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G = C2×C12⋊7D4  order 192 = 26·3

Direct product of C2 and C12⋊7D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C12⋊7D4, C23⋊5D12, C24.82D6, C12⋊15(C2×D4), (C2×C12)⋊37D4, (C23×C4)⋊9S3, C6⋊3(C4⋊D4), (C23×C12)⋊8C2, (C22×C6)⋊15D4, C22⋊3(C2×D12), (C22×C4)⋊47D6, D6⋊C4⋊42C22, (C2×D12)⋊50C22, (C22×D12)⋊12C2, (C2×C6).288C24, C4⋊Dic3⋊64C22, C6.134(C22×D4), C2.33(C22×D12), (C2×C12).705C23, (C22×C12)⋊60C22, C22.83(C4○D12), (S3×C23).75C22, C22.303(S3×C23), (C23×C6).110C22, C23.244(C22×S3), (C22×C6).417C23, (C22×S3).126C23, (C2×Dic3).150C23, (C22×Dic3).162C22, C3⋊4(C2×C4⋊D4), C4⋊4(C2×C3⋊D4), (C2×C6)⋊11(C2×D4), (C2×D6⋊C4)⋊14C2, C6.63(C2×C4○D4), (C2×C4)⋊16(C3⋊D4), (C2×C4⋊Dic3)⋊29C2, C2.71(C2×C4○D12), C2.7(C22×C3⋊D4), (C2×C3⋊D4)⋊42C22, (C22×C3⋊D4)⋊11C2, (C2×C6).114(C4○D4), (C2×C4).658(C22×S3), C22.104(C2×C3⋊D4), SmallGroup(192,1349)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C2×C12⋊7D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×C12⋊7D4
Lower central
C3 — C2×C6 — C2×C12⋊7D4

Subgroups: 1176 in 426 conjugacy classes, 143 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×10], C22 [×32], S3 [×4], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×8], C2×C4 [×18], D4 [×24], C23, C23 [×6], C23 [×20], Dic3 [×4], C12 [×4], C12 [×2], D6 [×20], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×24], C24, C24 [×2], D12 [×8], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×8], C2×C12 [×10], C22×S3 [×4], C22×S3 [×12], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], C4⋊Dic3 [×4], D6⋊C4 [×8], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×8], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], S3×C23 [×2], C23×C6, C2×C4⋊D4, C2×C4⋊Dic3, C2×D6⋊C4 [×2], C12⋊7D4 [×8], C22×D12, C22×C3⋊D4 [×2], C23×C12, C2×C12⋊7D4

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C4○D4 [×2], C24, D12 [×4], C3⋊D4 [×4], C22×S3 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×D12 [×6], C4○D12 [×2], C2×C3⋊D4 [×6], S3×C23, C2×C4⋊D4, C12⋊7D4 [×4], C22×D12, C2×C4○D12, C22×C3⋊D4, C2×C12⋊7D4

Generators and relations
 G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation
►On 96 points
Generators in S96
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 25)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(37 92)(38 93)(39 94)(40 95)(41 96)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 73)

(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 45 15 79)(2 44 16 78)(3 43 17 77)(4 42 18 76)(5 41 19 75)(6 40 20 74)(7 39 21 73)(8 38 22 84)(9 37 23 83)(10 48 24 82)(11 47 13 81)(12 46 14 80)(25 89 51 67)(26 88 52 66)(27 87 53 65)(28 86 54 64)(29 85 55 63)(30 96 56 62)(31 95 57 61)(32 94 58 72)(33 93 59 71)(34 92 60 70)(35 91 49 69)(36 90 50 68)

(1 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 60)(12 59)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)
 
G:=sub<Sym(96)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73), (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), (1,45,15,79)(2,44,16,78)(3,43,17,77)(4,42,18,76)(5,41,19,75)(6,40,20,74)(7,39,21,73)(8,38,22,84)(9,37,23,83)(10,48,24,82)(11,47,13,81)(12,46,14,80)(25,89,51,67)(26,88,52,66)(27,87,53,65)(28,86,54,64)(29,85,55,63)(30,96,56,62)(31,95,57,61)(32,94,58,72)(33,93,59,71)(34,92,60,70)(35,91,49,69)(36,90,50,68), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,60)(12,59)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)>;
 
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73), (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), (1,45,15,79)(2,44,16,78)(3,43,17,77)(4,42,18,76)(5,41,19,75)(6,40,20,74)(7,39,21,73)(8,38,22,84)(9,37,23,83)(10,48,24,82)(11,47,13,81)(12,46,14,80)(25,89,51,67)(26,88,52,66)(27,87,53,65)(28,86,54,64)(29,85,55,63)(30,96,56,62)(31,95,57,61)(32,94,58,72)(33,93,59,71)(34,92,60,70)(35,91,49,69)(36,90,50,68), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,60)(12,59)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79) );
 
G=PermutationGroup([(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,25),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,49),(37,92),(38,93),(39,94),(40,95),(41,96),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,73)], [(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)], [(1,45,15,79),(2,44,16,78),(3,43,17,77),(4,42,18,76),(5,41,19,75),(6,40,20,74),(7,39,21,73),(8,38,22,84),(9,37,23,83),(10,48,24,82),(11,47,13,81),(12,46,14,80),(25,89,51,67),(26,88,52,66),(27,87,53,65),(28,86,54,64),(29,85,55,63),(30,96,56,62),(31,95,57,61),(32,94,58,72),(33,93,59,71),(34,92,60,70),(35,91,49,69),(36,90,50,68)], [(1,58),(2,57),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,50),(10,49),(11,60),(12,59),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79)])
 

Matrix representation ►G ⊆ GL5(𝔽13)

120000
01000
00100
00010
00001
,
120000
00800
08000
000610
00033
,
10000
001200
01000
000012
000120
,
10000
01000
001200
000012
000120

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,6,3,0,0,0,10,3],[1,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
 

60 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O 3 4A···4H4I4J4K4L6A···6O12A···12P
order12···22222222234···444446···612···12
size11···122221212121222···2121212122···22···2

60 irreducible representations

dim1111111222222222
type+++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D6D6C4○D4C3⋊D4D12C4○D12
kernelC2×C12⋊7D4C2×C4⋊Dic3C2×D6⋊C4C12⋊7D4C22×D12C22×C3⋊D4C23×C12C23×C4C2×C12C22×C6C22×C4C24C2×C6C2×C4C23C22
# reps1128121144614888

In GAP, Magma, Sage, TeX 

C_2\times C_{12}\rtimes_7D_4
 
% in TeX
 
G:=Group("C2xC12:7D4");
 
// GroupNames label
 
G:=SmallGroup(192,1349);
 
// by ID
 
G=gap.SmallGroup(192,1349);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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