direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×S3×Q8, C42.230D6, C12⋊9(C2×Q8), (Q8×C12)⋊6C2, C4⋊C4.322D6, Dic3⋊7(C2×Q8), D6.14(C2×Q8), (C4×Dic6)⋊37C2, (Q8×Dic3)⋊31C2, Dic6⋊14(C2×C4), (S3×C42).5C2, C6.24(C23×C4), (C2×Q8).223D6, D6.36(C4○D4), C6.28(C22×Q8), C12.34(C22×C4), (C2×C6).115C24, D6.24(C22×C4), Dic6⋊C4⋊46C2, (C4×C12).167C22, (C2×C12).494C23, C22.34(S3×C23), (C6×Q8).215C22, C4⋊Dic3.365C22, Dic3.10(C22×C4), Dic3⋊C4.136C22, (C22×S3).255C23, (C4×Dic3).294C22, (C2×Dic6).289C22, (C2×Dic3).309C23, C3⋊3(C2×C4×Q8), C2.3(C2×S3×Q8), C4.34(S3×C2×C4), (C2×S3×Q8).9C2, C2.5(S3×C4○D4), (S3×C4⋊C4).14C2, (C3×Q8)⋊10(C2×C4), C2.26(S3×C22×C4), (C4×S3).17(C2×C4), C6.144(C2×C4○D4), (S3×C2×C4).294C22, (C3×C4⋊C4).343C22, (C2×C4).820(C22×S3), SmallGroup(192,1130)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 568 in 298 conjugacy classes, 169 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C4 [×14], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×29], Q8 [×4], Q8 [×12], C23, Dic3 [×8], Dic3 [×3], C12 [×8], C12 [×3], D6 [×6], C2×C6, C42 [×3], C42 [×9], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×7], C2×Q8, C2×Q8 [×11], Dic6 [×12], C4×S3 [×16], C4×S3 [×6], C2×Dic3, C2×Dic3 [×6], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8, C4×Q8 [×7], C22×Q8, C4×Dic3 [×9], Dic3⋊C4 [×6], C4⋊Dic3 [×3], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], S3×C2×C4, S3×C2×C4 [×6], S3×Q8 [×8], C6×Q8, C2×C4×Q8, C4×Dic6 [×3], S3×C42 [×3], Dic6⋊C4 [×3], S3×C4⋊C4 [×3], Q8×Dic3, Q8×C12, C2×S3×Q8, C4×S3×Q8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], Q8 [×4], C23 [×15], D6 [×7], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×S3 [×4], C22×S3 [×7], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, S3×C2×C4 [×6], S3×Q8 [×2], S3×C23, C2×C4×Q8, S3×C22×C4, C2×S3×Q8, S3×C4○D4, C4×S3×Q8
Generators and relations
G = < a,b,c,d,e | a4=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 96 points
(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 25 9)(2 26 10)(3 27 11)(4 28 12)(5 77 19)(6 78 20)(7 79 17)(8 80 18)(13 82 96)(14 83 93)(15 84 94)(16 81 95)(21 43 37)(22 44 38)(23 41 39)(24 42 40)(29 66 64)(30 67 61)(31 68 62)(32 65 63)(33 47 49)(34 48 50)(35 45 51)(36 46 52)(53 90 88)(54 91 85)(55 92 86)(56 89 87)(57 71 73)(58 72 74)(59 69 75)(60 70 76)
(9 25)(10 26)(11 27)(12 28)(13 82)(14 83)(15 84)(16 81)(17 79)(18 80)(19 77)(20 78)(21 43)(22 44)(23 41)(24 42)(29 66)(30 67)(31 68)(32 65)(33 49)(34 50)(35 51)(36 52)(53 90)(54 91)(55 92)(56 89)(57 73)(58 74)(59 75)(60 76)
