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G = C6.S3≀C2order 432 = 24·33

4th non-split extension by C6 of S3≀C2 acting via S3≀C2/C32⋊C4=C2

non-abelian, soluble

Aliases: C6.4S3≀C2, He3⋊C41C4, He31(C4⋊C4), He3⋊C2.Q8, C2.1(He3⋊D4), (C2×He3).4D4, C3.(C3⋊S3.Q8), (C2×He3⋊C4).2C2, He3⋊(C2×C4).2C2, He3⋊C2.3(C2×C4), (C2×He3⋊C2).1C22, SmallGroup(432,237)

Series: Derived Chief Lower central Upper central

C1C3He3He3⋊C2 — C6.S3≀C2
C1C3He3He3⋊C2C2×He3⋊C2He3⋊(C2×C4) — C6.S3≀C2
He3He3⋊C2 — C6.S3≀C2
C1C2

Generators and relations for C6.S3≀C2
 G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a2b, dbd-1=a4c, ebe-1=b-1, dcd-1=a2b-1, ece-1=a2c, ede-1=d-1 >

Subgroups: 523 in 71 conjugacy classes, 15 normal (11 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4 [×4], C22, S3 [×4], C6, C6 [×4], C2×C4 [×3], C32 [×2], Dic3 [×4], C12 [×4], D6 [×2], C2×C6, C4⋊C4, C3×S3 [×4], C3×C6 [×2], C4×S3 [×2], C2×Dic3 [×2], C2×C12, He3, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×2], C4⋊Dic3, He3⋊C2 [×2], C2×He3, S3×Dic3 [×2], C32⋊C12 [×2], He3⋊C4 [×2], C2×He3⋊C2, He3⋊(C2×C4) [×2], C2×He3⋊C4, C6.S3≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, S3≀C2, C3⋊S3.Q8, He3⋊D4, C6.S3≀C2

Character table of C6.S3≀C2

 class 12A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D6E12A12B12C12D12E12F12G12H
 size 1199212121818181818182121218181818181836363636
ρ111111111111111111111111111    trivial
ρ211111111-1-1-11-1111111111-1-1-1-1    linear of order 2
ρ31111111-11-1-1-1111111-1-1-1-11-11-1    linear of order 2
ρ41111111-1-111-1-111111-1-1-1-1-11-11    linear of order 2
ρ51-1-111111i-ii-1-i-1-1-1-111-11-1ii-i-i    linear of order 4
ρ61-1-111111-ii-i-1i-1-1-1-111-11-1-i-iii    linear of order 4
ρ71-1-11111-1ii-i1-i-1-1-1-11-11-11i-i-ii    linear of order 4
ρ81-1-11111-1-i-ii1i-1-1-1-11-11-11-iii-i    linear of order 4
ρ922-2-2222000000222-2-200000000    orthogonal lifted from D4
ρ102-22-2222000000-2-2-22-200000000    symplectic lifted from Q8, Schur index 2
ρ11440041-200-2-20041-20000000101    orthogonal lifted from S3≀C2
ρ12440041-200220041-20000000-10-1    orthogonal lifted from S3≀C2
ρ1344004-210-2000-24-210000001010    orthogonal lifted from S3≀C2
ρ1444004-210200024-21000000-10-10    orthogonal lifted from S3≀C2
ρ154-4004-2102i000-2i-42-1000000-i0i0    complex lifted from C3⋊S3.Q8
ρ164-40041-200-2i2i00-4-120000000-i0i    complex lifted from C3⋊S3.Q8
ρ174-40041-2002i-2i00-4-120000000i0-i    complex lifted from C3⋊S3.Q8
ρ184-4004-210-2i0002i-42-1000000i0-i0    complex lifted from C3⋊S3.Q8
ρ1966-2-2-300-2000-20-3001111110000    orthogonal lifted from He3⋊D4
ρ206622-300000000-300-1-1-333-30000    orthogonal lifted from He3⋊D4
ρ2166-2-2-300200020-30011-1-1-1-10000    orthogonal lifted from He3⋊D4
ρ226622-300000000-300-1-13-3-330000    orthogonal lifted from He3⋊D4
ρ236-62-2-300-200020300-111-11-10000    symplectic faithful, Schur index 2
ρ246-6-22-3000000003001-1-3-3330000    symplectic faithful, Schur index 2
ρ256-6-22-3000000003001-133-3-30000    symplectic faithful, Schur index 2
ρ266-62-2-3002000-20300-11-11-110000    symplectic faithful, Schur index 2

