direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.37C23, C4⋊Q8⋊20C6, (C4×Q8)⋊12C6, (C2×C12)⋊16Q8, C4.10(C6×Q8), (Q8×C12)⋊28C2, C12.99(C2×Q8), C22.3(C6×Q8), (C2×C42).23C6, C42.35(C2×C6), C42.C2⋊14C6, C22⋊Q8.10C6, C6.60(C22×Q8), (C2×C6).353C24, C12.275(C4○D4), C42⋊C2.11C6, (C4×C12).376C22, (C2×C12).662C23, C23.39(C22×C6), C22.27(C23×C6), (C6×Q8).269C22, (C22×C6).260C23, (C22×C12).598C22, C2.6(Q8×C2×C6), (C2×C4)⋊5(C3×Q8), (C2×C4×C12).43C2, (C3×C4⋊Q8)⋊41C2, C4⋊C4.67(C2×C6), C4.19(C3×C4○D4), C2.14(C6×C4○D4), (C2×C6).16(C2×Q8), C6.233(C2×C4○D4), (C2×Q8).68(C2×C6), C22⋊C4.14(C2×C6), (C2×C4).20(C22×C6), (C3×C42.C2)⋊31C2, (C3×C22⋊Q8).20C2, (C3×C4⋊C4).390C22, (C22×C4).132(C2×C6), (C3×C42⋊C2).25C2, (C3×C22⋊C4).148C22, SmallGroup(192,1422)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.37C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >
Subgroups: 274 in 222 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×Q8, C23.37C23, C2×C4×C12, C3×C42⋊C2, Q8×C12, C3×C22⋊Q8, C3×C42.C2, C3×C4⋊Q8, C3×C23.37C23
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C4○D4, C24, C3×Q8, C22×C6, C22×Q8, C2×C4○D4, C6×Q8, C3×C4○D4, C23×C6, C23.37C23, Q8×C2×C6, C6×C4○D4, C3×C23.37C23
►On 96 points
Generators in S96
(1 59 11)(2 60 12)(3 57 9)(4 58 10)(5 54 26)(6 55 27)(7 56 28)(8 53 25)(13 17 61)(14 18 62)(15 19 63)(16 20 64)(21 65 69)(22 66 70)(23 67 71)(24 68 72)(29 73 77)(30 74 78)(31 75 79)(32 76 80)(33 37 81)(34 38 82)(35 39 83)(36 40 84)(41 85 89)(42 86 90)(43 87 91)(44 88 92)(45 51 93)(46 52 94)(47 49 95)(48 50 96)
(1 3)(2 74)(4 76)(5 50)(6 8)(7 52)(9 11)(10 32)(12 30)(13 15)(14 36)(16 34)(17 19)(18 40)(20 38)(21 23)(22 44)(24 42)(25 27)(26 48)(28 46)(29 31)(33 35)(37 39)(41 43)(45 47)(49 51)(53 55)(54 96)(56 94)(57 59)(58 80)(60 78)(61 63)(62 84)(64 82)(65 67)(66 88)(68 86)(69 71)(70 92)(72 90)(73 75)(77 79)(81 83)(85 87)(89 91)(93 95)
(1 75)(2 76)(3 73)(4 74)(5 52)(6 49)(7 50)(8 51)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(53 93)(54 94)(55 95)(56 96)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)
(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 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 65 3 67)(2 68 4 66)(5 20 7 18)(6 19 8 17)(9 23 11 21)(10 22 12 24)(13 27 15 25)(14 26 16 28)(29 43 31 41)(30 42 32 44)(33 47 35 45)(34 46 36 48)(37 49 39 51)(38 52 40 50)(53 61 55 63)(54 64 56 62)(57 71 59 69)(58 70 60 72)(73 87 75 85)(74 86 76 88)(77 91 79 89)(78 90 80 92)(81 95 83 93)(82 94 84 96)
(1 35 75 15)(2 36 76 16)(3 33 73 13)(4 34 74 14)(5 70 52 90)(6 71 49 91)(7 72 50 92)(8 69 51 89)(9 81 29 61)(10 82 30 62)(11 83 31 63)(12 84 32 64)(17 57 37 77)(18 58 38 78)(19 59 39 79)(20 60 40 80)(21 93 41 53)(22 94 42 54)(23 95 43 55)(24 96 44 56)(25 65 45 85)(26 66 46 86)(27 67 47 87)(28 68 48 88)
G:=sub<Sym(96)| (1,59,11)(2,60,12)(3,57,9)(4,58,10)(5,54,26)(6,55,27)(7,56,28)(8,53,25)(13,17,61)(14,18,62)(15,19,63)(16,20,64)(21,65,69)(22,66,70)(23,67,71)(24,68,72)(29,73,77)(30,74,78)(31,75,79)(32,76,80)(33,37,81)(34,38,82)(35,39,83)(36,40,84)(41,85,89)(42,86,90)(43,87,91)(44,88,92)(45,51,93)(46,52,94)(47,49,95)(48,50,96), (1,3)(2,74)(4,76)(5,50)(6,8)(7,52)(9,11)(10,32)(12,30)(13,15)(14,36)(16,34)(17,19)(18,40)(20,38)(21,23)(22,44)(24,42)(25,27)(26,48)(28,46)(29,31)(33,35)(37,39)(41,43)(45,47)(49,51)(53,55)(54,96)(56,94)(57,59)(58,80)(60,78)(61,63)(62,84)(64,82)(65,67)(66,88)(68,86)(69,71)(70,92)(72,90)(73,75)(77,79)(81,83)(85,87)(89,91)(93,95), (1,75)(2,76)(3,73)(4,74)(5,52)(6,49)(7,50)(8,51)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(53,93)(54,94)(55,95)(56,96)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92), (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), (1,2,3,4)(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), (1,65,3,67)(2,68,4,66)(5,20,7,18)(6,19,8,17)(9,23,11,21)(10,22,12,24)(13,27,15,25)(14,26,16,28)(29,43,31,41)(30,42,32,44)(33,47,35,45)(34,46,36,48)(37,49,39,51)(38,52,40,50)(53,61,55,63)(54,64,56,62)(57,71,59,69)(58,70,60,72)(73,87,75,85)(74,86,76,88)(77,91,79,89)(78,90,80,92)(81,95,83,93)(82,94,84,96), (1,35,75,15)(2,36,76,16)(3,33,73,13)(4,34,74,14)(5,70,52,90)(6,71,49,91)(7,72,50,92)(8,69,51,89)(9,81,29,61)(10,82,30,62)(11,83,31,63)(12,84,32,64)(17,57,37,77)(18,58,38,78)(19,59,39,79)(20,60,40,80)(21,93,41,53)(22,94,42,54)(23,95,43,55)(24,96,44,56)(25,65,45,85)(26,66,46,86)(27,67,47,87)(28,68,48,88)>;
G:=Group( (1,59,11)(2,60,12)(3,57,9)(4,58,10)(5,54,26)(6,55,27)(7,56,28)(8,53,25)(13,17,61)(14,18,62)(15,19,63)(16,20,64)(21,65,69)(22,66,70)(23,67,71)(24,68,72)(29,73,77)(30,74,78)(31,75,79)(32,76,80)(33,37,81)(34,38,82)(35,39,83)(36,40,84)(41,85,89)(42,86,90)(43,87,91)(44,88,92)(45,51,93)(46,52,94)(47,49,95)(48,50,96), (1,3)(2,74)(4,76)(5,50)(6,8)(7,52)(9,11)(10,32)(12,30)(13,15)(14,36)(16,34)(17,19)(18,40)(20,38)(21,23)(22,44)(24,42)(25,27)(26,48)(28,46)(29,31)(33,35)(37,39)(41,43)(45,47)(49,51)(53,55)(54,96)(56,94)(57,59)(58,80)(60,78)(61,63)(62,84)(64,82)(65,67)(66,88)(68,86)(69,71)(70,92)(72,90)(73,75)(77,79)(81,83)(85,87)(89,91)(93,95), (1,75)(2,76)(3,73)(4,74)(5,52)(6,49)(7,50)(8,51)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(53,93)(54,94)(55,95)(56,96)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92), (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), (1,2,3,4)(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), (1,65,3,67)(2,68,4,66)(5,20,7,18)(6,19,8,17)(9,23,11,21)(10,22,12,24)(13,27,15,25)(14,26,16,28)(29,43,31,41)(30,42,32,44)(33,47,35,45)(34,46,36,48)(37,49,39,51)(38,52,40,50)(53,61,55,63)(54,64,56,62)(57,71,59,69)(58,70,60,72)(73,87,75,85)(74,86,76,88)(77,91,79,89)(78,90,80,92)(81,95,83,93)(82,94,84,96), (1,35,75,15)(2,36,76,16)(3,33,73,13)(4,34,74,14)(5,70,52,90)(6,71,49,91)(7,72,50,92)(8,69,51,89)(9,81,29,61)(10,82,30,62)(11,83,31,63)(12,84,32,64)(17,57,37,77)(18,58,38,78)(19,59,39,79)(20,60,40,80)(21,93,41,53)(22,94,42,54)(23,95,43,55)(24,96,44,56)(25,65,45,85)(26,66,46,86)(27,67,47,87)(28,68,48,88) );
G=PermutationGroup([[(1,59,11),(2,60,12),(3,57,9),(4,58,10),(5,54,26),(6,55,27),(7,56,28),(8,53,25),(13,17,61),(14,18,62),(15,19,63),(16,20,64),(21,65,69),(22,66,70),(23,67,71),(24,68,72),(29,73,77),(30,74,78),(31,75,79),(32,76,80),(33,37,81),(34,38,82),(35,39,83),(36,40,84),(41,85,89),(42,86,90),(43,87,91),(44,88,92),(45,51,93),(46,52,94),(47,49,95),(48,50,96)], [(1,3),(2,74),(4,76),(5,50),(6,8),(7,52),(9,11),(10,32),(12,30),(13,15),(14,36),(16,34),(17,19),(18,40),(20,38),(21,23),(22,44),(24,42),(25,27),(26,48),(28,46),(29,31),(33,35),(37,39),(41,43),(45,47),(49,51),(53,55),(54,96),(56,94),(57,59),(58,80),(60,78),(61,63),(62,84),(64,82),(65,67),(66,88),(68,86),(69,71),(70,92),(72,90),(73,75),(77,79),(81,83),(85,87),(89,91),(93,95)], [(1,75),(2,76),(3,73),(4,74),(5,52),(6,49),(7,50),(8,51),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(53,93),(54,94),(55,95),(56,96),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92)], [(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)], [(1,2,3,4),(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)], [(1,65,3,67),(2,68,4,66),(5,20,7,18),(6,19,8,17),(9,23,11,21),(10,22,12,24),(13,27,15,25),(14,26,16,28),(29,43,31,41),(30,42,32,44),(33,47,35,45),(34,46,36,48),(37,49,39,51),(38,52,40,50),(53,61,55,63),(54,64,56,62),(57,71,59,69),(58,70,60,72),(73,87,75,85),(74,86,76,88),(77,91,79,89),(78,90,80,92),(81,95,83,93),(82,94,84,96)], [(1,35,75,15),(2,36,76,16),(3,33,73,13),(4,34,74,14),(5,70,52,90),(6,71,49,91),(7,72,50,92),(8,69,51,89),(9,81,29,61),(10,82,30,62),(11,83,31,63),(12,84,32,64),(17,57,37,77),(18,58,38,78),(19,59,39,79),(20,60,40,80),(21,93,41,53),(22,94,42,54),(23,95,43,55),(24,96,44,56),(25,65,45,85),(26,66,46,86),(27,67,47,87),(28,68,48,88)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12H | 12I | ··· | 12AB | 12AC | ··· | 12AR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | Q8 | C4○D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C23.37C23 | C2×C4×C12 | C3×C42⋊C2 | Q8×C12 | C3×C22⋊Q8 | C3×C42.C2 | C3×C4⋊Q8 | C23.37C23 | C2×C42 | C42⋊C2 | C4×Q8 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C2×C12 | C12 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C23.37C23 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 9 | 0 | 0 |
| 9 | 10 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 1 | 5 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,12,0,0,0,0,1,5,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[3,9,0,0,9,10,0,0,0,0,8,1,0,0,2,5],[0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,5] >;
C3×C23.37C23 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{37}C_2^3 % in TeX
G:=Group("C3xC2^3.37C2^3"); // GroupNames label
G:=SmallGroup(192,1422);
// by ID
G=gap.SmallGroup(192,1422);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,2102,520,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=f^2=d,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
// generators/relations