direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C6×C8.C22, C24.46C23, C12.83C24, Q16⋊3(C2×C6), C4.67(C6×D4), (C6×Q16)⋊25C2, (C2×Q16)⋊11C6, (C2×SD16)⋊5C6, SD16⋊2(C2×C6), C8.1(C22×C6), C4.6(C23×C6), (C6×SD16)⋊16C2, C12.330(C2×D4), (C2×C12).526D4, M4(2)⋊4(C2×C6), (C2×M4(2))⋊4C6, (C6×M4(2))⋊9C2, (C6×Q8)⋊55C22, (C22×Q8)⋊17C6, D4.3(C22×C6), C23.56(C3×D4), C22.24(C6×D4), Q8.7(C22×C6), (C3×Q16)⋊17C22, (C3×D4).36C23, C6.204(C22×D4), (C22×C6).173D4, (C3×Q8).37C23, (C2×C24).208C22, (C2×C12).976C23, (C3×SD16)⋊18C22, (C6×D4).329C22, (C3×M4(2))⋊25C22, (C22×C12).466C22, (Q8×C2×C6)⋊21C2, C2.28(D4×C2×C6), (C2×C8).32(C2×C6), (C2×Q8)⋊17(C2×C6), C4○D4.20(C2×C6), (C2×C4○D4).18C6, (C6×C4○D4).26C2, (C2×D4).75(C2×C6), (C2×C4).137(C3×D4), (C2×C6).420(C2×D4), (C22×C4).82(C2×C6), (C2×C4).46(C22×C6), (C3×C4○D4).58C22, SmallGroup(192,1463)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 370 in 258 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C6, C6 [×2], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×6], Q8 [×7], C23, C23, C12 [×2], C12 [×2], C12 [×6], C2×C6, C2×C6 [×2], C2×C6 [×6], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×2], C24 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×11], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×6], C3×Q8 [×7], C22×C6, C22×C6, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C2×C24 [×2], C3×M4(2) [×4], C3×SD16 [×8], C3×Q16 [×8], C22×C12, C22×C12 [×2], C6×D4, C6×D4, C6×Q8, C6×Q8 [×6], C6×Q8 [×3], C3×C4○D4 [×4], C3×C4○D4 [×2], C2×C8.C22, C6×M4(2), C6×SD16 [×2], C6×Q16 [×2], C3×C8.C22 [×8], Q8×C2×C6, C6×C4○D4, C6×C8.C22
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C8.C22 [×2], C22×D4, C6×D4 [×6], C23×C6, C2×C8.C22, C3×C8.C22 [×2], D4×C2×C6, C6×C8.C22
Generators and relations
G = < a,b,c,d | a6=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
►On 96 points
(1 73 85 69 17 50)(2 74 86 70 18 51)(3 75 87 71 19 52)(4 76 88 72 20 53)(5 77 81 65 21 54)(6 78 82 66 22 55)(7 79 83 67 23 56)(8 80 84 68 24 49)(9 44 90 39 64 25)(10 45 91 40 57 26)(11 46 92 33 58 27)(12 47 93 34 59 28)(13 48 94 35 60 29)(14 41 95 36 61 30)(15 42 96 37 62 31)(16 43 89 38 63 32)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)(25 31)(27 29)(28 32)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(66 68)(67 71)(70 72)(74 76)(75 79)(78 80)(82 84)(83 87)(86 88)(89 93)(90 96)(92 94)
(1 28)(2 25)(3 30)(4 27)(5 32)(6 29)(7 26)(8 31)(9 74)(10 79)(11 76)(12 73)(13 78)(14 75)(15 80)(16 77)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)(41 87)(42 84)(43 81)(44 86)(45 83)(46 88)(47 85)(48 82)(49 62)(50 59)(51 64)(52 61)(53 58)(54 63)(55 60)(56 57)(65 89)(66 94)(67 91)(68 96)(69 93)(70 90)(71 95)(72 92)
G:=sub<Sym(96)| (1,73,85,69,17,50)(2,74,86,70,18,51)(3,75,87,71,19,52)(4,76,88,72,20,53)(5,77,81,65,21,54)(6,78,82,66,22,55)(7,79,83,67,23,56)(8,80,84,68,24,49)(9,44,90,39,64,25)(10,45,91,40,57,26)(11,46,92,33,58,27)(12,47,93,34,59,28)(13,48,94,35,60,29)(14,41,95,36,61,30)(15,42,96,37,62,31)(16,43,89,38,63,32), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,74)(10,79)(11,76)(12,73)(13,78)(14,75)(15,80)(16,77)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,87)(42,84)(43,81)(44,86)(45,83)(46,88)(47,85)(48,82)(49,62)(50,59)(51,64)(52,61)(53,58)(54,63)(55,60)(56,57)(65,89)(66,94)(67,91)(68,96)(69,93)(70,90)(71,95)(72,92)>;
