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## G = C24⋊30D4order 192 = 26·3

### 2nd semidirect product of C24 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24⋊30D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×C24⋊C2 — C24⋊30D4
 Lower central C3 — C6 — C2×C12 — C24⋊30D4
 Upper central C1 — C22 — C22×C4 — C22×C8

Generators and relations for C2430D4
G = < a,b,c | a24=b4=c2=1, bab-1=cac=a11, cbc=b-1 >

Subgroups: 392 in 124 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C24⋊C2, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×C24, C2×C24, C2×Dic6, C2×D12, C2×C3⋊D4, C22×C12, C88D4, C2.Dic12, C8⋊Dic3, C2.D24, C2×C24⋊C2, C12.48D4, C127D4, C22×C24, C2430D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C4○D8, C24⋊C2, C2×D12, C4○D12, C2×C3⋊D4, C88D4, C2×C24⋊C2, C4○D24, C127D4, C2430D4

Smallest permutation representation of C2430D4
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 85 37 52)(2 96 38 63)(3 83 39 50)(4 94 40 61)(5 81 41 72)(6 92 42 59)(7 79 43 70)(8 90 44 57)(9 77 45 68)(10 88 46 55)(11 75 47 66)(12 86 48 53)(13 73 25 64)(14 84 26 51)(15 95 27 62)(16 82 28 49)(17 93 29 60)(18 80 30 71)(19 91 31 58)(20 78 32 69)(21 89 33 56)(22 76 34 67)(23 87 35 54)(24 74 36 65)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(26 36)(27 47)(28 34)(29 45)(30 32)(31 43)(33 41)(35 39)(38 48)(40 46)(42 44)(49 76)(50 87)(51 74)(52 85)(53 96)(54 83)(55 94)(56 81)(57 92)(58 79)(59 90)(60 77)(61 88)(62 75)(63 86)(64 73)(65 84)(66 95)(67 82)(68 93)(69 80)(70 91)(71 78)(72 89)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A ··· 6G 8A ··· 8H 12A ··· 12H 24A ··· 24P order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 24 2 2 2 2 2 24 24 24 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 C4○D4 SD16 C3⋊D4 D12 D12 C4○D8 C4○D12 C24⋊C2 C4○D24 kernel C24⋊30D4 C2.Dic12 C8⋊Dic3 C2.D24 C2×C24⋊C2 C12.48D4 C12⋊7D4 C22×C24 C22×C8 C24 C2×C12 C22×C6 C2×C8 C22×C4 C12 C2×C6 C8 C2×C4 C23 C6 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 4 4 2 2 4 4 8 8

Matrix representation of C2430D4 in GL4(𝔽73) generated by

 27 19 0 0 0 46 0 0 0 0 48 37 0 0 36 11
,
 72 2 0 0 72 1 0 0 0 0 72 0 0 0 1 1
,
 1 0 0 0 1 72 0 0 0 0 1 0 0 0 72 72
G:=sub<GL(4,GF(73))| [27,0,0,0,19,46,0,0,0,0,48,36,0,0,37,11],[72,72,0,0,2,1,0,0,0,0,72,1,0,0,0,1],[1,1,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;

C2430D4 in GAP, Magma, Sage, TeX

C_{24}\rtimes_{30}D_4
% in TeX

G:=Group("C24:30D4");
// GroupNames label

G:=SmallGroup(192,673);
// by ID

G=gap.SmallGroup(192,673);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=c*a*c=a^11,c*b*c=b^-1>;
// generators/relations

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