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## G = C22×Q8⋊2S3order 192 = 26·3

### Direct product of C22 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C22×Q8⋊2S3
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C22×D12 — C22×Q8⋊2S3
 Lower central C3 — C6 — C12 — C22×Q8⋊2S3
 Upper central C1 — C23 — C22×C4 — C22×Q8

Generators and relations for C22×Q82S3
G = < a,b,c,d,e,f | a2=b2=c4=e3=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, fdf=c-1d, fef=e-1 >

Subgroups: 840 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], S3 [×4], C6, C6 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×10], Q8 [×4], Q8 [×6], C23, C23 [×10], C12, C12 [×3], C12 [×4], D6 [×16], C2×C6 [×7], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×9], C2×Q8 [×6], C2×Q8 [×3], C24, C3⋊C8 [×4], D12 [×4], D12 [×6], C2×C12 [×6], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×6], C22×S3 [×10], C22×C6, C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C2×C3⋊C8 [×6], Q82S3 [×16], C2×D12 [×6], C2×D12 [×3], C22×C12, C22×C12, C6×Q8 [×6], C6×Q8 [×3], S3×C23, C22×SD16, C22×C3⋊C8, C2×Q82S3 [×12], C22×D12, Q8×C2×C6, C22×Q82S3
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], SD16 [×4], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C2×SD16 [×6], C22×D4, Q82S3 [×4], C2×C3⋊D4 [×6], S3×C23, C22×SD16, C2×Q82S3 [×6], C22×C3⋊D4, C22×Q82S3

Smallest permutation representation of C22×Q82S3
On 96 points
Generators in S96
(1 60)(2 57)(3 58)(4 59)(5 51)(6 52)(7 49)(8 50)(9 55)(10 56)(11 53)(12 54)(13 63)(14 64)(15 61)(16 62)(17 67)(18 68)(19 65)(20 66)(21 71)(22 72)(23 69)(24 70)(25 75)(26 76)(27 73)(28 74)(29 79)(30 80)(31 77)(32 78)(33 83)(34 84)(35 81)(36 82)(37 87)(38 88)(39 85)(40 86)(41 91)(42 92)(43 89)(44 90)(45 95)(46 96)(47 93)(48 94)
(1 34)(2 35)(3 36)(4 33)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 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 96 3 94)(2 95 4 93)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)(21 83 23 81)(22 82 24 84)(25 92 27 90)(26 91 28 89)(29 88 31 86)(30 87 32 85)(33 69 35 71)(34 72 36 70)(37 78 39 80)(38 77 40 79)(41 74 43 76)(42 73 44 75)(45 59 47 57)(46 58 48 60)(49 68 51 66)(50 67 52 65)(53 64 55 62)(54 63 56 61)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 93 10)(6 94 11)(7 95 12)(8 96 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 37 42)(34 38 43)(35 39 44)(36 40 41)(45 54 49)(46 55 50)(47 56 51)(48 53 52)(57 61 66)(58 62 67)(59 63 68)(60 64 65)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 85 90)(82 86 91)(83 87 92)(84 88 89)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 29)(26 32)(27 31)(28 30)(33 35)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 53)(50 56)(51 55)(52 54)(57 59)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 77)(74 80)(75 79)(76 78)(81 83)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 8A ··· 8H 12A ··· 12L order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 12 12 12 12 2 2 2 2 2 4 4 4 4 2 ··· 2 6 ··· 6 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 C3⋊D4 C3⋊D4 Q8⋊2S3 kernel C22×Q8⋊2S3 C22×C3⋊C8 C2×Q8⋊2S3 C22×D12 Q8×C2×C6 C22×Q8 C2×C12 C22×C6 C22×C4 C2×Q8 C2×C6 C2×C4 C23 C22 # reps 1 1 12 1 1 1 3 1 1 6 8 6 2 4

Matrix representation of C22×Q82S3 in GL6(𝔽73)

 1 0 0 0 0 0 0 72 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
,
 72 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 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 30 60 0 0 0 0 13 43 0 0 0 0 0 0 6 6 0 0 0 0 6 67
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,30,13,0,0,0,0,60,43,0,0,0,0,0,0,6,6,0,0,0,0,6,67],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C22×Q82S3 in GAP, Magma, Sage, TeX

C_2^2\times Q_8\rtimes_2S_3
% in TeX

G:=Group("C2^2xQ8:2S3");
// GroupNames label

G:=SmallGroup(192,1366);
// by ID

G=gap.SmallGroup(192,1366);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,136,1684,235,102,6278]);
// Polycyclic

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

׿
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