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## G = C2×C32⋊7D4order 144 = 24·32

### Direct product of C2 and C32⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C32⋊7D4
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C32⋊7D4
 Lower central C32 — C3×C6 — C2×C32⋊7D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C327D4
G = < a,b,c,d,e | a2=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 514 in 162 conjugacy classes, 59 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×2], C22 [×6], S3 [×8], C6 [×12], C6 [×8], C2×C4, D4 [×4], C23, C23, C32, Dic3 [×8], D6 [×16], C2×C6 [×12], C2×C6 [×8], C2×D4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×4], C3⋊D4 [×16], C22×S3 [×4], C22×C6 [×4], C3⋊Dic3 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C2×C3⋊D4 [×4], C2×C3⋊Dic3, C327D4 [×4], C22×C3⋊S3, C2×C62, C2×C327D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C2×C3⋊S3 [×3], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C2×C327D4

Smallest permutation representation of C2×C327D4
On 72 points
Generators in S72
(1 44)(2 41)(3 42)(4 43)(5 67)(6 68)(7 65)(8 66)(9 30)(10 31)(11 32)(12 29)(13 72)(14 69)(15 70)(16 71)(17 35)(18 36)(19 33)(20 34)(21 39)(22 40)(23 37)(24 38)(25 48)(26 45)(27 46)(28 47)(49 56)(50 53)(51 54)(52 55)(57 63)(58 64)(59 61)(60 62)
(1 71 36)(2 33 72)(3 69 34)(4 35 70)(5 37 12)(6 9 38)(7 39 10)(8 11 40)(13 41 19)(14 20 42)(15 43 17)(16 18 44)(21 31 65)(22 66 32)(23 29 67)(24 68 30)(25 63 54)(26 55 64)(27 61 56)(28 53 62)(45 52 58)(46 59 49)(47 50 60)(48 57 51)
(1 8 27)(2 28 5)(3 6 25)(4 26 7)(9 63 69)(10 70 64)(11 61 71)(12 72 62)(13 60 29)(14 30 57)(15 58 31)(16 32 59)(17 52 21)(18 22 49)(19 50 23)(20 24 51)(33 53 37)(34 38 54)(35 55 39)(36 40 56)(41 47 67)(42 68 48)(43 45 65)(44 66 46)
(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)
(1 44)(2 43)(3 42)(4 41)(5 45)(6 48)(7 47)(8 46)(9 51)(10 50)(11 49)(12 52)(13 35)(14 34)(15 33)(16 36)(17 72)(18 71)(19 70)(20 69)(21 62)(22 61)(23 64)(24 63)(25 68)(26 67)(27 66)(28 65)(29 55)(30 54)(31 53)(32 56)(37 58)(38 57)(39 60)(40 59)

G:=sub<Sym(72)| (1,44)(2,41)(3,42)(4,43)(5,67)(6,68)(7,65)(8,66)(9,30)(10,31)(11,32)(12,29)(13,72)(14,69)(15,70)(16,71)(17,35)(18,36)(19,33)(20,34)(21,39)(22,40)(23,37)(24,38)(25,48)(26,45)(27,46)(28,47)(49,56)(50,53)(51,54)(52,55)(57,63)(58,64)(59,61)(60,62), (1,71,36)(2,33,72)(3,69,34)(4,35,70)(5,37,12)(6,9,38)(7,39,10)(8,11,40)(13,41,19)(14,20,42)(15,43,17)(16,18,44)(21,31,65)(22,66,32)(23,29,67)(24,68,30)(25,63,54)(26,55,64)(27,61,56)(28,53,62)(45,52,58)(46,59,49)(47,50,60)(48,57,51), (1,8,27)(2,28,5)(3,6,25)(4,26,7)(9,63,69)(10,70,64)(11,61,71)(12,72,62)(13,60,29)(14,30,57)(15,58,31)(16,32,59)(17,52,21)(18,22,49)(19,50,23)(20,24,51)(33,53,37)(34,38,54)(35,55,39)(36,40,56)(41,47,67)(42,68,48)(43,45,65)(44,66,46), (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), (1,44)(2,43)(3,42)(4,41)(5,45)(6,48)(7,47)(8,46)(9,51)(10,50)(11,49)(12,52)(13,35)(14,34)(15,33)(16,36)(17,72)(18,71)(19,70)(20,69)(21,62)(22,61)(23,64)(24,63)(25,68)(26,67)(27,66)(28,65)(29,55)(30,54)(31,53)(32,56)(37,58)(38,57)(39,60)(40,59)>;

