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## G = C3×C23⋊Q8order 192 = 26·3

### Direct product of C3 and C23⋊Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×C23⋊Q8
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — Q8×C2×C6 — C3×C23⋊Q8
 Lower central C1 — C23 — C3×C23⋊Q8
 Upper central C1 — C22×C6 — C3×C23⋊Q8

Generators and relations for C3×C23⋊Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 362 in 202 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C22×Q8, C3×C22⋊C4, C22×C12, C6×Q8, C23×C6, C23⋊Q8, C3×C2.C42, C6×C22⋊C4, Q8×C2×C6, C3×C23⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22≀C2, C22⋊Q8, C4.4D4, C6×D4, C6×Q8, C3×C4○D4, C23⋊Q8, C3×C22≀C2, C3×C22⋊Q8, C3×C4.4D4, C3×C23⋊Q8

Smallest permutation representation of C3×C23⋊Q8
On 96 points
Generators in S96
(1 37 33)(2 38 34)(3 39 35)(4 40 36)(5 96 92)(6 93 89)(7 94 90)(8 95 91)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 49 53)(22 50 54)(23 51 55)(24 52 56)(25 68 29)(26 65 30)(27 66 31)(28 67 32)(41 73 45)(42 74 46)(43 75 47)(44 76 48)(57 61 69)(58 62 70)(59 63 71)(60 64 72)(77 85 81)(78 86 82)(79 87 83)(80 88 84)
(1 3)(2 44)(4 42)(5 86)(6 30)(7 88)(8 32)(9 11)(10 22)(12 24)(13 15)(14 54)(16 56)(17 19)(18 50)(20 52)(21 23)(25 60)(26 93)(27 58)(28 95)(29 72)(31 70)(33 35)(34 48)(36 46)(37 39)(38 76)(40 74)(41 43)(45 47)(49 51)(53 55)(57 83)(59 81)(61 79)(62 66)(63 77)(64 68)(65 89)(67 91)(69 87)(71 85)(73 75)(78 92)(80 90)(82 96)(84 94)
(1 9)(2 10)(3 11)(4 12)(5 31)(6 32)(7 29)(8 30)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 43)(22 44)(23 41)(24 42)(25 94)(26 95)(27 96)(28 93)(45 55)(46 56)(47 53)(48 54)(49 75)(50 76)(51 73)(52 74)(57 81)(58 82)(59 83)(60 84)(61 77)(62 78)(63 79)(64 80)(65 91)(66 92)(67 89)(68 90)(69 85)(70 86)(71 87)(72 88)
(1 41)(2 42)(3 43)(4 44)(5 72)(6 69)(7 70)(8 71)(9 23)(10 24)(11 21)(12 22)(13 55)(14 56)(15 53)(16 54)(17 51)(18 52)(19 49)(20 50)(25 82)(26 83)(27 84)(28 81)(29 86)(30 87)(31 88)(32 85)(33 45)(34 46)(35 47)(36 48)(37 73)(38 74)(39 75)(40 76)(57 93)(58 94)(59 95)(60 96)(61 89)(62 90)(63 91)(64 92)(65 79)(66 80)(67 77)(68 78)
(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 61 3 63)(2 64 4 62)(5 76 7 74)(6 75 8 73)(9 77 11 79)(10 80 12 78)(13 81 15 83)(14 84 16 82)(17 85 19 87)(18 88 20 86)(21 65 23 67)(22 68 24 66)(25 56 27 54)(26 55 28 53)(29 52 31 50)(30 51 32 49)(33 57 35 59)(34 60 36 58)(37 69 39 71)(38 72 40 70)(41 89 43 91)(42 92 44 90)(45 93 47 95)(46 96 48 94)

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

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

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

66 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3A 3B 4A ··· 4L 6A ··· 6N 6O 6P 6Q 6R 12A ··· 12X order 1 2 ··· 2 2 2 3 3 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 1 1 4 ··· 4 1 ··· 1 4 4 4 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 kernel C3×C23⋊Q8 C3×C2.C42 C6×C22⋊C4 Q8×C2×C6 C23⋊Q8 C2.C42 C2×C22⋊C4 C22×Q8 C2×C12 C22×C6 C2×C6 C2×C4 C23 C22 # reps 1 3 3 1 2 6 6 2 6 2 6 12 4 12

Matrix representation of C3×C23⋊Q8 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 3
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 12 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 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 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 8 0 0 0 0 0 8 5
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 10 0 0 0 0 0 8 0 0 0 0 0 0 6 1 0 0 0 0 2 7

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,10,8,0,0,0,0,0,0,6,2,0,0,0,0,1,7] >;

C3×C23⋊Q8 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes Q_8
% in TeX

G:=Group("C3xC2^3:Q8");
// GroupNames label

G:=SmallGroup(192,826);
// by ID

G=gap.SmallGroup(192,826);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,176,1094,1059]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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