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G = C24.56D6  order 192 = 26·3

3rd non-split extension by C24 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.56D6, C23.47D12, C23.55(C4×S3), (C22×C4).39D6, (C22×C6).61D4, (C22×Dic3)⋊8C4, C22.41(C2×D12), C6.C42⋊10C2, C22.22(D6⋊C4), C23.57(C3⋊D4), C3⋊2(C23.34D4), (C23×C6).26C22, (C23×Dic3).3C2, C6.25(C42⋊C2), (C22×C6).318C23, C23.286(C22×S3), (C22×C12).22C22, C22.42(D4⋊2S3), C2.3(C23.21D6), C2.1(C23.23D6), C6.70(C22.D4), C2.12(C23.16D6), (C22×Dic3).183C22, C2.7(C2×D6⋊C4), (C2×C6).149(C2×D4), (C2×C22⋊C4).5S3, (C6×C22⋊C4).6C2, C6.34(C2×C22⋊C4), C22.122(S3×C2×C4), (C22×C6).46(C2×C4), C22.46(C2×C3⋊D4), (C2×C6).139(C4○D4), (C2×C6).13(C22⋊C4), (C2×C6).104(C22×C4), (C2×Dic3).93(C2×C4), (C2×C6.D4).5C2, SmallGroup(192,502)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.56D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C23×Dic3 — C24.56D6
Lower central
C3 — C2×C6 — C24.56D6
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C24.56D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=bcd, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 520 in 218 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×C12, C23×C6, C23.34D4, C6.C42, C2×C6.D4, C6×C22⋊C4, C23×Dic3, C24.56D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C42⋊C2, C22.D4, D6⋊C4, S3×C2×C4, C2×D12, D4⋊2S3, C2×C3⋊D4, C23.34D4, C23.16D6, C23.21D6, C2×D6⋊C4, C23.23D6, C24.56D6

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

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E···4L4M4N4O4P6A···6G6H6I6J6K12A···12H
order12···22222344444···444446···6666612···12
size11···12222244446···6121212122···244444···4

48 irreducible representations

dim111111222222224
type++++++++++-
imageC1C2C2C2C2C4S3D4D6D6C4○D4C4×S3D12C3⋊D4D4⋊2S3
kernelC24.56D6C6.C42C2×C6.D4C6×C22⋊C4C23×Dic3C22×Dic3C2×C22⋊C4C22×C6C22×C4C24C2×C6C23C23C23C22
# reps141118142184444

Matrix representation of C24.56D6 ►in GL8(𝔽13)

10000000
812000000
00100000
00010000
000012000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000012000
000001200
00000010
00000001
,
120000000
012000000
001200000
000120000
00001000
00000100
00000010
00000001
,
120000000
012000000
00100000
00010000
000012000
000001200
00000010
00000001
,
811000000
125000000
00810000
00250000
00000100
000012000
00000001
000000121
,
1210000000
01000000
001260000
00410000
00000500
00008000
000000121
00000001

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

C24.56D6 in GAP, Magma, Sage, TeX 

C_2^4._{56}D_6
 
% in TeX
 
G:=Group("C2^4.56D6");
 
// GroupNames label
 
G:=SmallGroup(192,502);
 
// by ID
 
G=gap.SmallGroup(192,502);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,422,387,58,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*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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