(1 61 39 45)(2 62 40 46)(3 63 37 47)(4 64 38 48)(5 70 96 86)(6 71 93 87)(7 72 94 88)(8 69 95 85)(9 67 41 35)(10 68 42 36)(11 65 43 33)(12 66 44 34)(13 55 77 76)(14 56 78 73)(15 53 79 74)(16 54 80 75)(17 58 84 90)(18 59 81 91)(19 60 82 92)(20 57 83 89)(21 49 27 32)(22 50 28 29)(23 51 25 30)(24 52 26 31)
(1 85 39 69)(2 86 40 70)(3 87 37 71)(4 88 38 72)(5 62 96 46)(6 63 93 47)(7 64 94 48)(8 61 95 45)(9 91 41 59)(10 92 42 60)(11 89 43 57)(12 90 44 58)(13 52 77 31)(14 49 78 32)(15 50 79 29)(16 51 80 30)(17 66 84 34)(18 67 81 35)(19 68 82 36)(20 65 83 33)(21 73 27 56)(22 74 28 53)(23 75 25 54)(24 76 26 55)
G:=sub<Sym(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), (1,25,9)(2,26,10)(3,27,11)(4,28,12)(5,77,19)(6,78,20)(7,79,17)(8,80,18)(13,82,96)(14,83,93)(15,84,94)(16,81,95)(21,43,37)(22,44,38)(23,41,39)(24,42,40)(29,66,64)(30,67,61)(31,68,62)(32,65,63)(33,47,49)(34,48,50)(35,45,51)(36,46,52)(53,90,88)(54,91,85)(55,92,86)(56,89,87)(57,71,73)(58,72,74)(59,69,75)(60,70,76), (9,25)(10,26)(11,27)(12,28)(13,82)(14,83)(15,84)(16,81)(17,79)(18,80)(19,77)(20,78)(21,43)(22,44)(23,41)(24,42)(29,66)(30,67)(31,68)(32,65)(33,49)(34,50)(35,51)(36,52)(53,90)(54,91)(55,92)(56,89)(57,73)(58,74)(59,75)(60,76), (1,61,39,45)(2,62,40,46)(3,63,37,47)(4,64,38,48)(5,70,96,86)(6,71,93,87)(7,72,94,88)(8,69,95,85)(9,67,41,35)(10,68,42,36)(11,65,43,33)(12,66,44,34)(13,55,77,76)(14,56,78,73)(15,53,79,74)(16,54,80,75)(17,58,84,90)(18,59,81,91)(19,60,82,92)(20,57,83,89)(21,49,27,32)(22,50,28,29)(23,51,25,30)(24,52,26,31), (1,85,39,69)(2,86,40,70)(3,87,37,71)(4,88,38,72)(5,62,96,46)(6,63,93,47)(7,64,94,48)(8,61,95,45)(9,91,41,59)(10,92,42,60)(11,89,43,57)(12,90,44,58)(13,52,77,31)(14,49,78,32)(15,50,79,29)(16,51,80,30)(17,66,84,34)(18,67,81,35)(19,68,82,36)(20,65,83,33)(21,73,27,56)(22,74,28,53)(23,75,25,54)(24,76,26,55)>;
G:=Group( (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), (1,25,9)(2,26,10)(3,27,11)(4,28,12)(5,77,19)(6,78,20)(7,79,17)(8,80,18)(13,82,96)(14,83,93)(15,84,94)(16,81,95)(21,43,37)(22,44,38)(23,41,39)(24,42,40)(29,66,64)(30,67,61)(31,68,62)(32,65,63)(33,47,49)(34,48,50)(35,45,51)(36,46,52)(53,90,88)(54,91,85)(55,92,86)(56,89,87)(57,71,73)(58,72,74)(59,69,75)(60,70,76), (9,25)(10,26)(11,27)(12,28)(13,82)(14,83)(15,84)(16,81)(17,79)(18,80)(19,77)(20,78)(21,43)(22,44)(23,41)(24,42)(29,66)(30,67)(31,68)(32,65)(33,49)(34,50)(35,51)(36,52)(53,90)(54,91)(55,92)(56,89)(57,73)(58,74)(59,75)(60,76), (1,61,39,45)(2,62,40,46)(3,63,37,47)(4,64,38,48)(5,70,96,86)(6,71,93,87)(7,72,94,88)(8,69,95,85)(9,67,41,35)(10,68,42,36)(11,65,43,33)(12,66,44,34)(13,55,77,76)(14,56,78,73)(15,53,79,74)(16,54,80,75)(17,58,84,