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G = C4○D4⋊C12order 192 = 26·3

The semidirect product of C4○D4 and C12 acting via C12/C2=C6

non-abelian, soluble

Aliases: C4○D4⋊C12, C4⋊C4.1A4, C4.A45C4, C4.1(C4×A4), (C4×Q8).1C6, Q8.3(C2×C12), C2.2(D4.A4), C2.1(Q8.A4), C23.33C23⋊C3, SL2(𝔽3)⋊7(C2×C4), (C4×SL2(𝔽3))⋊4C2, C22.16(C22×A4), (C2×SL2(𝔽3)).26C22, C2.8(C2×C4×A4), (C2×C4).3(C2×A4), (C2×C4.A4).7C2, (C2×C4○D4).1C6, (C2×Q8).37(C2×C6), SmallGroup(192,999)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C4○D4⋊C12
C1C2Q8C2×Q8C2×SL2(𝔽3)C2×C4.A4 — C4○D4⋊C12
Q8 — C4○D4⋊C12
C1C22C4⋊C4

Generators and relations for C4○D4⋊C12
 G = < a,b,c,d | a4=c2=d12=1, b2=a2, dcd-1=ab=ba, ac=ca, dad-1=a-1, cbc=a2b, dbd-1=a-1bc >

Subgroups: 253 in 84 conjugacy classes, 27 normal (15 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8, Q8, C23, C12 [×4], C2×C6, C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C4○D4 [×2], SL2(𝔽3), C2×C12 [×3], C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×C4○D4, C3×C4⋊C4, C2×SL2(𝔽3), C4.A4 [×2], C23.33C23, C4×SL2(𝔽3) [×2], C2×C4.A4, C4○D4⋊C12
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, C12 [×2], A4, C2×C6, C2×C12, C2×A4 [×3], C4×A4 [×2], C22×A4, C2×C4×A4, Q8.A4, D4.A4, C4○D4⋊C12

Smallest permutation representation of C4○D4⋊C12
On 64 points
Generators in S64
(1 9 6 13)(2 14 7 10)(3 11 8 15)(4 16 5 12)(17 55 47 37)(18 38 48 56)(19 57 49 39)(20 40 50 58)(21 59 51 29)(22 30 52 60)(23 61 41 31)(24 32 42 62)(25 63 43 33)(26 34 44 64)(27 53 45 35)(28 36 46 54)
(1 57 6 39)(2 54 7 36)(3 63 8 33)(4 60 5 30)(9 49 13 19)(10 46 14 28)(11 43 15 25)(12 52 16 22)(17 21 47 51)(18 44 48 26)(20 24 50 42)(23 27 41 45)(29 55 59 37)(31 35 61 53)(32 58 62 40)(34 38 64 56)
(1 23)(2 50)(3 17)(4 44)(5 26)(6 41)(7 20)(8 47)(9 61)(10 40)(11 55)(12 34)(13 31)(14 58)(15 37)(16 64)(18 60)(19 53)(21 33)(22 38)(24 54)(25 59)(27 39)(28 32)(29 43)(30 48)(35 49)(36 42)(45 57)(46 62)(51 63)(52 56)
(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)

G:=sub<Sym(64)| (1,9,6,13)(2,14,7,10)(3,11,8,15)(4,16,5,12)(17,55,47,37)(18,38,48,56)(19,57,49,39)(20,40,50,58)(21,59,51,29)(22,30,52,60)(23,61,41,31)(24,32,42,62)(25,63,43,33)(26,34,44,64)(27,53,45,35)(28,36,46,54), (1,57,6,39)(2,54,7,36)(3,63,8,33)(4,60,5,30)(9,49,13,19)(10,46,14,28)(11,43,15,25)(12,52,16,22)(17,21,47,51)(18,44,48,26)(20,24,50,42)(23,27,41,45)(29,55,59,37)(31,35,61,53)(32,58,62,40)(34,38,64,56), (1,23)(2,50)(3,17)(4,44)(5,26)(6,41)(7,20)(8,47)(9,61)(10,40)(11,55)(12,34)(13,31)(14,58)(15,37)(16,64)(18,60)(19,53)(21,33)(22,38)(24,54)(25,59)(27,39)(28,32)(29,43)(30,48)(35,49)(36,42)(45,57)(46,62)(51,63)(52,56), (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)>;