Smallest permutation representation of C6.S3≀C2
On 72 points
Generators in S72
(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 17 19)(2 18 20)(3 13 21)(4 14 22)(5 15 23)(6 16 24)(7 66 68)(8 61 69)(9 62 70)(10 63 71)(11 64 72)(12 65 67)(25 40 32)(26 41 33)(27 42 34)(28 37 35)(29 38 36)(30 39 31)(43 54 57)(44 49 58)(45 50 59)(46 51 60)(47 52 55)(48 53 56)
(1 15 19)(2 16 20)(3 17 21)(4 18 22)(5 13 23)(6 14 24)(7 72 64)(8 67 65)(9 68 66)(10 69 61)(11 70 62)(12 71 63)(25 42 34)(26 37 35)(27 38 36)(28 39 31)(29 40 32)(30 41 33)(43 59 54)(44 60 49)(45 55 50)(46 56 51)(47 57 52)(48 58 53)
(1 70 4 67)(2 71 5 68)(3 72 6 69)(7 22 63 17)(8 23 64 18)(9 24 65 13)(10 19 66 14)(11 20 61 15)(12 21 62 16)(25 57 37 46)(26 58 38 47)(27 59 39 48)(28 60 40 43)(29 55 41 44)(30 56 42 45)(31 49 34 52)(32 50 35 53)(33 51 36 54)
(1 50 4 53)(2 49 5 52)(3 54 6 51)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 43 16 46)(14 48 17 45)(15 47 18 44)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 65 28 62)(26 64 29 61)(27 63 30 66)(31 68 34 71)(32 67 35 70)(33 72 36 69)

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

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

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

Matrix representation of C6.S3≀C2 in GL10(𝔽13)

12000000000
01200000000
00120000000
00012000000
0000900000
0000090000
0000009000
00001000300
0000910030
0000901003
,
01200000000
11200000000
0901000000
401212000000
0000306000
00003010000
000012110000
0000901102
00000000012
00000000112
,
01200000000
11200000000
401212000000
0910000000
0000306000
00000010000
00000310000
0000901102
00000001012
000011204912
,
0058000000
66105000000
1170000000
1970000000
000081111000
0000605000
0000650000
00006101081111
0000270650
0000207605
,
00121000000
441112000000
101090000000
91090000000
0000755399
000047012100
000040712010
0000733522
000012010007
000012100070

G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,9,0,0,10,9,9,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[0,1,0,4,0,0,0,0,0,0,12,12,9,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,3,3,12,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,10,10,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,12,12],[0,1,4,0,0,0,0,0,0,0,12,12,0,9,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,1,0,0,0,0,0,0,3,0,0,12,0,0,0,0,6,10,10,1,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,2,12,12],[0,6,1,1,0,0,0,0,0,0,0,6,1,9,0,0,0,0,0,0,5,10,7,7,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,0,0,0,8,6,6,6,2,2,0,0,0,0,11,0,5,10,7,0,0,0,0,0,11,5,0,10,0,7,0,0,0,0,0,0,0,8,6,6,0,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,11,0,5],[0,4,10,9,0,0,0,0,0,0,0,4,10,10,0,0,0,0,0,0,12,11,9,9,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,7,4,4,7,12,12,0,0,0,0,5,7,0,3,0,10,0,0,0,0,5,0,7,3,10,0,0,0,0,0,3,12,12,5,0,0,0,0,0,0,9,10,0,2,0,7,0,0,0,0,9,0,10,2,7,0] >;

C6.S3≀C2 in GAP, Magma, Sage, TeX

C_6.S_3\wr C_2
% in TeX

G:=Group("C6.S3wrC2");
// GroupNames label

G:=SmallGroup(432,237);
// by ID

G=gap.SmallGroup(432,237);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,1124,851,298,348,1027,537,14118,7069]);
// Polycyclic

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

Export

Character table of C6.S3≀C2 in TeX

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