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G = C24.76D6order 192 = 26·3

5th non-split extension by C24 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.76D6, C23.33D12, (C2×C12)⋊35D4, (C23×C4)⋊5S3, C6.85(C4×D4), (C23×C12)⋊1C2, C223(D6⋊C4), C6.70C22≀C2, C23.43(C4×S3), C2.5(C127D4), C6.80(C4⋊D4), C22.61(C2×D12), (C22×C6).193D4, (C22×C4).424D6, C6.C4225C2, C2.2(C244S3), C23.90(C3⋊D4), C34(C23.23D4), C22.64(C4○D12), (S3×C23).24C22, (C22×C6).364C23, (C23×C6).100C22, C23.314(C22×S3), (C22×C12).485C22, C6.69(C22.D4), C2.5(C23.28D6), (C22×Dic3).67C22, (C2×C3⋊D4)⋊7C4, (C2×D6⋊C4)⋊11C2, C2.36(C2×D6⋊C4), C2.29(C4×C3⋊D4), (C2×C4)⋊15(C3⋊D4), (C2×C6)⋊5(C22⋊C4), (C22×S3)⋊5(C2×C4), (C2×Dic3)⋊9(C2×C4), (C2×C6).550(C2×D4), C6.65(C2×C22⋊C4), C22.150(S3×C2×C4), (C2×C6).92(C4○D4), (C2×C6.D4)⋊7C2, (C22×C3⋊D4).7C2, C22.88(C2×C3⋊D4), (C2×C6).143(C22×C4), (C22×C6).100(C2×C4), SmallGroup(192,772)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C24.76D6
C1C3C6C2×C6C22×C6S3×C23C22×C3⋊D4 — C24.76D6
C3C2×C6 — C24.76D6
C1C23C23×C4

Generators and relations for C24.76D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=db=bd, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 760 in 286 conjugacy classes, 87 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C3, C4 [×8], C22 [×3], C22 [×8], C22 [×22], S3 [×2], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×4], C2×C4 [×22], D4 [×8], C23, C23 [×6], C23 [×12], Dic3 [×4], C12 [×4], D6 [×10], C2×C6 [×3], C2×C6 [×8], C2×C6 [×12], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, C2×Dic3 [×2], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×4], C2×C12 [×12], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2.C42 [×2], C2×C22⋊C4 [×3], C23×C4, C22×D4, D6⋊C4 [×4], C6.D4 [×2], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], C22×C12 [×6], S3×C23, C23×C6, C23.23D4, C6.C42 [×2], C2×D6⋊C4 [×2], C2×C6.D4, C22×C3⋊D4, C23×C12, C24.76D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×6], C22×S3, C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C4○D12 [×2], C2×C3⋊D4 [×3], C23.23D4, C2×D6⋊C4, C4×C3⋊D4 [×2], C23.28D6, C127D4 [×2], C244S3, C24.76D6

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

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

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

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

60 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M 3 4A···4H4I···4N6A···6O12A···12P
order12···222222234···44···46···612···12
size11···12222121222···212···122···22···2

60 irreducible representations

dim111111122222222222
type++++++++++++
imageC1C2C2C2C2C2C4S3D4D4D6D6C4○D4C3⋊D4C4×S3D12C3⋊D4C4○D12
kernelC24.76D6C6.C42C2×D6⋊C4C2×C6.D4C22×C3⋊D4C23×C12C2×C3⋊D4C23×C4C2×C12C22×C6C22×C4C24C2×C6C2×C4C23C23C23C22
# reps122111814421484448

Matrix representation of C24.76D6 in GL6(𝔽13)

100000
010000
00121100
000100
000010
000001
,
1200000
0120000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
0000120
0000012
,
550000
800000
008300
000500
0000112
0000119
,
550000
080000
0051000
008800
0000211
0000911

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,8,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,3,5,0,0,0,0,0,0,11,11,0,0,0,0,2,9],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,5,8,0,0,0,0,10,8,0,0,0,0,0,0,2,9,0,0,0,0,11,11] >;

C24.76D6 in GAP, Magma, Sage, TeX

C_2^4._{76}D_6
% in TeX

G:=Group("C2^4.76D6");
// GroupNames label

G:=SmallGroup(192,772);
// by ID

G=gap.SmallGroup(192,772);
# by ID

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

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

׿
×
𝔽