Copied to
clipboard

## G = (C3×D4).32D4order 192 = 26·3

### 2nd non-split extension by C3×D4 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C3×D4).32D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — C2×D4.S3 — (C3×D4).32D4
 Lower central C3 — C6 — C2×C12 — (C3×D4).32D4
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for (C3×D4).32D4
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=dbd-1=ebe-1=b-1, dcd-1=ece-1=bc, ede-1=d-1 >

Subgroups: 360 in 152 conjugacy classes, 47 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×5], C22, C22 [×7], C6 [×3], C6 [×3], C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×2], D4 [×5], Q8 [×2], Q8 [×3], C23, C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×7], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4 [×4], C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×7], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×C6, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, D4.S3 [×2], C3⋊Q16 [×2], C6.D4, C2×Dic6, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], D4.7D4, C12.55D4, D4⋊Dic3, Q82Dic3, C12.48D4, C2×D4.S3, C2×C3⋊Q16, C6×C4○D4, (C3×D4).32D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C4○D8, C8.C22, C2×C3⋊D4 [×3], D4.7D4, Q8.13D6, Q8.14D6, C244S3, (C3×D4).32D4

Smallest permutation representation of (C3×D4).32D4
On 96 points
Generators in S96
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 95 10)(6 96 11)(7 93 12)(8 94 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 44 39)(34 41 40)(35 42 37)(36 43 38)(45 56 51)(46 53 52)(47 54 49)(48 55 50)(57 63 68)(58 64 65)(59 61 66)(60 62 67)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 92 87)(82 89 88)(83 90 85)(84 91 86)
(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)(2 48)(3 47)(4 46)(5 66)(6 65)(7 68)(8 67)(9 62)(10 61)(11 64)(12 63)(13 55)(14 54)(15 53)(16 56)(17 51)(18 50)(19 49)(20 52)(21 34)(22 33)(23 36)(24 35)(25 40)(26 39)(27 38)(28 37)(29 42)(30 41)(31 44)(32 43)(57 93)(58 96)(59 95)(60 94)(69 81)(70 84)(71 83)(72 82)(73 87)(74 86)(75 85)(76 88)(77 89)(78 92)(79 91)(80 90)
(1 69 22 57)(2 72 23 60)(3 71 24 59)(4 70 21 58)(5 53 85 41)(6 56 86 44)(7 55 87 43)(8 54 88 42)(9 49 89 37)(10 52 90 40)(11 51 91 39)(12 50 92 38)(13 76 32 67)(14 75 29 66)(15 74 30 65)(16 73 31 68)(17 78 26 63)(18 77 27 62)(19 80 28 61)(20 79 25 64)(33 96 45 84)(34 95 46 83)(35 94 47 82)(36 93 48 81)
(1 60 3 58)(2 59 4 57)(5 56 7 54)(6 55 8 53)(9 52 11 50)(10 51 12 49)(13 66 15 68)(14 65 16 67)(17 62 19 64)(18 61 20 63)(21 69 23 71)(22 72 24 70)(25 78 27 80)(26 77 28 79)(29 74 31 76)(30 73 32 75)(33 81 35 83)(34 84 36 82)(37 90 39 92)(38 89 40 91)(41 86 43 88)(42 85 44 87)(45 93 47 95)(46 96 48 94)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D ··· 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 1 1 4 4 4 2 2 2 2 2 4 4 24 24 2 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C3⋊D4 C3⋊D4 C4○D8 C8.C22 Q8.13D6 Q8.14D6 kernel (C3×D4).32D4 C12.55D4 D4⋊Dic3 Q8⋊2Dic3 C12.48D4 C2×D4.S3 C2×C3⋊Q16 C6×C4○D4 C2×C4○D4 C2×C12 C3×D4 C3×Q8 C22×C6 C22×C4 C2×D4 C2×Q8 C2×C4 D4 Q8 C23 C6 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 4 4 2 4 1 2 2

Matrix representation of (C3×D4).32D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 8 64 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 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 44 27
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 42 1 0 0 0 0 0 0 1 51 0 0 0 0 0 72
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 8 56 0 0 0 0 8 65 0 0 0 0 0 0 63 1 0 0 0 0 47 10
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 65 17 0 0 0 0 65 8 0 0 0 0 0 0 51 46 0 0 0 0 45 22

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,8,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,44,0,0,0,0,0,27],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,42,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,51,72],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,8,8,0,0,0,0,56,65,0,0,0,0,0,0,63,47,0,0,0,0,1,10],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,65,65,0,0,0,0,17,8,0,0,0,0,0,0,51,45,0,0,0,0,46,22] >;

(C3×D4).32D4 in GAP, Magma, Sage, TeX

(C_3\times D_4)._{32}D_4
% in TeX

G:=Group("(C3xD4).32D4");
// GroupNames label

G:=SmallGroup(192,798);
// by ID

G=gap.SmallGroup(192,798);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,184,1684,851,102,6278]);
// Polycyclic

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

׿
×
𝔽