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## G = C24⋊21D4order 192 = 26·3

### 21st semidirect product of C24 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24⋊21D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C4×C3⋊D4 — C24⋊21D4
 Lower central C3 — C2×C6 — C24⋊21D4
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C2421D4
G = < a,b,c | a24=b4=c2=1, bab-1=a17, cac=a5, cbc=b-1 >

Subgroups: 280 in 122 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, C3×M4(2), S3×C2×C4, C2×C3⋊D4, C22×C12, C86D4, C8×Dic3, Dic3⋊C8, D6⋊C8, C12.55D4, C2×C8⋊S3, C4×C3⋊D4, C6×M4(2), C2421D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×M4(2), C8○D4, S3×C2×C4, C4○D12, C2×C3⋊D4, C86D4, S3×M4(2), D12.C4, C4×C3⋊D4, C2421D4

Smallest permutation representation of C2421D4
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 35 62 96)(2 28 63 89)(3 45 64 82)(4 38 65 75)(5 31 66 92)(6 48 67 85)(7 41 68 78)(8 34 69 95)(9 27 70 88)(10 44 71 81)(11 37 72 74)(12 30 49 91)(13 47 50 84)(14 40 51 77)(15 33 52 94)(16 26 53 87)(17 43 54 80)(18 36 55 73)(19 29 56 90)(20 46 57 83)(21 39 58 76)(22 32 59 93)(23 25 60 86)(24 42 61 79)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 94)(26 75)(27 80)(28 85)(29 90)(30 95)(31 76)(32 81)(33 86)(34 91)(35 96)(36 77)(37 82)(38 87)(39 92)(40 73)(41 78)(42 83)(43 88)(44 93)(45 74)(46 79)(47 84)(48 89)(49 69)(51 55)(52 60)(53 65)(54 70)(57 61)(58 66)(59 71)(63 67)(64 72)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D4 D6 D6 M4(2) C4○D4 C3⋊D4 C4×S3 C4×S3 C8○D4 C4○D12 S3×M4(2) D12.C4 kernel C24⋊21D4 C8×Dic3 Dic3⋊C8 D6⋊C8 C12.55D4 C2×C8⋊S3 C4×C3⋊D4 C6×M4(2) Dic3⋊C4 D6⋊C4 C6.D4 C2×C3⋊D4 C2×M4(2) C24 C2×C8 C22×C4 Dic3 C12 C8 C2×C4 C23 C6 C4 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 1 4 2 4 2 2 4 4 2 2

Matrix representation of C2421D4 in GL6(𝔽73)

 0 46 0 0 0 0 27 46 0 0 0 0 0 0 46 3 0 0 0 0 40 27 0 0 0 0 0 0 72 48 0 0 0 0 42 1
,
 72 1 0 0 0 0 0 1 0 0 0 0 0 0 10 7 0 0 0 0 69 63 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 18 72 0 0 0 0 0 0 1 0 0 0 0 0 70 72

G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,46,46,0,0,0,0,0,0,46,40,0,0,0,0,3,27,0,0,0,0,0,0,72,42,0,0,0,0,48,1],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,10,69,0,0,0,0,7,63,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,18,0,0,0,0,0,72,0,0,0,0,0,0,1,70,0,0,0,0,0,72] >;

C2421D4 in GAP, Magma, Sage, TeX

C_{24}\rtimes_{21}D_4
% in TeX

G:=Group("C24:21D4");
// GroupNames label

G:=SmallGroup(192,687);
// by ID

G=gap.SmallGroup(192,687);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,758,387,58,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

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