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G = C3×C22.56C24  order 192 = 26·3

Direct product of C3 and C22.56C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C22.56C24, C6.1712+ 1+4, C6.1232- 1+4, C4⋊D4⋊18C6, C22⋊Q8⋊19C6, C4.4D4⋊16C6, C42.54(C2×C6), C42.C2⋊11C6, (C2×C6).382C24, (C2×C12).683C23, (C4×C12).295C22, (C6×D4).224C22, C22.D4⋊14C6, C22.56(C23×C6), C23.25(C22×C6), (C6×Q8).187C22, (C22×C6).108C23, C2.23(C3×2+ 1+4), C2.15(C3×2- 1+4), (C22×C12).462C22, C4⋊C4.34(C2×C6), (C3×C4⋊D4)⋊45C2, (C2×D4).37(C2×C6), C22⋊C4.7(C2×C6), (C3×C22⋊Q8)⋊46C2, (C2×Q8).31(C2×C6), (C3×C4.4D4)⋊36C2, (C2×C4).42(C22×C6), (C22×C4).78(C2×C6), (C3×C42.C2)⋊28C2, (C3×C4⋊C4).251C22, (C3×C22.D4)⋊33C2, (C3×C22⋊C4).92C22, SmallGroup(192,1451)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C22.56C24
Chief series
C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4.4D4 — C3×C22.56C24
Lower central
C1 — C22 — C3×C22.56C24
Upper central
C1 — C2×C6 — C3×C22.56C24

Generators and relations for C3×C22.56C24
 G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=g2=1, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Subgroups: 362 in 220 conjugacy classes, 142 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C22.56C24, C3×C4⋊D4, C3×C22⋊Q8, C3×C22.D4, C3×C4.4D4, C3×C42.C2, C3×C22.56C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2+ 1+4, 2- 1+4, C23×C6, C22.56C24, C3×2+ 1+4, C3×2- 1+4, C3×C22.56C24

Smallest permutation representation of C3×C22.56C24
►On 96 points
Generators in S96
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 26 21)(10 27 22)(11 28 23)(12 25 24)(13 94 84)(14 95 81)(15 96 82)(16 93 83)(29 40 44)(30 37 41)(31 38 42)(32 39 43)(33 51 48)(34 52 45)(35 49 46)(36 50 47)(53 62 66)(54 63 67)(55 64 68)(56 61 65)(57 76 70)(58 73 71)(59 74 72)(60 75 69)(77 88 91)(78 85 92)(79 86 89)(80 87 90)

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

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

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

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

(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)

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

57 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A···4K6A···6F6G···6N12A···12V
order12222222334···46···66···612···12
size11114444114···41···14···44···4

57 irreducible representations

dim1111111111114444
type+++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C62+ 1+42- 1+4C3×2+ 1+4C3×2- 1+4
kernelC3×C22.56C24C3×C4⋊D4C3×C22⋊Q8C3×C22.D4C3×C4.4D4C3×C42.C2C22.56C24C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C6C6C2C2
# reps1444212888422142

Matrix representation of C3×C22.56C24 ►in GL9(𝔽13)

300000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
0000012000
0000001200
0000000120
0000000012
,
100000000
0120000000
0012000000
0001200000
0000120000
0000012000
0000001200
0000000120
0000000012
,
100000000
000100000
000010000
010000000
001000000
000006200
000002700
0000000711
0000000116
,
100000000
001000000
010000000
000010000
000100000
000000010
000000001
000001000
000000100
,
1200000000
0120000000
001000000
000100000
0000120000
000000100
0000012000
000000001
0000000120
,
1200000000
010000000
001000000
0001200000
0000120000
000001000
000000100
0000000120
0000000012

G:=sub<GL(9,GF(13))| [3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,0,11,6],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12] >;
 

C3×C22.56C24 in GAP, Magma, Sage, TeX 

C_3\times C_2^2._{56}C_2^4
 
% in TeX
 
G:=Group("C3xC2^2.56C2^4");
 
// GroupNames label
 
G:=SmallGroup(192,1451);
 
// by ID
 
G=gap.SmallGroup(192,1451);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,1563,268,4259,794]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=e^2=g^2=1,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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