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G = (C2×D4).D6order 192 = 26·3

2nd non-split extension by C2×D4 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D4).2D6, (C2×C12).2D4, (C2×C4).2D12, C23⋊C4.1S3, C23.7(C4×S3), (C22×C6).9D4, C6.D42C4, C6.8(C23⋊C4), (C6×D4).2C22, C22.9(D6⋊C4), C12.D4.1C2, (C22×Dic3)⋊1C4, C23.2(C3⋊D4), C31(C23.D4), C23.23D6.1C2, C2.9(C23.6D6), (C3×C23⋊C4).1C2, (C22×C6).2(C2×C4), (C2×C6).2(C22⋊C4), SmallGroup(192,31)

Series: Derived Chief Lower central Upper central

C1C22×C6 — (C2×D4).D6
C1C3C6C2×C6C22×C6C6×D4C23.23D6 — (C2×D4).D6
C3C6C2×C6C22×C6 — (C2×D4).D6
C1C2C22C2×D4C23⋊C4

Generators and relations for (C2×D4).D6
 G = < a,b,c,d,e | a2=b4=c2=1, d6=ab2c, e2=ba=dbd-1=ab, ece-1=ac=ca, dad-1=eae-1=ab2, cbc=ebe-1=b-1, dcd-1=b2c, ede-1=bcd5 >

Subgroups: 224 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×4], C6, C6 [×3], C8, C2×C4, C2×C4 [×4], D4, C23 [×2], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×4], C22⋊C4 [×3], C4⋊C4, M4(2), C22×C4, C2×D4, C3⋊C8, C2×Dic3 [×3], C2×C12, C2×C12, C3×D4, C22×C6 [×2], C23⋊C4, C4.D4, C22.D4, C4.Dic3, Dic3⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C6×D4, C23.D4, C12.D4, C3×C23⋊C4, C23.23D6, (C2×D4).D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C23.D4, C23.6D6, (C2×D4).D6

Character table of (C2×D4).D6

 class 12A2B2C2D34A4B4C4D4E4F6A6B6C6D6E8A8B12A12B12C12D12E
 size 11244248812122424448242488888
ρ1111111111111111111111111    trivial
ρ2111111111-1-1-111111-1-111111    linear of order 2
ρ31111111-1-1-1-1-11111111-1-11-1-1    linear of order 2
ρ41111111-1-111111111-1-1-1-11-1-1    linear of order 2
ρ5111-111-1-ii11-11-1-111-iiii-1-i-i    linear of order 4
ρ6111-111-1-ii-1-111-1-111i-iii-1-i-i    linear of order 4
ρ7111-111-1i-i-1-111-1-111-ii-i-i-1ii    linear of order 4
ρ8111-111-1i-i11-11-1-111i-i-i-i-1ii    linear of order 4
ρ9222-2-222000002-2-22-20000200    orthogonal lifted from D4
ρ1022222-1222000-1-1-1-1-100-1-1-1-1-1    orthogonal lifted from S3
ρ112222-22-2000002222-20000-200    orthogonal lifted from D4
ρ1222222-12-2-2000-1-1-1-1-10011-111    orthogonal lifted from D6
ρ13222-2-2-1200000-111-1100-33-13-3    orthogonal lifted from D12
ρ14222-2-2-1200000-111-11003-3-1-33    orthogonal lifted from D12
ρ15222-22-1-22i-2i000-111-1-100ii1-i-i    complex lifted from C4×S3
ρ16222-22-1-2-2i2i000-111-1-100-i-i1ii    complex lifted from C4×S3
ρ172222-2-1-200000-1-1-1-1100-3--31-3--3    complex lifted from C3⋊D4
ρ182222-2-1-200000-1-1-1-1100--3-31--3-3    complex lifted from C3⋊D4
ρ1944-4004000000400-400000000    orthogonal lifted from C23⋊C4
ρ2044-400-2000000-2-2-32-3200000000    complex lifted from C23.6D6
ρ2144-400-2000000-22-3-2-3200000000    complex lifted from C23.6D6
ρ224-40004000-2i2i0-400000000000    complex lifted from C23.D4
ρ234-400040002i-2i0-400000000000    complex lifted from C23.D4
ρ248-8000-4000000400000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C2×D4).D6
On 48 points
Generators in S48
(2 41)(4 43)(6 45)(8 47)(10 37)(12 39)(14 27)(16 29)(18 31)(20 33)(22 35)(24 25)
(1 13 40 26)(2 14 41 27)(3 28 42 15)(4 29 43 16)(5 17 44 30)(6 18 45 31)(7 32 46 19)(8 33 47 20)(9 21 48 34)(10 22 37 35)(11 36 38 23)(12 25 39 24)
(1 46)(2 8)(3 48)(4 10)(5 38)(6 12)(7 40)(9 42)(11 44)(13 32)(14 20)(15 34)(16 22)(17 36)(18 24)(19 26)(21 28)(23 30)(25 31)(27 33)(29 35)(37 43)(39 45)(41 47)
(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 12 13 24 40 39 26 25)(2 23 27 11 41 36 14 38)(3 37 28 22 42 10 15 35)(4 21 16 48 43 34 29 9)(5 8 17 20 44 47 30 33)(6 19 31 7 45 32 18 46)

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

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

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

Matrix representation of (C2×D4).D6 in GL6(𝔽73)

100000
010000
001000
000100
0000720
0000072
,
100000
010000
0046000
0002700
0000270
0000046
,
100000
010000
0004600
0027000
0000027
0000460
,
0720000
1720000
000010
000001
0002700
0046000
,
1720000
0720000
0000460
0000046
001000
0007200

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,46,0,0,0,0,0,0,46,0,0] >;

(C2×D4).D6 in GAP, Magma, Sage, TeX

(C_2\times D_4).D_6
% in TeX

G:=Group("(C2xD4).D6");
// GroupNames label

G:=SmallGroup(192,31);
// by ID

G=gap.SmallGroup(192,31);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,346,297,851,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^6=a*b^2*c,e^2=b*a=d*b*d^-1=a*b,e*c*e^-1=a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c=e*b*e^-1=b^-1,d*c*d^-1=b^2*c,e*d*e^-1=b*c*d^5>;
// generators/relations

Export

Character table of (C2×D4).D6 in TeX

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