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G = C2×C322Q8order 144 = 24·32

Direct product of C2 and C322Q8

direct product, metabelian, supersoluble, monomial

Aliases: C2×C322Q8, C61Dic6, Dic3.7D6, C62.12C22, (C3×C6)⋊2Q8, C324(C2×Q8), C22.12S32, C32(C2×Dic6), (C2×C6).17D6, (C3×C6).16C23, C6.16(C22×S3), (C2×Dic3).3S3, (C6×Dic3).4C2, C3⋊Dic3.15C22, (C3×Dic3).7C22, C2.16(C2×S32), (C2×C3⋊Dic3).8C2, SmallGroup(144,152)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C322Q8
C1C3C32C3×C6C3×Dic3C322Q8 — C2×C322Q8
C32C3×C6 — C2×C322Q8
C1C22

Generators and relations for C2×C322Q8
 G = < a,b,c,d,e | a2=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 224 in 84 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×6], C22, C6 [×6], C6 [×3], C2×C4 [×3], Q8 [×4], C32, Dic3 [×4], Dic3 [×6], C12 [×4], C2×C6 [×2], C2×C6, C2×Q8, C3×C6, C3×C6 [×2], Dic6 [×8], C2×Dic3 [×2], C2×Dic3 [×3], C2×C12 [×2], C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C2×Dic6 [×2], C322Q8 [×4], C6×Dic3 [×2], C2×C3⋊Dic3, C2×C322Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], Q8 [×2], C23, D6 [×6], C2×Q8, Dic6 [×4], C22×S3 [×2], S32, C2×Dic6 [×2], C322Q8 [×2], C2×S32, C2×C322Q8

Character table of C2×C322Q8

 class 12A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D6E6F6G6H6I12A12B12C12D12E12F12G12H
 size 11112246666181822222244466666666
ρ1111111111111111111111111111111    trivial
ρ21-11-11111-1-111-1-1-11-1-11-11-11-1-1111-1-1    linear of order 2
ρ31-11-111111-1-1-11-1-11-1-11-11-1-1-1-1-11111    linear of order 2
ρ411111111-11-1-1-1111111111-111-111-1-1    linear of order 2
ρ51111111-11-11-1-11111111111-1-11-1-111    linear of order 2
ρ61-11-1111-1-111-11-1-11-1-11-11-11111-1-1-1-1    linear of order 2
ρ71111111-1-1-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81-11-1111-111-11-1-1-11-1-11-11-1-111-1-1-111    linear of order 2
ρ92-22-2-12-1020-2001-2-11-221-11100100-1-1    orthogonal lifted from D6
ρ102222-12-1020200-12-1-122-1-1-1-100-100-1-1    orthogonal lifted from S3
ρ112-22-22-1-1-202000-212-21-11-110-1-101100    orthogonal lifted from D6
ρ122-22-22-1-120-2000-212-21-11-110110-1-100    orthogonal lifted from D6
ρ1322222-1-1-20-20002-122-1-1-1-1-101101100    orthogonal lifted from D6
ρ1422222-1-12020002-122-1-1-1-1-10-1-10-1-100    orthogonal lifted from S3
ρ152-22-2-12-10-202001-2-11-221-11-100-10011    orthogonal lifted from D6
ρ162222-12-10-20-200-12-1-122-1-1-110010011    orthogonal lifted from D6
ρ172-2-2222200000022-2-2-2-22-2-200000000    symplectic lifted from Q8, Schur index 2
ρ1822-2-2222000000-2-2-222-2-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ1922-2-22-1-1000000-21-22-1111-103-303-300    symplectic lifted from Dic6, Schur index 2
ρ202-2-222-1-10000002-1-2-211-1110-3303-300    symplectic lifted from Dic6, Schur index 2
ρ212-2-22-12-1000000-1211-2-2-111-3003003-3    symplectic lifted from Dic6, Schur index 2
ρ2222-2-2-12-10000001-21-12-211-1300-3003-3    symplectic lifted from Dic6, Schur index 2
ρ232-2-222-1-10000002-1-2-211-11103-30-3300    symplectic lifted from Dic6, Schur index 2
ρ2422-2-2-12-10000001-21-12-211-1-300300-33    symplectic lifted from Dic6, Schur index 2
ρ252-2-22-12-1000000-1211-2-2-111300-300-33    symplectic lifted from Dic6, Schur index 2
ρ2622-2-22-1-1000000-21-22-1111-10-330-3300    symplectic lifted from Dic6, Schur index 2
ρ274444-2-21000000-2-2-2-2-2-211100000000    orthogonal lifted from S32
ρ284-44-4-2-2100000022-222-2-11-100000000    orthogonal lifted from C2×S32
ρ2944-4-4-2-21000000222-2-22-1-1100000000    symplectic lifted from C322Q8, Schur index 2
ρ304-4-44-2-21000000-2-222221-1-100000000    symplectic lifted from C322Q8, Schur index 2

Smallest permutation representation of C2×C322Q8
On 48 points
Generators in S48
(1 5)(2 6)(3 7)(4 8)(9 34)(10 35)(11 36)(12 33)(13 38)(14 39)(15 40)(16 37)(17 29)(18 30)(19 31)(20 32)(21 27)(22 28)(23 25)(24 26)(41 48)(42 45)(43 46)(44 47)
(1 39 35)(2 36 40)(3 37 33)(4 34 38)(5 14 10)(6 11 15)(7 16 12)(8 9 13)(17 22 41)(18 42 23)(19 24 43)(20 44 21)(25 30 45)(26 46 31)(27 32 47)(28 48 29)
(1 35 39)(2 36 40)(3 33 37)(4 34 38)(5 10 14)(6 11 15)(7 12 16)(8 9 13)(17 41 22)(18 42 23)(19 43 24)(20 44 21)(25 30 45)(26 31 46)(27 32 47)(28 29 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 47 3 45)(2 46 4 48)(5 44 7 42)(6 43 8 41)(9 17 11 19)(10 20 12 18)(13 22 15 24)(14 21 16 23)(25 39 27 37)(26 38 28 40)(29 36 31 34)(30 35 32 33)

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

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

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

C2×C322Q8 is a maximal subgroup of
C62.4D4  Dic35Dic6  C62.8C23  C62.9C23  C62.10C23  D6⋊Dic6  Dic3.D12  C62.35C23  D62Dic6  C62.65C23  D64Dic6  C62.85C23  C123Dic6  C12⋊Dic6  C62.95C23  C62.101C23  C623Q8  C624Q8  C62.15D4  C2×S3×Dic6  Dic6.24D6
C2×C322Q8 is a maximal quotient of
C62.39C23  C62.42C23  C123Dic6  C12⋊Dic6  C623Q8  C624Q8

Matrix representation of C2×C322Q8 in GL6(𝔽13)

100000
010000
0012000
0001200
000010
000001
,
100000
010000
001000
000100
000001
00001212
,
100000
010000
0012100
0012000
000010
000001
,
010000
1200000
0012000
0001200
000010
00001212
,
4100000
1090000
000100
001000
000010
000001

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[4,10,0,0,0,0,10,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×C322Q8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C2xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(144,152);
// by ID

G=gap.SmallGroup(144,152);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,55,490,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

Export

Character table of C2×C322Q8 in TeX

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