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G = D12.35C23order 192 = 26·3

16th non-split extension by D12 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.40C24, D12.35C23, 2- 1+46S3, Dic6.35C23, C35(Q8○D8), Q8○D1210C2, C4○D4.33D6, (C3×D4).39D4, C3⋊C8.19C23, (C3×Q8).39D4, C12.272(C2×D4), C4.40(S3×C23), D4⋊S3.2C22, (C2×Q8).117D6, Q8.14D612C2, Q8.13D611C2, D4.21(C3⋊D4), D4.Dic313C2, Q8.28(C3⋊D4), (C3×D4).28C23, D4.28(C22×S3), C6.174(C22×D4), D4.S3.3C22, Q8.38(C22×S3), (C3×Q8).28C23, Q8.11D612C2, C3⋊Q16.4C22, (C2×C12).121C23, C4○D12.34C22, (C3×2- 1+4)⋊3C2, (C6×Q8).154C22, Q82S3.3C22, C4.Dic3.32C22, (C2×Dic6).205C22, (C2×C6).88(C2×D4), C4.78(C2×C3⋊D4), (C2×C3⋊Q16)⋊32C2, C22.9(C2×C3⋊D4), (C2×C3⋊C8).185C22, C2.47(C22×C3⋊D4), (C2×C4).105(C22×S3), (C3×C4○D4).30C22, SmallGroup(192,1397)

Series: Derived Chief Lower central Upper central

C1C12 — D12.35C23
C1C3C6C12D12C4○D12Q8○D12 — D12.35C23
C3C6C12 — D12.35C23
C1C2C4○D42- 1+4

Generators and relations for D12.35C23
 G = < a,b,c,d,e | a12=b2=e2=1, c2=d2=a6, bab=a-1, ac=ca, ad=da, eae=a7, bc=cb, bd=db, ebe=a3b, dcd-1=a6c, ce=ec, de=ed >

Subgroups: 536 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3, C6, C6 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×12], D4, D4 [×3], D4 [×7], Q8, Q8 [×3], Q8 [×9], Dic3 [×3], C12, C12 [×3], C12 [×3], D6, C2×C6 [×3], C2×C6, C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×3], C4○D4 [×9], C3⋊C8, C3⋊C8 [×3], Dic6 [×3], Dic6 [×3], C4×S3 [×3], D12, C2×Dic3 [×3], C3⋊D4 [×3], C2×C12 [×3], C2×C12 [×6], C3×D4, C3×D4 [×3], C3×D4 [×3], C3×Q8, C3×Q8 [×3], C3×Q8 [×3], C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- 1+4, 2- 1+4, C2×C3⋊C8 [×3], C4.Dic3 [×3], D4⋊S3, D4.S3 [×3], Q82S3 [×3], C3⋊Q16 [×9], C2×Dic6 [×3], C4○D12 [×3], D42S3 [×3], S3×Q8, C6×Q8 [×3], C6×Q8, C3×C4○D4, C3×C4○D4 [×3], C3×C4○D4 [×3], Q8○D8, Q8.11D6 [×3], C2×C3⋊Q16 [×3], D4.Dic3, Q8.13D6 [×3], Q8.14D6 [×3], Q8○D12, C3×2- 1+4, D12.35C23
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, Q8○D8, C22×C3⋊D4, D12.35C23

Smallest permutation representation of D12.35C23
On 96 points
Generators in S96
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 15)(16 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 45)(38 44)(39 43)(40 42)(46 48)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 63)(64 72)(65 71)(66 70)(67 69)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 52 7 58)(2 53 8 59)(3 54 9 60)(4 55 10 49)(5 56 11 50)(6 57 12 51)(13 67 19 61)(14 68 20 62)(15 69 21 63)(16 70 22 64)(17 71 23 65)(18 72 24 66)(25 73 31 79)(26 74 32 80)(27 75 33 81)(28 76 34 82)(29 77 35 83)(30 78 36 84)(37 88 43 94)(38 89 44 95)(39 90 45 96)(40 91 46 85)(41 92 47 86)(42 93 48 87)
(1 34 7 28)(2 35 8 29)(3 36 9 30)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)(49 73 55 79)(50 74 56 80)(51 75 57 81)(52 76 58 82)(53 77 59 83)(54 78 60 84)(61 91 67 85)(62 92 68 86)(63 93 69 87)(64 94 70 88)(65 95 71 89)(66 96 72 90)
(1 16)(2 23)(3 18)(4 13)(5 20)(6 15)(7 22)(8 17)(9 24)(10 19)(11 14)(12 21)(25 40)(26 47)(27 42)(28 37)(29 44)(30 39)(31 46)(32 41)(33 48)(34 43)(35 38)(36 45)(49 61)(50 68)(51 63)(52 70)(53 65)(54 72)(55 67)(56 62)(57 69)(58 64)(59 71)(60 66)(73 91)(74 86)(75 93)(76 88)(77 95)(78 90)(79 85)(80 92)(81 87)(82 94)(83 89)(84 96)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J6A6B···6F8A8B8C8D8E12A···12J
order12222223444444444466···68888812···12
size112224122222244412121224···4661212124···4

39 irreducible representations

dim11111111222222248
type+++++++++++++--
imageC1C2C2C2C2C2C2C2S3D4D4D6D6C3⋊D4C3⋊D4Q8○D8D12.35C23
kernelD12.35C23Q8.11D6C2×C3⋊Q16D4.Dic3Q8.13D6Q8.14D6Q8○D12C3×2- 1+42- 1+4C3×D4C3×Q8C2×Q8C4○D4D4Q8C3C1
# reps13313311131346221

Matrix representation of D12.35C23 in GL6(𝔽73)

0720000
1720000
000100
0072000
00225101
005151720
,
1720000
0720000
000100
001000
00595101
00221410
,
100000
010000
006664360
00714036
00635479
001270269
,
7200000
0720000
006138670
004947067
006271235
006462426
,
100000
010000
00575700
00571600
005685757
0068685716

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,22,51,0,0,1,0,51,51,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,59,22,0,0,1,0,51,14,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,71,63,12,0,0,64,4,54,70,0,0,36,0,7,2,0,0,0,36,9,69],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,61,49,6,6,0,0,38,47,27,46,0,0,67,0,12,24,0,0,0,67,35,26],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,57,57,5,68,0,0,57,16,68,68,0,0,0,0,57,57,0,0,0,0,57,16] >;

D12.35C23 in GAP, Magma, Sage, TeX

D_{12}._{35}C_2^3
% in TeX

G:=Group("D12.35C2^3");
// GroupNames label

G:=SmallGroup(192,1397);
// by ID

G=gap.SmallGroup(192,1397);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,1684,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=e^2=1,c^2=d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^7,b*c=c*b,b*d=d*b,e*b*e=a^3*b,d*c*d^-1=a^6*c,c*e=e*c,d*e=e*d>;
// generators/relations

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