G:=Group( (1,73,85,69,17,50)(2,74,86,70,18,51)(3,75,87,71,19,52)(4,76,88,72,20,53)(5,77,81,65,21,54)(6,78,82,66,22,55)(7,79,83,67,23,56)(8,80,84,68,24,49)(9,44,90,39,64,25)(10,45,91,40,57,26)(11,46,92,33,58,27)(12,47,93,34,59,28)(13,48,94,35,60,29)(14,41,95,36,61,30)(15,42,96,37,62,31)(16,43,89,38,63,32), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,74)(10,79)(11,76)(12,73)(13,78)(14,75)(15,80)(16,77)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,87)(42,84)(43,81)(44,86)(45,83)(46,88)(47,85)(48,82)(49,62)(50,59)(51,64)(52,61)(53,58)(54,63)(55,60)(56,57)(65,89)(66,94)(67,91)(68,96)(69,93)(70,90)(71,95)(72,92) );
G=PermutationGroup([(1,73,85,69,17,50),(2,74,86,70,18,51),(3,75,87,71,19,52),(4,76,88,72,20,53),(5,77,81,65,21,54),(6,78,82,66,22,55),(7,79,83,67,23,56),(8,80,84,68,24,49),(9,44,90,39,64,25),(10,45,91,40,57,26),(11,46,92,33,58,27),(12,47,93,34,59,28),(13,48,94,35,60,29),(14,41,95,36,61,30),(15,42,96,37,62,31),(16,43,89,38,63,32)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(18,20),(19,23),(22,24),(25,31),(27,29),(28,32),(33,35),(34,38),(37,39),(42,44),(43,47),(46,48),(49,55),(51,53),(52,56),(58,60),(59,63),(62,64),(66,68),(67,71),(70,72),(74,76),(75,79),(78,80),(82,84),(83,87),(86,88),(89,93),(90,96),(92,94)], [(1,28),(2,25),(3,30),(4,27),(5,32),(6,29),(7,26),(8,31),(9,74),(10,79),(11,76),(12,73),(13,78),(14,75),(15,80),(16,77),(17,34),(18,39),(19,36),(20,33),(21,38),(22,35),(23,40),(24,37),(41,87),(42,84),(43,81),(44,86),(45,83),(46,88),(47,85),(48,82),(49,62),(50,59),(51,64),(52,61),(53,58),(54,63),(55,60),(56,57),(65,89),(66,94),(67,91),(68,96),(69,93),(70,90),(71,95),(72,92)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 30 | 12 | 0 | 0 | 0 | 0 |
| 4 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 13 | 53 | 20 |
| 0 | 0 | 60 | 7 | 0 | 20 |
| 0 | 0 | 7 | 13 | 0 | 6 |
| 0 | 0 | 7 | 6 | 60 | 6 |
| 1 | 58 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 | 72 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 72 | 1 | 72 | 2 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,4,0,0,0,0,12,43,0,0,0,0,0,0,60,60,7,7,0,0,13,7,13,6,0,0,53,0,0,60,0,0,20,20,6,6],[1,0,0,0,0,0,58,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,72,72,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,2,0,72] >;
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12T | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C8.C22 | C3×C8.C22 |
| kernel | C6×C8.C22 | C6×M4(2) | C6×SD16 | C6×Q16 | C3×C8.C22 | Q8×C2×C6 | C6×C4○D4 | C2×C8.C22 | C2×M4(2) | C2×SD16 | C2×Q16 | C8.C22 | C22×Q8 | C2×C4○D4 | C2×C12 | C22×C6 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 3 | 1 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_6\times C_8.C_2^2
% in TeX
G:=Group("C6xC8.C2^2"); // GroupNames label
G:=SmallGroup(192,1463);
// by ID
G=gap.SmallGroup(192,1463);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations