?
G = C6.102+ (1+4) order 192 = 26·3
10th non-split extension by C6 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.102+ (1+4), C3⋊D4⋊4Q8, C12⋊Q8⋊10C2, D6.7(C2×Q8), C4⋊C4.264D6, C4.D12⋊9C2, D6⋊Q8⋊2C2, C3⋊1(D4⋊3Q8), C22.8(S3×Q8), Dic3.Q8⋊1C2, (C2×C6).55C24, Dic3.8(C2×Q8), Dic6⋊C4⋊9C2, C4.92(C4○D12), C6.26(C22×Q8), D6⋊C4.94C22, (C22×C4).196D6, C12.194(C4○D4), C2.13(D4⋊6D6), C12.48D4⋊18C2, (C2×C12).138C23, C22.89(S3×C23), C4⋊Dic3.192C22, C23.237(C22×S3), (C22×C6).404C23, Dic3⋊C4.149C22, (C22×S3).160C23, (C22×C12).103C22, (C2×Dic6).141C22, (C2×Dic3).191C23, (C4×Dic3).192C22, C6.D4.88C22, C2.9(C2×S3×Q8), (C6×C4⋊C4)⋊17C2, (S3×C4⋊C4)⋊10C2, (C2×C4⋊C4)⋊20S3, C6.22(C2×C4○D4), (C4×C3⋊D4).3C2, (C2×C6).95(C2×Q8), C2.24(C2×C4○D12), (S3×C2×C4).191C22, (C3×C4⋊C4).297C22, (C2×C4).573(C22×S3), (C2×C3⋊D4).147C22, SmallGroup(192,1070)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 520 in 228 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×13], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×4], Q8 [×4], C23, C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×5], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×Q8 [×3], Dic6 [×4], C4×S3 [×4], C2×Dic3 [×3], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×S3, C22×C6, C2×C4⋊C4, C2×C4⋊C4, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×8], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×2], C6.D4, C6.D4 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], C2×C3⋊D4, C22×C12, C22×C12 [×2], D4⋊3Q8, Dic6⋊C4, C12⋊Q8, Dic3.Q8 [×2], S3×C4⋊C4, D6⋊Q8 [×2], C4.D12, C12.48D4, C12.48D4 [×2], C4×C3⋊D4, C4×C3⋊D4 [×2], C6×C4⋊C4, C6.102+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C4○D12 [×2], S3×Q8 [×2], S3×C23, D4⋊3Q8, C2×C4○D12, D4⋊6D6, C2×S3×Q8, C6.102+ (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a3b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, ede-1=b2d >
►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 12 17 91)(2 11 18 96)(3 10 13 95)(4 9 14 94)(5 8 15 93)(6 7 16 92)(19 79 26 86)(20 84 27 85)(21 83 28 90)(22 82 29 89)(23 81 30 88)(24 80 25 87)(31 74 38 67)(32 73 39 72)(33 78 40 71)(34 77 41 70)(35 76 42 69)(36 75 37 68)(43 58 50 65)(44 57 51 64)(45 56 52 63)(46 55 53 62)(47 60 54 61)(48 59 49 66)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 87)(8 88)(9 89)(10 90)(11 85)(12 86)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 77 17 70)(2 78 18 71)(3 73 13 72)(4 74 14 67)(5 75 15 68)(6 76 16 69)(7 39 92 32)(8 40 93 33)(9 41 94 34)(10 42 95 35)(11 37 96 36)(12 38 91 31)(19 62 26 55)(20 63 27 56)(21 64 28 57)(22 65 29 58)(23 66 30 59)(24 61 25 60)(43 86 50 79)(44 87 51 80)(45 88 52 81)(46 89 53 82)(47 90 54 83)(48 85 49 84)
(1 86 4 89)(2 87 5 90)(3 88 6 85)(7 20 10 23)(8 21 11 24)(9 22 12 19)(13 81 16 84)(14 82 17 79)(15 83 18 80)(25 93 28 96)(26 94 29 91)(27 95 30 92)(31 62 34 65)(32 63 35 66)(33 64 36 61)(37 60 40 57)(38 55 41 58)(39 56 42 59)(43 74 46 77)(44 75 47 78)(45 76 48 73)(49 72 52 69)(50 67 53 70)(51 68 54 71)
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), (1,12,17,91)(2,11,18,96)(3,10,13,95)(4,9,14,94)(5,8,15,93)(6,7,16,92)(19,79,26,86)(20,84,27,85)(21,83,28,90)(22,82,29,89)(23,81,30,88)(24,80,25,87)(31,74,38,67)(32,73,39,72)(33,78,40,71)(34,77,41,70)(35,76,42,69)(36,75,37,68)(43,58,50,65)(44,57,51,64)(45,56,52,63)(46,55,53,62)(47,60,54,61)(48,59,49,66), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,87)(8,88)(9,89)(10,90)(11,85)(12,86)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,77,17,70)(2,78,18,71)(3,73,13,72)(4,74,14,67)(5,75,15,68)(6,76,16,69)(7,39,92,32)(8,40,93,33)(9,41,94,34)(10,42,95,35)(11,37,96,36)(12,38,91,31)(19,62,26,55)(20,63,27,56)(21,64,28,57)(22,65,29,58)(23,66,30,59)(24,61,25,60)(43,86,50,79)(44,87,51,80)(45,88,52,81)(46,89,53,82)(47,90,54,83)(48,85,49,84), (1,86,4,89)(2,87,5,90)(3,88,6,85)(7,20,10,23)(8,21,11,24)(9,22,12,19)(13,81,16,84)(14,82,17,79)(15,83,18,80)(25,93,28,96)(26,94,29,91)(27,95,30,92)(31,62,34,65)(32,63,35,66)(33,64,36,61)(37,60,40,57)(38,55,41,58)(39,56,42,59)(43,74,46,77)(44,75,47,78)(45,76,48,73)(49,72,52,69)(50,67,53,70)(51,68,54,71)>;
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), (1,12,17,91)(2,11,18,96)(3,10,13,95)(4,9,14,94)(5,8,15,93)(6,7,16,92)(19,79,26,86)(20,84,27,85)(21,83,28,90)(22,82,29,89)(23,81,30,88)(24,80,25,87)(31,74,38,67)(32,73,39,72)(33,78,40,71)(34,77,41,70)(35,76,42,69)(36,75,37,68)(43,58,50,65)(44,57,51,64)(45,56,52,63)(46,55,53,62)(47,60,54,61)(48,59,49,66), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,87)(8,88)(9,89)(10,90)(11,85)(12,86)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,77,17,70)(2,78,18,71)(3,73,13,72)(4,74,14,67)(5,75,15,68)(6,76,16,69)(7,39,92,32)(8,40,93,33)(9,41,94,34)(10,42,95,35)(11,37,96,36)(12,38,91,31)(19,62,26,55)(20,63,27,56)(21,64,28,57)(22,65,29,58)(23,66,30,59)(24,61,25,60)(43,86,50,79)(44,87,51,80)(45,88,52,81)(46,89,53,82)(47,90,54,83)(48,85,49,84), (1,86,4,89)(2,87,5,90)(3,88,6,85)(7,20,10,23)(8,21,11,24)(9,22,12,19)(13,81,16,84)(14,82,17,79)(15,83,18,80)(25,93,28,96)(26,94,29,91)(27,95,30,92)(31,62,34,65)(32,63,35,66)(33,64,36,61)(37,60,40,57)(38,55,41,58)(39,56,42,59)(43,74,46,77)(44,75,47,78)(45,76,48,73)(49,72,52,69)(50,67,53,70)(51,68,54,71) );
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)], [(1,12,17,91),(2,11,18,96),(3,10,13,95),(4,9,14,94),(5,8,15,93),(6,7,16,92),(19,79,26,86),(20,84,27,85),(21,83,28,90),(22,82,29,89),(23,81,30,88),(24,80,25,87),(31,74,38,67),(32,73,39,72),(33,78,40,71),(34,77,41,70),(35,76,42,69),(36,75,37,68),(43,58,50,65),(44,57,51,64),(45,56,52,63),(46,55,53,62),(47,60,54,61),(48,59,49,66)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(7,87),(8,88),(9,89),(10,90),(11,85),(12,86),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(1,77,17,70),(2,78,18,71),(3,73,13,72),(4,74,14,67),(5,75,15,68),(6,76,16,69),(7,39,92,32),(8,40,93,33),(9,41,94,34),(10,42,95,35),(11,37,96,36),(12,38,91,31),(19,62,26,55),(20,63,27,56),(21,64,28,57),(22,65,29,58),(23,66,30,59),(24,61,25,60),(43,86,50,79),(44,87,51,80),(45,88,52,81),(46,89,53,82),(47,90,54,83),(48,85,49,84)], [(1,86,4,89),(2,87,5,90),(3,88,6,85),(7,20,10,23),(8,21,11,24),(9,22,12,19),(13,81,16,84),(14,82,17,79),(15,83,18,80),(25,93,28,96),(26,94,29,91),(27,95,30,92),(31,62,34,65),(32,63,35,66),(33,64,36,61),(37,60,40,57),(38,55,41,58),(39,56,42,59),(43,74,46,77),(44,75,47,78),(45,76,48,73),(49,72,52,69),(50,67,53,70),(51,68,54,71)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 10 | 6 | 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 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[4,1,0,0,0,0,11,9,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,3,0,0,0,0,10,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,3,5],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,11,1] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | S3×Q8 | D4⋊6D6 |
| kernel | C6.102+ (1+4) | Dic6⋊C4 | C12⋊Q8 | Dic3.Q8 | S3×C4⋊C4 | D6⋊Q8 | C4.D12 | C12.48D4 | C4×C3⋊D4 | C6×C4⋊C4 | C2×C4⋊C4 | C3⋊D4 | C4⋊C4 | C22×C4 | C12 | C4 | C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 1 | 1 | 4 | 4 | 3 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_6._{10}2_+^{(1+4)} % in TeX
G:=Group("C6.10ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1070);
// by ID
G=gap.SmallGroup(192,1070);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,1571,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=b^2,e^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^3*b^-1,d*b*d^-1=a^3*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations