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## G = C4⋊C4.230D6order 192 = 26·3

### 8th non-split extension by C4⋊C4 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C4⋊C4.230D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.48D4 — C4⋊C4.230D6
 Lower central C3 — C6 — C2×C12 — C4⋊C4.230D6
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for C4⋊C4.230D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-1 >

Subgroups: 248 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×6], C22×C6, C22⋊C8, Q8⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C22×C12, C22×C12, C23.48D4, C6.Q16 [×2], C6.SD16 [×2], C12.55D4, C12.48D4, C6×C4⋊C4, C4⋊C4.230D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C2×Q16, C8⋊C22, C3⋊Q16 [×2], C4○D12 [×2], C2×C3⋊D4, C23.48D4, C23.28D6, D126C22, C2×C3⋊Q16, C4⋊C4.230D6

Smallest permutation representation of C4⋊C4.230D6
On 96 points
Generators in S96
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 13 10 16)(8 14 11 17)(9 15 12 18)(25 46 28 43)(26 47 29 44)(27 48 30 45)(31 40 34 37)(32 41 35 38)(33 42 36 39)(49 73 64 71)(50 74 65 72)(51 75 66 67)(52 76 61 68)(53 77 62 69)(54 78 63 70)(55 93 79 87)(56 94 80 88)(57 95 81 89)(58 96 82 90)(59 91 83 85)(60 92 84 86)
(1 34 10 25)(2 35 11 26)(3 36 12 27)(4 31 7 28)(5 32 8 29)(6 33 9 30)(13 46 22 37)(14 47 23 38)(15 48 24 39)(16 43 19 40)(17 44 20 41)(18 45 21 42)(49 91 61 88)(50 92 62 89)(51 93 63 90)(52 94 64 85)(53 95 65 86)(54 96 66 87)(55 70 82 75)(56 71 83 76)(57 72 84 77)(58 67 79 78)(59 68 80 73)(60 69 81 74)
(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 63 10 51)(2 65 11 53)(3 61 12 49)(4 54 7 66)(5 50 8 62)(6 52 9 64)(13 75 22 70)(14 77 23 72)(15 73 24 68)(16 67 19 78)(17 69 20 74)(18 71 21 76)(25 55 34 82)(26 57 35 84)(27 59 36 80)(28 79 31 58)(29 81 32 60)(30 83 33 56)(37 96 46 87)(38 92 47 89)(39 94 48 85)(40 90 43 93)(41 86 44 95)(42 88 45 91)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 Q16 C3⋊D4 C3⋊D4 C4○D12 C8⋊C22 C3⋊Q16 D12⋊6C22 kernel C4⋊C4.230D6 C6.Q16 C6.SD16 C12.55D4 C12.48D4 C6×C4⋊C4 C2×C4⋊C4 C2×C12 C22×C6 C4⋊C4 C22×C4 C12 C2×C6 C2×C4 C23 C4 C6 C22 C2 # reps 1 2 2 1 1 1 1 1 1 2 1 4 4 2 2 8 1 2 2

Matrix representation of C4⋊C4.230D6 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 0
,
 27 0 0 0 0 27 0 0 0 0 72 61 0 0 61 1
,
 8 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 0 64 0 0 65 0 0 0 0 0 62 30 0 0 30 11
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,0],[27,0,0,0,0,27,0,0,0,0,72,61,0,0,61,1],[8,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,65,0,0,64,0,0,0,0,0,62,30,0,0,30,11] >;

C4⋊C4.230D6 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{230}D_6
% in TeX

G:=Group("C4:C4.230D6");
// GroupNames label

G:=SmallGroup(192,529);
// by ID

G=gap.SmallGroup(192,529);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,268,1123,297,136,6278]);
// Polycyclic

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

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