G:=Group( (1,44)(2,41)(3,42)(4,43)(5,67)(6,68)(7,65)(8,66)(9,30)(10,31)(11,32)(12,29)(13,72)(14,69)(15,70)(16,71)(17,35)(18,36)(19,33)(20,34)(21,39)(22,40)(23,37)(24,38)(25,48)(26,45)(27,46)(28,47)(49,56)(50,53)(51,54)(52,55)(57,63)(58,64)(59,61)(60,62), (1,71,36)(2,33,72)(3,69,34)(4,35,70)(5,37,12)(6,9,38)(7,39,10)(8,11,40)(13,41,19)(14,20,42)(15,43,17)(16,18,44)(21,31,65)(22,66,32)(23,29,67)(24,68,30)(25,63,54)(26,55,64)(27,61,56)(28,53,62)(45,52,58)(46,59,49)(47,50,60)(48,57,51), (1,8,27)(2,28,5)(3,6,25)(4,26,7)(9,63,69)(10,70,64)(11,61,71)(12,72,62)(13,60,29)(14,30,57)(15,58,31)(16,32,59)(17,52,21)(18,22,49)(19,50,23)(20,24,51)(33,53,37)(34,38,54)(35,55,39)(36,40,56)(41,47,67)(42,68,48)(43,45,65)(44,66,46), (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), (1,44)(2,43)(3,42)(4,41)(5,45)(6,48)(7,47)(8,46)(9,51)(10,50)(11,49)(12,52)(13,35)(14,34)(15,33)(16,36)(17,72)(18,71)(19,70)(20,69)(21,62)(22,61)(23,64)(24,63)(25,68)(26,67)(27,66)(28,65)(29,55)(30,54)(31,53)(32,56)(37,58)(38,57)(39,60)(40,59) );

G=PermutationGroup([(1,44),(2,41),(3,42),(4,43),(5,67),(6,68),(7,65),(8,66),(9,30),(10,31),(11,32),(12,29),(13,72),(14,69),(15,70),(16,71),(17,35),(18,36),(19,33),(20,34),(21,39),(22,40),(23,37),(24,38),(25,48),(26,45),(27,46),(28,47),(49,56),(50,53),(51,54),(52,55),(57,63),(58,64),(59,61),(60,62)], [(1,71,36),(2,33,72),(3,69,34),(4,35,70),(5,37,12),(6,9,38),(7,39,10),(8,11,40),(13,41,19),(14,20,42),(15,43,17),(16,18,44),(21,31,65),(22,66,32),(23,29,67),(24,68,30),(25,63,54),(26,55,64),(27,61,56),(28,53,62),(45,52,58),(46,59,49),(47,50,60),(48,57,51)], [(1,8,27),(2,28,5),(3,6,25),(4,26,7),(9,63,69),(10,70,64),(11,61,71),(12,72,62),(13,60,29),(14,30,57),(15,58,31),(16,32,59),(17,52,21),(18,22,49),(19,50,23),(20,24,51),(33,53,37),(34,38,54),(35,55,39),(36,40,56),(41,47,67),(42,68,48),(43,45,65),(44,66,46)], [(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)], [(1,44),(2,43),(3,42),(4,41),(5,45),(6,48),(7,47),(8,46),(9,51),(10,50),(11,49),(12,52),(13,35),(14,34),(15,33),(16,36),(17,72),(18,71),(19,70),(20,69),(21,62),(22,61),(23,64),(24,63),(25,68),(26,67),(27,66),(28,65),(29,55),(30,54),(31,53),(32,56),(37,58),(38,57),(39,60),(40,59)])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A ··· 6AB order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 ··· 6 size 1 1 1 1 2 2 18 18 2 2 2 2 18 18 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 C3⋊D4 kernel C2×C32⋊7D4 C2×C3⋊Dic3 C32⋊7D4 C22×C3⋊S3 C2×C62 C22×C6 C3×C6 C2×C6 C6 # reps 1 1 4 1 1 4 2 12 16

Matrix representation of C2×C327D4 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 0 1 0 0 12 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 1 0 0 12 0
,
 11 9 0 0 11 2 0 0 0 0 9 2 0 0 11 4
,
 1 0 0 0 12 12 0 0 0 0 0 12 0 0 12 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[11,11,0,0,9,2,0,0,0,0,9,11,0,0,2,4],[1,12,0,0,0,12,0,0,0,0,0,12,0,0,12,0] >;

C2×C327D4 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_7D_4
% in TeX

G:=Group("C2xC3^2:7D4");
// GroupNames label

G:=SmallGroup(144,177);
// by ID

G=gap.SmallGroup(144,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,964,3461]);
// Polycyclic

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

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