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G = C24:33D4order 192 = 26·3

5th semidirect product of C24 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24:33D4, D6:C8:3C2, C3:6(C8:9D4), D6:C4.3C4, C6.78(C4xD4), C8:15(C3:D4), Dic3:C8:3C2, (C22xC8):15S3, C24:C4:23C2, (C2xC8).294D6, (C2xC6):5M4(2), (C22xC24):19C2, C6.19(C8oD4), C12.437(C2xD4), Dic3:C4.3C4, C23.40(C4xS3), C22:3(C8:S3), C2.19(C8oD12), C6.D4.9C4, (C22xC4).418D6, C6.13(C2xM4(2)), C4.136(C4oD12), C12.252(C4oD4), C12.55D4:26C2, (C2xC24).355C22, (C2xC12).861C23, (C22xC12).561C22, (C4xDic3).187C22, (C2xC4).94(C4xS3), (C2xC3:D4).9C4, (C2xC8:S3):23C2, C2.23(C4xC3:D4), C2.15(C2xC8:S3), (C4xC3:D4).15C2, C4.127(C2xC3:D4), C22.142(S3xC2xC4), (C2xC12).210(C2xC4), (C2xC3:C8).205C22, (S3xC2xC4).184C22, (C22xC6).96(C2xC4), (C22xS3).24(C2xC4), (C2xC4).803(C22xS3), (C2xC6).131(C22xC4), (C2xDic3).32(C2xC4), SmallGroup(192,670)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C24:33D4
C1C3C6C12C2xC12S3xC2xC4C4xC3:D4 — C24:33D4
C3C2xC6 — C24:33D4
C1C2xC4C22xC8

Generators and relations for C24:33D4
 G = < a,b,c | a24=b4=c2=1, bab-1=cac=a5, cbc=b-1 >

Subgroups: 280 in 124 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C3:C8, C24, C24, C4xS3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xC6, C8:C4, C22:C8, C4:C8, C4xD4, C22xC8, C2xM4(2), C8:S3, C2xC3:C8, C4xDic3, Dic3:C4, D6:C4, C6.D4, C2xC24, C2xC24, S3xC2xC4, C2xC3:D4, C22xC12, C8:9D4, Dic3:C8, C24:C4, D6:C8, C12.55D4, C2xC8:S3, C4xC3:D4, C22xC24, C24:33D4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, M4(2), C22xC4, C2xD4, C4oD4, C4xS3, C3:D4, C22xS3, C4xD4, C2xM4(2), C8oD4, C8:S3, S3xC2xC4, C4oD12, C2xC3:D4, C8:9D4, C2xC8:S3, C8oD12, C4xC3:D4, C24:33D4

Smallest permutation representation of C24:33D4
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 36 95 66)(2 41 96 71)(3 46 73 52)(4 27 74 57)(5 32 75 62)(6 37 76 67)(7 42 77 72)(8 47 78 53)(9 28 79 58)(10 33 80 63)(11 38 81 68)(12 43 82 49)(13 48 83 54)(14 29 84 59)(15 34 85 64)(16 39 86 69)(17 44 87 50)(18 25 88 55)(19 30 89 60)(20 35 90 65)(21 40 91 70)(22 45 92 51)(23 26 93 56)(24 31 94 61)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 59)(26 64)(27 69)(28 50)(29 55)(30 60)(31 65)(32 70)(33 51)(34 56)(35 61)(36 66)(37 71)(38 52)(39 57)(40 62)(41 67)(42 72)(43 53)(44 58)(45 63)(46 68)(47 49)(48 54)(73 81)(74 86)(75 91)(76 96)(78 82)(79 87)(80 92)(84 88)(85 93)(90 94)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I6A···6G8A···8H8I8J8K8L12A···12H24A···24P
order122222234444444446···68···8888812···1224···24
size1111221221111221212122···22···2121212122···22···2

60 irreducible representations

dim1111111111112222222222222
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D4D6D6C4oD4M4(2)C3:D4C4xS3C4xS3C8oD4C4oD12C8:S3C8oD12
kernelC24:33D4Dic3:C8C24:C4D6:C8C12.55D4C2xC8:S3C4xC3:D4C22xC24Dic3:C4D6:C4C6.D4C2xC3:D4C22xC8C24C2xC8C22xC4C12C2xC6C8C2xC4C23C6C4C22C2
# reps1111111122221221244224488

Matrix representation of C24:33D4 in GL4(F73) generated by

72000
07200
00670
0033
,
1200
727200
001330
004360
,
1000
727200
0001
0010
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,6,3,0,0,70,3],[1,72,0,0,2,72,0,0,0,0,13,43,0,0,30,60],[1,72,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;

C24:33D4 in GAP, Magma, Sage, TeX

C_{24}\rtimes_{33}D_4
% in TeX

G:=Group("C24:33D4");
// GroupNames label

G:=SmallGroup(192,670);
// by ID

G=gap.SmallGroup(192,670);
# by ID

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

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

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