4C4C4C4
# reps1411112124444888

Matrix representation of C4.5D24 in GL4(𝔽73) generated by

665900
14700
0010
0001
,
623700
362500
006850
002318
,
623700
481100
006850
00555
G:=sub<GL(4,GF(73))| [66,14,0,0,59,7,0,0,0,0,1,0,0,0,0,1],[62,36,0,0,37,25,0,0,0,0,68,23,0,0,50,18],[62,48,0,0,37,11,0,0,0,0,68,55,0,0,50,5] >;

C4.5D24 in GAP, Magma, Sage, TeX

C_4._5D_{24}
% in TeX

G:=Group("C4.5D24");
// GroupNames label

G:=SmallGroup(192,253);
// by ID

G=gap.SmallGroup(192,253);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,142,1123,136,6278]);
// Polycyclic

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

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