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## G = C4○D4⋊4Dic3order 192 = 26·3

### 2nd semidirect product of C4○D4 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4○D4⋊4Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C23.26D6 — C4○D4⋊4Dic3
 Lower central C3 — C6 — C12 — C4○D4⋊4Dic3
 Upper central C1 — C2×C4 — C22×C4 — C2×C4○D4

Generators and relations for C4○D44Dic3
G = < a,b,c,d,e | a4=c2=d6=1, b2=a2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, ebe-1=a-1bc, cd=dc, ece-1=a2c, ede-1=d-1 >

Subgroups: 328 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C23.24D4, D4⋊Dic3, Q82Dic3, C22×C3⋊C8, C23.26D6, C6×C4○D4, C4○D44Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C4○D8, C6.D4, C22×Dic3, C2×C3⋊D4, C23.24D4, Q8.13D6, C2×C6.D4, C4○D44Dic3

Smallest permutation representation of C4○D44Dic3
On 96 points
Generators in S96
(1 40 16 36)(2 41 17 31)(3 42 18 32)(4 37 13 33)(5 38 14 34)(6 39 15 35)(7 74 93 69)(8 75 94 70)(9 76 95 71)(10 77 96 72)(11 78 91 67)(12 73 92 68)(19 52 29 43)(20 53 30 44)(21 54 25 45)(22 49 26 46)(23 50 27 47)(24 51 28 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)
(1 40 16 36)(2 41 17 31)(3 42 18 32)(4 37 13 33)(5 38 14 34)(6 39 15 35)(7 57 93 62)(8 58 94 63)(9 59 95 64)(10 60 96 65)(11 55 91 66)(12 56 92 61)(19 43 29 52)(20 44 30 53)(21 45 25 54)(22 46 26 49)(23 47 27 50)(24 48 28 51)(67 79 78 90)(68 80 73 85)(69 81 74 86)(70 82 75 87)(71 83 76 88)(72 84 77 89)
(1 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 57)(8 58)(9 59)(10 60)(11 55)(12 56)(13 54)(14 49)(15 50)(16 51)(17 52)(18 53)(19 41)(20 42)(21 37)(22 38)(23 39)(24 40)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)(61 92)(62 93)(63 94)(64 95)(65 96)(66 91)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)
(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 57 4 60)(2 56 5 59)(3 55 6 58)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 65 16 62)(14 64 17 61)(15 63 18 66)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 80 34 83)(32 79 35 82)(33 84 36 81)(37 89 40 86)(38 88 41 85)(39 87 42 90)(43 92 46 95)(44 91 47 94)(45 96 48 93)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 D6 Dic3 C3⋊D4 C3⋊D4 C4○D8 Q8.13D6 kernel C4○D4⋊4Dic3 D4⋊Dic3 Q8⋊2Dic3 C22×C3⋊C8 C23.26D6 C6×C4○D4 C3×C4○D4 C2×C4○D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C6 C2 # reps 1 2 2 1 1 1 8 1 3 1 1 1 1 4 6 2 8 4

Matrix representation of C4○D44Dic3 in GL5(𝔽73)

 1 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 27 0 0 0 0 30 46 0 0 0 0 0 72 0 0 0 0 0 1
,
 1 0 0 0 0 0 27 39 0 0 0 30 46 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 64
,
 46 0 0 0 0 0 0 15 0 0 0 39 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,27,30,0,0,0,0,46,0,0,0,0,0,72,0,0,0,0,0,1],[1,0,0,0,0,0,27,30,0,0,0,39,46,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[46,0,0,0,0,0,0,39,0,0,0,15,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C4○D44Dic3 in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes_4{\rm Dic}_3
% in TeX

G:=Group("C4oD4:4Dic3");
// GroupNames label

G:=SmallGroup(192,792);
// by ID

G=gap.SmallGroup(192,792);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^6=1,b^2=a^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,b*d=d*b,e*b*e^-1=a^-1*b*c,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=d^-1>;
// generators/relations

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