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G = C4⋊C4.233D6order 192 = 26·3

11st non-split extension by C4⋊C4 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.233D6, C6.83(C4○D8), (C2×C12).449D4, C6.Q1628C2, (C22×C6).72D4, C4.90(C4○D12), C6.SD1627C2, C42⋊C2.5S3, C12.Q828C2, (C22×C4).124D6, C12.178(C4○D4), (C2×C12).326C23, C2.6(Q8.14D6), C2.8(Q8.13D6), C12.55D4.5C2, C36(C23.20D4), C23.26(C3⋊D4), C12.48D4.11C2, C6.105(C8.C22), C4⋊Dic3.134C22, (C2×Dic6).98C22, (C22×C12).146C22, C6.65(C22.D4), C2.16(C23.28D6), (C2×C6).455(C2×D4), (C2×C3⋊C8).86C22, (C2×C4).214(C3⋊D4), (C3×C4⋊C4).264C22, (C3×C42⋊C2).5C2, (C2×C4).426(C22×S3), C22.144(C2×C3⋊D4), SmallGroup(192,555)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C4⋊C4.233D6
C1C3C6C12C2×C12C4⋊Dic3C12.48D4 — C4⋊C4.233D6
C3C6C2×C12 — C4⋊C4.233D6
C1C22C22×C4C42⋊C2

Generators and relations for C4⋊C4.233D6
 G = < a,b,c,d | a4=b4=1, c6=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c5 >

Subgroups: 232 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6 [×3], C6, C8 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C22×C6, C22⋊C8, Q8⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C2×Dic6, C22×C12, C23.20D4, C6.Q16, C12.Q8, C6.SD16 [×2], C12.55D4, C12.48D4, C3×C42⋊C2, C4⋊C4.233D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C4○D8, C8.C22, C4○D12 [×2], C2×C3⋊D4, C23.20D4, C23.28D6, Q8.13D6, Q8.14D6, C4⋊C4.233D6

Smallest permutation representation of C4⋊C4.233D6
On 96 points
Generators in S96
(1 43 7 37)(2 44 8 38)(3 45 9 39)(4 46 10 40)(5 47 11 41)(6 48 12 42)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(49 79 55 73)(50 80 56 74)(51 81 57 75)(52 82 58 76)(53 83 59 77)(54 84 60 78)(61 70 67 64)(62 71 68 65)(63 72 69 66)(85 88 91 94)(86 89 92 95)(87 90 93 96)
(1 59 40 80)(2 60 41 81)(3 49 42 82)(4 50 43 83)(5 51 44 84)(6 52 45 73)(7 53 46 74)(8 54 47 75)(9 55 48 76)(10 56 37 77)(11 57 38 78)(12 58 39 79)(13 93 62 27)(14 94 63 28)(15 95 64 29)(16 96 65 30)(17 85 66 31)(18 86 67 32)(19 87 68 33)(20 88 69 34)(21 89 70 35)(22 90 71 36)(23 91 72 25)(24 92 61 26)
(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 31 7 25)(2 96 8 90)(3 29 9 35)(4 94 10 88)(5 27 11 33)(6 92 12 86)(13 81 19 75)(14 59 20 53)(15 79 21 73)(16 57 22 51)(17 77 23 83)(18 55 24 49)(26 39 32 45)(28 37 34 43)(30 47 36 41)(38 87 44 93)(40 85 46 91)(42 95 48 89)(50 66 56 72)(52 64 58 70)(54 62 60 68)(61 82 67 76)(63 80 69 74)(65 78 71 84)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12N
order12222344444444446666688881212121212···12
size111142222244442424222441212121222224···4

39 irreducible representations

dim11111112222222222444
type++++++++++++--
imageC1C2C2C2C2C2C2S3D4D4D6D6C4○D4C3⋊D4C3⋊D4C4○D8C4○D12C8.C22Q8.13D6Q8.14D6
kernelC4⋊C4.233D6C6.Q16C12.Q8C6.SD16C12.55D4C12.48D4C3×C42⋊C2C42⋊C2C2×C12C22×C6C4⋊C4C22×C4C12C2×C4C23C6C4C6C2C2
# reps11121111112142248122

Matrix representation of C4⋊C4.233D6 in GL6(𝔽73)

4600000
0270000
0072000
0007200
0000720
0000072
,
0460000
4600000
00436000
00133000
0000460
00006327
,
4600000
0460000
000100
0072100
000010
00002272
,
0510000
1000000
00703100
0028300
0000512
00001422

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,0,0,0,0,46,63,0,0,0,0,0,27],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,22,0,0,0,0,0,72],[0,10,0,0,0,0,51,0,0,0,0,0,0,0,70,28,0,0,0,0,31,3,0,0,0,0,0,0,51,14,0,0,0,0,2,22] >;

C4⋊C4.233D6 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{233}D_6
% in TeX

G:=Group("C4:C4.233D6");
// GroupNames label

G:=SmallGroup(192,555);
// by ID

G=gap.SmallGroup(192,555);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,232,254,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations

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