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G = C6×C22⋊Q8order 192 = 26·3

Direct product of C6 and C22⋊Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×C22⋊Q8
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C6×Q8 — C3×C22⋊Q8 — C6×C22⋊Q8
 Lower central C1 — C22 — C6×C22⋊Q8
 Upper central C1 — C22×C6 — C6×C22⋊Q8

Generators and relations for C6×C22⋊Q8
G = < a,b,c,d,e | a6=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C22×C12, C6×Q8, C6×Q8, C23×C6, C2×C22⋊Q8, C6×C22⋊C4, C6×C4⋊C4, C6×C4⋊C4, C3×C22⋊Q8, C23×C12, Q8×C2×C6, C6×C22⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C24, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C6×D4, C6×Q8, C3×C4○D4, C23×C6, C2×C22⋊Q8, C3×C22⋊Q8, D4×C2×C6, Q8×C2×C6, C6×C4○D4, C6×C22⋊Q8

Smallest permutation representation of C6×C22⋊Q8
On 96 points
Generators in S96
(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)
(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 96)(20 91)(21 92)(22 93)(23 94)(24 95)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(79 87)(80 88)(81 89)(82 90)(83 85)(84 86)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 96)(20 91)(21 92)(22 93)(23 94)(24 95)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(55 63)(56 64)(57 65)(58 66)(59 61)(60 62)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(79 87)(80 88)(81 89)(82 90)(83 85)(84 86)
(1 47 39 58)(2 48 40 59)(3 43 41 60)(4 44 42 55)(5 45 37 56)(6 46 38 57)(7 81 20 70)(8 82 21 71)(9 83 22 72)(10 84 23 67)(11 79 24 68)(12 80 19 69)(13 89 91 76)(14 90 92 77)(15 85 93 78)(16 86 94 73)(17 87 95 74)(18 88 96 75)(25 52 36 65)(26 53 31 66)(27 54 32 61)(28 49 33 62)(29 50 34 63)(30 51 35 64)
(1 71 39 82)(2 72 40 83)(3 67 41 84)(4 68 42 79)(5 69 37 80)(6 70 38 81)(7 46 20 57)(8 47 21 58)(9 48 22 59)(10 43 23 60)(11 44 24 55)(12 45 19 56)(13 52 91 65)(14 53 92 66)(15 54 93 61)(16 49 94 62)(17 50 95 63)(18 51 96 64)(25 76 36 89)(26 77 31 90)(27 78 32 85)(28 73 33 86)(29 74 34 87)(30 75 35 88)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 4A ··· 4H 4I ··· 4P 6A ··· 6N 6O ··· 6V 12A ··· 12P 12Q ··· 12AF order 1 2 ··· 2 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 2 2 2 2 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 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 C6 C6 C6 C6 C6 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 kernel C6×C22⋊Q8 C6×C22⋊C4 C6×C4⋊C4 C3×C22⋊Q8 C23×C12 Q8×C2×C6 C2×C22⋊Q8 C2×C22⋊C4 C2×C4⋊C4 C22⋊Q8 C23×C4 C22×Q8 C2×C12 C22×C6 C2×C6 C2×C4 C23 C22 # reps 1 2 3 8 1 1 2 4 6 16 2 2 4 4 4 8 8 8

Matrix representation of C6×C22⋊Q8 in GL5(𝔽13)

 12 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 3 0 0 0 0 0 3
,
 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 5 0 0 0 0 0 8 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 12 0

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

C6×C22⋊Q8 in GAP, Magma, Sage, TeX

C_6\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C6xC2^2:Q8");
// GroupNames label

G:=SmallGroup(192,1412);
// by ID

G=gap.SmallGroup(192,1412);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,344,2102]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^2=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,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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