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G = C24.21D6order 192 = 26·3

10th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.21D6, C23.19D12, (C2×C12).51D4, (C22×C6).65D4, C6.58(C4⋊D4), C2.6(C127D4), (C22×C4).107D6, C6.C4215C2, C6.38(C4.4D4), C22.125(C2×D12), C34(C23.11D4), (C23×C6).36C22, C6.16(C422C2), C2.6(C23.12D6), C22.98(C4○D12), C23.380(C22×S3), (C22×C6).328C23, (C22×C12).60C22, C22.96(D42S3), C6.73(C22.D4), C2.8(C23.23D6), C2.14(C23.8D6), C2.16(C23.21D6), (C22×Dic3).42C22, (C2×C4⋊Dic3)⋊12C2, (C2×C6).432(C2×D4), (C2×C4).30(C3⋊D4), (C2×C22⋊C4).15S3, (C6×C22⋊C4).16C2, (C2×C6).144(C4○D4), C22.126(C2×C3⋊D4), (C2×C6.D4).15C2, SmallGroup(192,512)

Series: Derived Chief Lower central Upper central

C1C22×C6 — C24.21D6
C1C3C6C2×C6C22×C6C22×Dic3C2×C4⋊Dic3 — C24.21D6
C3C22×C6 — C24.21D6
C1C23C2×C22⋊C4

Generators and relations for C24.21D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 440 in 170 conjugacy classes, 59 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C3, C4 [×7], C22 [×3], C22 [×4], C22 [×10], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×2], C2×C4 [×17], C23, C23 [×2], C23 [×6], Dic3 [×4], C12 [×3], C2×C6 [×3], C2×C6 [×4], C2×C6 [×10], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, C2×Dic3 [×12], C2×C12 [×2], C2×C12 [×5], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42 [×3], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊Dic3 [×2], C6.D4 [×4], C3×C22⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23.11D4, C6.C42, C6.C42 [×2], C2×C4⋊Dic3, C2×C6.D4 [×2], C6×C22⋊C4, C24.21D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4 [×5], D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C2×D12, C4○D12, D42S3 [×4], C2×C3⋊D4, C23.11D4, C23.8D6 [×2], C23.21D6 [×2], C127D4, C23.23D6, C23.12D6, C24.21D6

Smallest permutation representation of C24.21D6
On 96 points
Generators in S96
(1 7)(2 55)(3 9)(4 57)(5 11)(6 59)(8 49)(10 51)(12 53)(13 19)(14 40)(15 21)(16 42)(17 23)(18 44)(20 46)(22 48)(24 38)(25 95)(26 63)(27 85)(28 65)(29 87)(30 67)(31 89)(32 69)(33 91)(34 71)(35 93)(36 61)(37 43)(39 45)(41 47)(50 56)(52 58)(54 60)(62 77)(64 79)(66 81)(68 83)(70 73)(72 75)(74 92)(76 94)(78 96)(80 86)(82 88)(84 90)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 68)(26 69)(27 70)(28 71)(29 72)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 60)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 60)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 73)(34 74)(35 75)(36 76)(61 94)(62 95)(63 96)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(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 60 33)(2 84 49 32)(3 83 50 31)(4 82 51 30)(5 81 52 29)(6 80 53 28)(7 79 54 27)(8 78 55 26)(9 77 56 25)(10 76 57 36)(11 75 58 35)(12 74 59 34)(13 89 45 68)(14 88 46 67)(15 87 47 66)(16 86 48 65)(17 85 37 64)(18 96 38 63)(19 95 39 62)(20 94 40 61)(21 93 41 72)(22 92 42 71)(23 91 43 70)(24 90 44 69)

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

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

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

42 conjugacy classes

class 1 2A···2G2H2I 3 4A4B4C4D4E···4L6A···6G6H6I6J6K12A···12H
order12···222344444···46···6666612···12
size11···1442444412···122···244444···4

42 irreducible representations

dim111112222222224
type+++++++++++-
imageC1C2C2C2C2S3D4D4D6D6C4○D4C3⋊D4D12C4○D12D42S3
kernelC24.21D6C6.C42C2×C4⋊Dic3C2×C6.D4C6×C22⋊C4C2×C22⋊C4C2×C12C22×C6C22×C4C24C2×C6C2×C4C23C22C22
# reps1312112221104444

Matrix representation of C24.21D6 in GL6(𝔽13)

1240000
010000
0012000
0010100
0000120
0000012
,
1200000
0120000
001000
000100
000010
000001
,
1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012000
0001200
000010
000001
,
560000
080000
0012500
0010100
000070
000092
,
4120000
290000
005100
000800
000029
0000411

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,4,1,0,0,0,0,0,0,12,10,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,6,8,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,7,9,0,0,0,0,0,2],[4,2,0,0,0,0,12,9,0,0,0,0,0,0,5,0,0,0,0,0,1,8,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;

C24.21D6 in GAP, Magma, Sage, TeX

C_2^4._{21}D_6
% in TeX

G:=Group("C2^4.21D6");
// GroupNames label

G:=SmallGroup(192,512);
// by ID

G=gap.SmallGroup(192,512);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,387,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;
// generators/relations

׿
×
𝔽