G:=Group( (1,9,6,13)(2,14,7,10)(3,11,8,15)(4,16,5,12)(17,55,47,37)(18,38,48,56)(19,57,49,39)(20,40,50,58)(21,59,51,29)(22,30,52,60)(23,61,41,31)(24,32,42,62)(25,63,43,33)(26,34,44,64)(27,53,45,35)(28,36,46,54), (1,57,6,39)(2,54,7,36)(3,63,8,33)(4,60,5,30)(9,49,13,19)(10,46,14,28)(11,43,15,25)(12,52,16,22)(17,21,47,51)(18,44,48,26)(20,24,50,42)(23,27,41,45)(29,55,59,37)(31,35,61,53)(32,58,62,40)(34,38,64,56), (1,23)(2,50)(3,17)(4,44)(5,26)(6,41)(7,20)(8,47)(9,61)(10,40)(11,55)(12,34)(13,31)(14,58)(15,37)(16,64)(18,60)(19,53)(21,33)(22,38)(24,54)(25,59)(27,39)(28,32)(29,43)(30,48)(35,49)(36,42)(45,57)(46,62)(51,63)(52,56), (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) );

G=PermutationGroup([(1,9,6,13),(2,14,7,10),(3,11,8,15),(4,16,5,12),(17,55,47,37),(18,38,48,56),(19,57,49,39),(20,40,50,58),(21,59,51,29),(22,30,52,60),(23,61,41,31),(24,32,42,62),(25,63,43,33),(26,34,44,64),(27,53,45,35),(28,36,46,54)], [(1,57,6,39),(2,54,7,36),(3,63,8,33),(4,60,5,30),(9,49,13,19),(10,46,14,28),(11,43,15,25),(12,52,16,22),(17,21,47,51),(18,44,48,26),(20,24,50,42),(23,27,41,45),(29,55,59,37),(31,35,61,53),(32,58,62,40),(34,38,64,56)], [(1,23),(2,50),(3,17),(4,44),(5,26),(6,41),(7,20),(8,47),(9,61),(10,40),(11,55),(12,34),(13,31),(14,58),(15,37),(16,64),(18,60),(19,53),(21,33),(22,38),(24,54),(25,59),(27,39),(28,32),(29,43),(30,48),(35,49),(36,42),(45,57),(46,62),(51,63),(52,56)], [(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)])

38 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4F4G···4L6A···6F12A···12L
order122222334···44···46···612···12
size111166442···26···64···48···8

38 irreducible representations

dim111111113334444
type++++++-
imageC1C2C2C3C4C6C6C12A4C2×A4C4×A4Q8.A4Q8.A4D4.A4D4.A4
kernelC4○D4⋊C12C4×SL2(𝔽3)C2×C4.A4C23.33C23C4.A4C4×Q8C2×C4○D4C4○D4C4⋊C4C2×C4C4C2C2C2C2
# reps121244281341212

Matrix representation of C4○D4⋊C12 in GL7(𝔽13)

1000000
0100000
0010000
0005000
0000500
0000080
0000008
,
0100000
1000000
1212120000
00001200
0001000
00000012
0000010
,
1212120000
0010000
0100000
0006200
0002700
00000711
00000116
,
5000000
0050000
8880000
0000010
00000109
0001000
00010900

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8],[0,1,12,0,0,0,0,1,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,6,2,0,0,0,0,0,2,7,0,0,0,0,0,0,0,7,11,0,0,0,0,0,11,6],[5,0,8,0,0,0,0,0,0,8,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,9,0,0,0,1,10,0,0,0,0,0,0,9,0,0] >;

C4○D4⋊C12 in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes C_{12}
% in TeX

G:=Group("C4oD4:C12");
// GroupNames label

G:=SmallGroup(192,999);
// by ID

G=gap.SmallGroup(192,999);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,672,1373,92,438,172,775,285,124]);
// Polycyclic

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

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