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G = C24.74D6order 192 = 26·3

3rd non-split extension by C24 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.74D6, (C22×C12)⋊10C4, (C23×C12).3C2, (C23×C4).11S3, (C22×C4)⋊9Dic3, (C22×C4).423D6, (C22×C6).192D4, C6.C4223C2, C33(C23.34D4), C23.89(C3⋊D4), (C23×C6).98C22, C23.35(C2×Dic3), C6.48(C42⋊C2), C22.62(C4○D12), C23.312(C22×S3), (C22×C6).362C23, (C22×C12).483C22, C2.4(C23.28D6), C6.68(C22.D4), C22.49(C22×Dic3), C2.11(C23.26D6), C22.19(C6.D4), (C22×Dic3).65C22, (C2×C6).548(C2×D4), C6.69(C2×C22⋊C4), (C2×C12).281(C2×C4), (C2×C6).90(C4○D4), (C2×C4).66(C2×Dic3), C2.5(C2×C6.D4), C22.86(C2×C3⋊D4), (C22×C6).135(C2×C4), (C2×C6).192(C22×C4), (C2×C6).109(C22⋊C4), (C2×C6.D4).17C2, SmallGroup(192,770)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C24.74D6
C1C3C6C2×C6C22×C6C22×Dic3C2×C6.D4 — C24.74D6
C3C2×C6 — C24.74D6
C1C23C23×C4

Generators and relations for C24.74D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=bcd, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 472 in 218 conjugacy classes, 87 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×8], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×24], C23, C23 [×6], C23 [×4], Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×6], C22×C4 [×8], C24, C2×Dic3 [×12], C2×C12 [×4], C2×C12 [×12], C22×C6, C22×C6 [×6], C22×C6 [×4], C2.C42 [×4], C2×C22⋊C4 [×2], C23×C4, C6.D4 [×4], C22×Dic3 [×4], C22×C12 [×6], C22×C12 [×4], C23×C6, C23.34D4, C6.C42 [×4], C2×C6.D4 [×2], C23×C12, C24.74D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C6.D4 [×4], C4○D12 [×4], C22×Dic3, C2×C3⋊D4 [×2], C23.34D4, C23.26D6 [×2], C23.28D6 [×4], C2×C6.D4, C24.74D6

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

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

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

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

60 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A···4H4I···4P6A···6O12A···12P
order12···2222234···44···46···612···12
size11···1222222···212···122···22···2

60 irreducible representations

dim1111122222222
type++++++-++
imageC1C2C2C2C4S3D4Dic3D6D6C4○D4C3⋊D4C4○D12
kernelC24.74D6C6.C42C2×C6.D4C23×C12C22×C12C23×C4C22×C6C22×C4C22×C4C24C2×C6C23C22
# reps14218144218816

Matrix representation of C24.74D6 in GL5(𝔽13)

10000
012000
00100
00010
00001
,
120000
012000
001200
00010
00001
,
10000
012000
001200
000120
000012
,
10000
012000
001200
00010
00001
,
10000
05000
00800
00090
000010
,
80000
00500
05000
00003
00040

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,9,0,0,0,0,0,10],[8,0,0,0,0,0,0,5,0,0,0,5,0,0,0,0,0,0,0,4,0,0,0,3,0] >;

C24.74D6 in GAP, Magma, Sage, TeX

C_2^4._{74}D_6
% in TeX

G:=Group("C2^4.74D6");
// GroupNames label

G:=SmallGroup(192,770);
// by ID

G=gap.SmallGroup(192,770);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,422,184,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,f^2=b*c*d,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,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

׿
×
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