90)(18,59,81,91)(19,60,82,92)(20,57,83,89)(21,49,27,32)(22,50,28,29)(23,51,25,30)(24,52,26,31), (1,85,39,69)(2,86,40,70)(3,87,37,71)(4,88,38,72)(5,62,96,46)(6,63,93,47)(7,64,94,48)(8,61,95,45)(9,91,41,59)(10,92,42,60)(11,89,43,57)(12,90,44,58)(13,52,77,31)(14,49,78,32)(15,50,79,29)(16,51,80,30)(17,66,84,34)(18,67,81,35)(19,68,82,36)(20,65,83,33)(21,73,27,56)(22,74,28,53)(23,75,25,54)(24,76,26,55) );
G=PermutationGroup([(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)], [(1,25,9),(2,26,10),(3,27,11),(4,28,12),(5,77,19),(6,78,20),(7,79,17),(8,80,18),(13,82,96),(14,83,93),(15,84,94),(16,81,95),(21,43,37),(22,44,38),(23,41,39),(24,42,40),(29,66,64),(30,67,61),(31,68,62),(32,65,63),(33,47,49),(34,48,50),(35,45,51),(36,46,52),(53,90,88),(54,91,85),(55,92,86),(56,89,87),(57,71,73),(58,72,74),(59,69,75),(60,70,76)], [(9,25),(10,26),(11,27),(12,28),(13,82),(14,83),(15,84),(16,81),(17,79),(18,80),(19,77),(20,78),(21,43),(22,44),(23,41),(24,42),(29,66),(30,67),(31,68),(32,65),(33,49),(34,50),(35,51),(36,52),(53,90),(54,91),(55,92),(56,89),(57,73),(58,74),(59,75),(60,76)], [(1,61,39,45),(2,62,40,46),(3,63,37,47),(4,64,38,48),(5,70,96,86),(6,71,93,87),(7,72,94,88),(8,69,95,85),(9,67,41,35),(10,68,42,36),(11,65,43,33),(12,66,44,34),(13,55,77,76),(14,56,78,73),(15,53,79,74),(16,54,80,75),(17,58,84,90),(18,59,81,91),(19,60,82,92),(20,57,83,89),(21,49,27,32),(22,50,28,29),(23,51,25,30),(24,52,26,31)], [(1,85,39,69),(2,86,40,70),(3,87,37,71),(4,88,38,72),(5,62,96,46),(6,63,93,47),(7,64,94,48),(8,61,95,45),(9,91,41,59),(10,92,42,60),(11,89,43,57),(12,90,44,58),(13,52,77,31),(14,49,78,32),(15,50,79,29),(16,51,80,30),(17,66,84,34),(18,67,81,35),(19,68,82,36),(20,65,83,33),(21,73,27,56),(22,74,28,53),(23,75,25,54),(24,76,26,55)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 12 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 3 |
| 0 | 0 | 3 | 4 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,5,0,0,0,0,5],[0,1,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,9,3,0,0,3,4] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R | 4S | 4T | 4U | ··· | 4AF | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | C4×S3 | S3×Q8 | S3×C4○D4 |
| kernel | C4×S3×Q8 | C4×Dic6 | S3×C42 | Dic6⋊C4 | S3×C4⋊C4 | Q8×Dic3 | Q8×C12 | C2×S3×Q8 | S3×Q8 | C4×Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | D6 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\times S_3\times Q_8
% in TeX
G:=Group("C4xS3xQ8"); // GroupNames label
G:=SmallGroup(192,1130);
// by ID
G=gap.SmallGroup(192,1130);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations