direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.24D4, C4○D4⋊3C12, (C22×C8)⋊8C6, C4.53(C6×D4), (C22×C24)⋊8C2, D4.5(C2×C12), D4⋊C4⋊20C6, C42⋊C2⋊2C6, Q8⋊C4⋊20C6, C12.460(C2×D4), (C2×C12).518D4, C4.3(C22×C12), Q8.10(C2×C12), C22.43(C6×D4), C23.28(C3×D4), C6.114(C4○D8), (C22×C6).127D4, C12.148(C22×C4), (C2×C12).892C23, (C2×C24).358C22, (C6×D4).287C22, (C6×Q8).251C22, C12.116(C22⋊C4), (C22×C12).583C22, (C3×C4○D4)⋊7C4, C2.1(C3×C4○D8), C4⋊C4.38(C2×C6), (C2×C8).61(C2×C6), (C2×C4).49(C2×C12), (C2×C4○D4).10C6, (C6×C4○D4).18C2, (C2×D4).45(C2×C6), (C3×D4).27(C2×C4), (C2×C4).122(C3×D4), (C2×C6).619(C2×D4), C4.32(C3×C22⋊C4), C2.19(C6×C22⋊C4), (C3×Q8).29(C2×C4), (C2×Q8).48(C2×C6), (C3×D4⋊C4)⋊43C2, (C2×C12).270(C2×C4), (C3×Q8⋊C4)⋊43C2, C6.107(C2×C22⋊C4), (C2×C4).67(C22×C6), C22.4(C3×C22⋊C4), (C3×C42⋊C2)⋊23C2, (C3×C4⋊C4).359C22, (C2×C6).31(C22⋊C4), (C22×C4).119(C2×C6), SmallGroup(192,849)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.24D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 258 in 158 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C23.24D4, C3×D4⋊C4, C3×Q8⋊C4, C3×C42⋊C2, C22×C24, C6×C4○D4, C3×C23.24D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C4○D8, C3×C22⋊C4, C22×C12, C6×D4, C23.24D4, C6×C22⋊C4, C3×C4○D8, C3×C23.24D4
►On 96 points
Generators in S96
(1 58 15)(2 59 16)(3 60 9)(4 61 10)(5 62 11)(6 63 12)(7 64 13)(8 57 14)(17 25 68)(18 26 69)(19 27 70)(20 28 71)(21 29 72)(22 30 65)(23 31 66)(24 32 67)(33 76 84)(34 77 85)(35 78 86)(36 79 87)(37 80 88)(38 73 81)(39 74 82)(40 75 83)(41 50 92)(42 51 93)(43 52 94)(44 53 95)(45 54 96)(46 55 89)(47 56 90)(48 49 91)
(1 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 89)(16 90)(17 79)(18 80)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 87)(26 88)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(1 79)(2 80)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(57 86)(58 87)(59 88)(60 81)(61 82)(62 83)(63 84)(64 85)(65 94)(66 95)(67 96)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 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 8 79 78)(2 77 80 7)(3 6 73 76)(4 75 74 5)(9 12 38 33)(10 40 39 11)(13 16 34 37)(14 36 35 15)(17 20 46 41)(18 48 47 19)(21 24 42 45)(22 44 43 23)(25 28 55 50)(26 49 56 27)(29 32 51 54)(30 53 52 31)(57 87 86 58)(59 85 88 64)(60 63 81 84)(61 83 82 62)(65 95 94 66)(67 93 96 72)(68 71 89 92)(69 91 90 70)
G:=sub<Sym(96)| (1,58,15)(2,59,16)(3,60,9)(4,61,10)(5,62,11)(6,63,12)(7,64,13)(8,57,14)(17,25,68)(18,26,69)(19,27,70)(20,28,71)(21,29,72)(22,30,65)(23,31,66)(24,32,67)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(41,50,92)(42,51,93)(43,52,94)(44,53,95)(45,54,96)(46,55,89)(47,56,90)(48,49,91), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,79)(18,80)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (1,8,79,78)(2,77,80,7)(3,6,73,76)(4,75,74,5)(9,12,38,33)(10,40,39,11)(13,16,34,37)(14,36,35,15)(17,20,46,41)(18,48,47,19)(21,24,42,45)(22,44,43,23)(25,28,55,50)(26,49,56,27)(29,32,51,54)(30,53,52,31)(57,87,86,58)(59,85,88,64)(60,63,81,84)(61,83,82,62)(65,95,94,66)(67,93,96,72)(68,71,89,92)(69,91,90,70)>;
G:=Group( (1,58,15)(2,59,16)(3,60,9)(4,61,10)(5,62,11)(6,63,12)(7,64,13)(8,57,14)(17,25,68)(18,26,69)(19,27,70)(20,28,71)(21,29,72)(22,30,65)(23,31,66)(24,32,67)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(41,50,92)(42,51,93)(43,52,94)(44,53,95)(45,54,96)(46,55,89)(47,56,90)(48,49,91), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,79)(18,80)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (1,8,79,78)(2,77,80,7)(3,6,73,76)(4,75,74,5)(9,12,38,33)(10,40,39,11)(13,16,34,37)(14,36,35,15)(17,20,46,41)(18,48,47,19)(21,24,42,45)(22,44,43,23)(25,28,55,50)(26,49,56,27)(29,32,51,54)(30,53,52,31)(57,87,86,58)(59,85,88,64)(60,63,81,84)(61,83,82,62)(65,95,94,66)(67,93,96,72)(68,71,89,92)(69,91,90,70) );
G=PermutationGroup([[(1,58,15),(2,59,16),(3,60,9),(4,61,10),(5,62,11),(6,63,12),(7,64,13),(8,57,14),(17,25,68),(18,26,69),(19,27,70),(20,28,71),(21,29,72),(22,30,65),(23,31,66),(24,32,67),(33,76,84),(34,77,85),(35,78,86),(36,79,87),(37,80,88),(38,73,81),(39,74,82),(40,75,83),(41,50,92),(42,51,93),(43,52,94),(44,53,95),(45,54,96),(46,55,89),(47,56,90),(48,49,91)], [(1,46),(2,47),(3,48),(4,41),(5,42),(6,43),(7,44),(8,45),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,89),(16,90),(17,79),(18,80),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,87),(26,88),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59)], [(1,79),(2,80),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,38),(10,39),(11,40),(12,33),(13,34),(14,35),(15,36),(16,37),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(57,86),(58,87),(59,88),(60,81),(61,82),(62,83),(63,84),(64,85),(65,94),(66,95),(67,96),(68,89),(69,90),(70,91),(71,92),(72,93)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96)], [(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)], [(1,8,79,78),(2,77,80,7),(3,6,73,76),(4,75,74,5),(9,12,38,33),(10,40,39,11),(13,16,34,37),(14,36,35,15),(17,20,46,41),(18,48,47,19),(21,24,42,45),(22,44,43,23),(25,28,55,50),(26,49,56,27),(29,32,51,54),(30,53,52,31),(57,87,86,58),(59,85,88,64),(60,63,81,84),(61,83,82,62),(65,95,94,66),(67,93,96,72),(68,71,89,92),(69,91,90,70)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | ··· | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12X | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | D4 | C3×D4 | C3×D4 | C4○D8 | C3×C4○D8 |
| kernel | C3×C23.24D4 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C42⋊C2 | C22×C24 | C6×C4○D4 | C23.24D4 | C3×C4○D4 | D4⋊C4 | Q8⋊C4 | C42⋊C2 | C22×C8 | C2×C4○D4 | C4○D4 | C2×C12 | C22×C6 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 4 | 2 | 2 | 2 | 16 | 3 | 1 | 6 | 2 | 8 | 16 |
Matrix representation of C3×C23.24D4 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 27 | 54 |
| 0 | 46 | 46 |
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 27 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 67 | 61 |
| 27 | 0 | 0 |
| 0 | 12 | 12 |
| 0 | 67 | 61 |
G:=sub<GL(3,GF(73))| [64,0,0,0,64,0,0,0,64],[72,0,0,0,27,46,0,54,46],[72,0,0,0,72,0,0,0,72],[1,0,0,0,72,0,0,0,72],[27,0,0,0,0,67,0,12,61],[27,0,0,0,12,67,0,12,61] >;
C3×C23.24D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{24}D_4 % in TeX
G:=Group("C3xC2^3.24D4"); // GroupNames label
G:=SmallGroup(192,849);
// by ID
G=gap.SmallGroup(192,849);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,520,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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*d*e^3>;
// generators/relations