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G = D12.32C23order 192 = 26·3

13rd non-split extension by D12 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.37C24, D12.32C23, 2+ 1+48S3, Dic6.32C23, C4○D48D6, C35(D4○D8), D4○D129C2, (C2×D4)⋊16D6, (C3×D4).36D4, C3⋊C8.16C23, (C3×Q8).36D4, D4⋊D611C2, D4⋊S320C22, Q8.13D68C2, C12.269(C2×D4), (C6×D4)⋊24C22, C4.37(S3×C23), D126C2211C2, D4.18(C3⋊D4), C4○D1210C22, (C2×D12)⋊39C22, D4.Dic310C2, D4.S319C22, Q8.25(C3⋊D4), D4.25(C22×S3), C3⋊Q1621C22, (C3×D4).25C23, C6.171(C22×D4), (C3×Q8).25C23, Q8.35(C22×S3), (C2×C12).118C23, Q82S319C22, C4.Dic316C22, (C3×2+ 1+4)⋊2C2, (C2×D4⋊S3)⋊32C2, (C2×C3⋊C8)⋊24C22, (C2×C6).85(C2×D4), C4.75(C2×C3⋊D4), (C3×C4○D4)⋊8C22, C22.6(C2×C3⋊D4), C2.44(C22×C3⋊D4), (C2×C4).102(C22×S3), SmallGroup(192,1394)

Series: Derived Chief Lower central Upper central

C1C12 — D12.32C23
C1C3C6C12D12C2×D12D4○D12 — D12.32C23
C3C6C12 — D12.32C23
C1C2C4○D42+ 1+4

Generators and relations for D12.32C23
 G = < a,b,c,d,e | a12=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a7, cbc=a6b, dbd=a10b, ebe=a3b, cd=dc, ce=ec, ede=a9d >

Subgroups: 728 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×3], C6, C6 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×6], D4 [×6], D4 [×15], Q8 [×2], Q8, C23 [×6], Dic3, C12, C12 [×3], C12, D6 [×6], C2×C6 [×3], C2×C6 [×6], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×3], C2×D4 [×9], C4○D4, C4○D4 [×3], C4○D4 [×5], C3⋊C8, C3⋊C8 [×3], Dic6, C4×S3 [×3], D12 [×3], D12 [×3], C3⋊D4 [×3], C2×C12 [×3], C2×C12 [×3], C3×D4 [×6], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×3], C22×C6 [×3], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4, 2+ 1+4, C2×C3⋊C8 [×3], C4.Dic3 [×3], D4⋊S3 [×9], D4.S3 [×3], Q82S3 [×3], C3⋊Q16, C2×D12 [×3], C4○D12 [×3], S3×D4 [×3], Q83S3, C6×D4 [×3], C6×D4 [×3], C3×C4○D4, C3×C4○D4 [×3], C3×C4○D4, D4○D8, C2×D4⋊S3 [×3], D126C22 [×3], D4.Dic3, D4⋊D6 [×3], Q8.13D6 [×3], D4○D12, C3×2+ 1+4, D12.32C23
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, D4○D8, C22×C3⋊D4, D12.32C23

Smallest permutation representation of D12.32C23
On 48 points
Generators in S48
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 31)(26 30)(27 29)(32 36)(33 35)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)
(1 22)(2 17)(3 24)(4 19)(5 14)(6 21)(7 16)(8 23)(9 18)(10 13)(11 20)(12 15)(25 40)(26 47)(27 42)(28 37)(29 44)(30 39)(31 46)(32 41)(33 48)(34 43)(35 38)(36 45)

G:=sub<Sym(48)| (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (2,12)(3,11)(4,10)(5,9)(6,8)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,22)(2,17)(3,24)(4,19)(5,14)(6,21)(7,16)(8,23)(9,18)(10,13)(11,20)(12,15)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45)>;

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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (2,12)(3,11)(4,10)(5,9)(6,8)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,22)(2,17)(3,24)(4,19)(5,14)(6,21)(7,16)(8,23)(9,18)(10,13)(11,20)(12,15)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45) );

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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,36),(11,35),(12,34),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21),(25,31),(26,30),(27,29),(32,36),(33,35),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48)], [(1,22),(2,17),(3,24),(4,19),(5,14),(6,21),(7,16),(8,23),(9,18),(10,13),(11,20),(12,15),(25,40),(26,47),(27,42),(28,37),(29,44),(30,39),(31,46),(32,41),(33,48),(34,43),(35,38),(36,45)])

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F6A6B···6J8A8B8C8D8E12A···12F
order12222222222344444466···68888812···12
size112224441212122222241224···4661212124···4

39 irreducible representations

dim11111111222222248
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6C3⋊D4C3⋊D4D4○D8D12.32C23
kernelD12.32C23C2×D4⋊S3D126C22D4.Dic3D4⋊D6Q8.13D6D4○D12C3×2+ 1+42+ 1+4C3×D4C3×Q8C2×D4C4○D4D4Q8C3C1
# reps13313311131346221

Matrix representation of D12.32C23 in GL6(𝔽73)

72720000
100000
0007200
001000
00727212
00107272
,
72720000
010000
000010
00727212
001000
0072011
,
7200000
0720000
001000
000100
0000720
0011072
,
7200000
110000
000100
001000
00117271
000001
,
7200000
0720000
00575700
00571600
005757032
00570160

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,1,0,0,72,0,72,0,0,0,0,0,1,72,0,0,0,0,2,72],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,72,1,72,0,0,0,72,0,0,0,0,1,1,0,1,0,0,0,2,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,72,0,0,0,0,0,71,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,57,57,57,0,0,57,16,57,0,0,0,0,0,0,16,0,0,0,0,32,0] >;

D12.32C23 in GAP, Magma, Sage, TeX

D_{12}._{32}C_2^3
% in TeX

G:=Group("D12.32C2^3");
// GroupNames label

G:=SmallGroup(192,1394);
// by ID

G=gap.SmallGroup(192,1394);
# by ID

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

G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^7,c*b*c=a^6*b,d*b*d=a^10*b,e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=a^9*d>;
// generators/relations

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