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G = C3211SD16order 144 = 24·32

2nd semidirect product of C32 and SD16 acting via SD16/Q8=C2

metabelian, supersoluble, monomial

Aliases: C12.18D6, C3211SD16, (C3×Q8)⋊3S3, Q82(C3⋊S3), (C3×C6).36D4, C324C85C2, (Q8×C32)⋊2C2, C12⋊S3.3C2, C33(Q82S3), C6.24(C3⋊D4), (C3×C12).14C22, C2.6(C327D4), C4.3(C2×C3⋊S3), SmallGroup(144,98)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C3211SD16
C1C3C32C3×C6C3×C12C12⋊S3 — C3211SD16
C32C3×C6C3×C12 — C3211SD16
C1C2C4Q8

Generators and relations for C3211SD16
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c3 >

Subgroups: 234 in 60 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, D4, Q8, C32, C12 [×4], C12 [×4], D6 [×4], SD16, C3⋊S3, C3×C6, C3⋊C8 [×4], D12 [×4], C3×Q8 [×4], C3×C12, C3×C12, C2×C3⋊S3, Q82S3 [×4], C324C8, C12⋊S3, Q8×C32, C3211SD16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], SD16, C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, Q82S3 [×4], C327D4, C3211SD16

Character table of C3211SD16

 class 12A2B3A3B3C3D4A4B6A6B6C6D8A8B12A12B12C12D12E12F12G12H12I12J12K12L
 size 113622222422221818444444444444
ρ1111111111111111111111111111    trivial
ρ211-111111-11111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ311111111-11111-1-11-1-1-1-1-1-1-1-1111    linear of order 2
ρ411-11111111111-1-1111111111111    linear of order 2
ρ5220-1-12-122-1-1-1200222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62202-1-1-122-12-1-100-1-1-1-12-1-1-12-12-1    orthogonal lifted from S3
ρ72202222-20222200-200000000-2-2-2    orthogonal lifted from D4
ρ8220-1-12-12-2-1-1-12002-2-2111111-1-1-1    orthogonal lifted from D6
ρ9220-12-1-12-2-1-12-100-11111-21-21-1-12    orthogonal lifted from D6
ρ10220-1-1-12222-1-1-100-1-1-12-1-12-1-12-1-1    orthogonal lifted from S3
ρ112202-1-1-12-2-12-1-100-1111-2111-2-12-1    orthogonal lifted from D6
ρ12220-1-1-122-22-1-1-100-111-211-2112-1-1    orthogonal lifted from D6
ρ13220-12-1-122-1-12-100-1-1-1-1-12-12-1-1-12    orthogonal lifted from S3
ρ14220-12-1-1-20-1-12-1001-3--3-3--30--30-311-2    complex lifted from C3⋊D4
ρ152202-1-1-1-20-12-1-1001--3-3-30--3--3-301-21    complex lifted from C3⋊D4
ρ16220-1-12-1-20-1-1-1200-200--3--3--3-3-3-3111    complex lifted from C3⋊D4
ρ17220-12-1-1-20-1-12-1001--3-3--3-30-30--311-2    complex lifted from C3⋊D4
ρ18220-1-1-12-202-1-1-1001-3--30-3--30-3--3-211    complex lifted from C3⋊D4
ρ192202-1-1-1-20-12-1-1001-3--3--30-3-3--301-21    complex lifted from C3⋊D4
ρ20220-1-1-12-202-1-1-1001--3-30--3-30--3-3-211    complex lifted from C3⋊D4
ρ21220-1-12-1-20-1-1-1200-200-3-3-3--3--3--3111    complex lifted from C3⋊D4
ρ222-20222200-2-2-2-2--2-2000000000000    complex lifted from SD16
ρ232-20222200-2-2-2-2-2--2000000000000    complex lifted from SD16
ρ244-40-2-24-200222-400000000000000    orthogonal lifted from Q82S3
ρ254-40-2-2-2400-422200000000000000    orthogonal lifted from Q82S3
ρ264-40-24-2-20022-4200000000000000    orthogonal lifted from Q82S3
ρ274-404-2-2-2002-42200000000000000    orthogonal lifted from Q82S3

Smallest permutation representation of C3211SD16
On 72 points
Generators in S72
(1 11 56)(2 49 12)(3 13 50)(4 51 14)(5 15 52)(6 53 16)(7 9 54)(8 55 10)(17 30 48)(18 41 31)(19 32 42)(20 43 25)(21 26 44)(22 45 27)(23 28 46)(24 47 29)(33 62 69)(34 70 63)(35 64 71)(36 72 57)(37 58 65)(38 66 59)(39 60 67)(40 68 61)
(1 22 34)(2 35 23)(3 24 36)(4 37 17)(5 18 38)(6 39 19)(7 20 40)(8 33 21)(9 43 68)(10 69 44)(11 45 70)(12 71 46)(13 47 72)(14 65 48)(15 41 66)(16 67 42)(25 61 54)(26 55 62)(27 63 56)(28 49 64)(29 57 50)(30 51 58)(31 59 52)(32 53 60)
(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)
(2 4)(3 7)(6 8)(9 50)(10 53)(11 56)(12 51)(13 54)(14 49)(15 52)(16 55)(17 35)(18 38)(19 33)(20 36)(21 39)(22 34)(23 37)(24 40)(25 72)(26 67)(27 70)(28 65)(29 68)(30 71)(31 66)(32 69)(41 59)(42 62)(43 57)(44 60)(45 63)(46 58)(47 61)(48 64)

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

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

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

C3211SD16 is a maximal subgroup of
S3×Q82S3  D126D6  Dic6.22D6  D12.13D6  SD16×C3⋊S3  C247D6  C24.35D6  C24.28D6  C62.134D4  C62.73D4  C62.74D4  He310SD16  C36.20D6  C32.3GL2(𝔽3)  C3313SD16  C3315SD16  C3327SD16  C325GL2(𝔽3)
C3211SD16 is a maximal quotient of
C12.10Dic6  C62.113D4  C62.117D4  C36.20D6  He311SD16  C3313SD16  C3315SD16  C3327SD16

Matrix representation of C3211SD16 in GL6(𝔽73)

010000
72720000
00727200
001000
000010
000001
,
100000
010000
000100
00727200
000010
000001
,
100000
72720000
00431300
00433000
00006155
000040
,
100000
72720000
0072000
001100
000010
00004872

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,43,43,0,0,0,0,13,30,0,0,0,0,0,0,61,4,0,0,0,0,55,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,0,72] >;

C3211SD16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_{11}{\rm SD}_{16}
% in TeX

G:=Group("C3^2:11SD16");
// GroupNames label

G:=SmallGroup(144,98);
// by ID

G=gap.SmallGroup(144,98);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,55,218,116,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations

Export

Character table of C3211SD16 in TeX

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