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G = (C2×D12)⋊13C4order 192 = 26·3

9th semidirect product of C2×D12 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D12)⋊13C4, (C2×C4).48D12, C42⋊C25S3, C4.53(D6⋊C4), (C2×Dic6)⋊13C4, (C2×C12).145D4, C22⋊C4.83D6, C22.15(C2×D12), C23.6D67C2, (C22×C4).130D6, C12.49(C22⋊C4), C23.83(C22×S3), C23.26D614C2, (C22×C6).112C23, C32(C23.C23), (C22×C12).155C22, C6.D4.78C22, (S3×C2×C4)⋊3C4, (C2×C4).47(C4×S3), C2.21(C2×D6⋊C4), C22.19(S3×C2×C4), (C2×C12).95(C2×C4), (C2×C4○D12).9C2, (C2×C6).462(C2×D4), C6.48(C2×C22⋊C4), (C3×C42⋊C2)⋊5C2, (C2×C4).46(C3⋊D4), (C22×S3).4(C2×C4), (C2×C6).13(C22×C4), (C2×Dic3).4(C2×C4), C22.28(C2×C3⋊D4), (C2×C3⋊D4).84C22, (C3×C22⋊C4).94C22, SmallGroup(192,565)

Series: Derived Chief Lower central Upper central

C1C2×C6 — (C2×D12)⋊13C4
C1C3C6C2×C6C22×C6C2×C3⋊D4C2×C4○D12 — (C2×D12)⋊13C4
C3C6C2×C6 — (C2×D12)⋊13C4
C1C4C22×C4C42⋊C2

Generators and relations for (C2×D12)⋊13C4
 G = < a,b,c,d | a2=b12=c2=d4=1, ab=ba, dcd-1=ac=ca, dad-1=ab6, cbc=b-1, bd=db >

Subgroups: 456 in 158 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C3, C4 [×4], C4 [×6], C22 [×3], C22 [×5], S3 [×2], C6, C6 [×3], C2×C4 [×6], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×4], C12 [×4], C12 [×2], D6 [×4], C2×C6 [×3], C2×C6, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×2], C22×S3 [×2], C22×C6, C23⋊C4 [×4], C42⋊C2, C42⋊C2, C2×C4○D4, C4×Dic3, C4⋊Dic3, C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, C23.C23, C23.6D6 [×4], C23.26D6, C3×C42⋊C2, C2×C4○D12, (C2×D12)⋊13C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.C23, C2×D6⋊C4, (C2×D12)⋊13C4

Smallest permutation representation of (C2×D12)⋊13C4
On 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)
(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 38)(2 37)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 36)(21 35)(22 34)(23 33)(24 32)
(1 48 13 35)(2 37 14 36)(3 38 15 25)(4 39 16 26)(5 40 17 27)(6 41 18 28)(7 42 19 29)(8 43 20 30)(9 44 21 31)(10 45 22 32)(11 46 23 33)(12 47 24 34)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J···4O6A6B6C6D6E12A12B12C12D12E···12N
order122222234444444444···4666661212121212···12
size112221212211222444412···122224422224···4

42 irreducible representations

dim11111111222222244
type++++++++++
imageC1C2C2C2C2C4C4C4S3D4D6D6C4×S3D12C3⋊D4C23.C23(C2×D12)⋊13C4
kernel(C2×D12)⋊13C4C23.6D6C23.26D6C3×C42⋊C2C2×C4○D12C2×Dic6S3×C2×C4C2×D12C42⋊C2C2×C12C22⋊C4C22×C4C2×C4C2×C4C2×C4C3C1
# reps14111242142144424

Matrix representation of (C2×D12)⋊13C4 in GL4(𝔽13) generated by

2983
41158
0024
00911
,
101000
3700
00710
00310
,
0566
5060
0988
9405
,
8041
0834
4050
9405
G:=sub<GL(4,GF(13))| [2,4,0,0,9,11,0,0,8,5,2,9,3,8,4,11],[10,3,0,0,10,7,0,0,0,0,7,3,0,0,10,10],[0,5,0,9,5,0,9,4,6,6,8,0,6,0,8,5],[8,0,4,9,0,8,0,4,4,3,5,0,1,4,0,5] >;

(C2×D12)⋊13C4 in GAP, Magma, Sage, TeX

(C_2\times D_{12})\rtimes_{13}C_4
% in TeX

G:=Group("(C2xD12):13C4");
// GroupNames label

G:=SmallGroup(192,565);
// by ID

G=gap.SmallGroup(192,565);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,58,1123,438,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,d*a*d^-1=a*b^6,c*b*c=b^-1,b*d=d*b>;
// generators/relations

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