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G = (C22×C12)⋊C4order 192 = 26·3

2nd semidirect product of C22×C12 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D4).8D6, (C2×C12).4D4, (C22×C12)⋊2C4, C22⋊C42Dic3, (C6×D4).6C22, (C22×C4)⋊4Dic3, (C22×C6).15D4, C6.22(C23⋊C4), C12.D4.2C2, C33(C23.D4), C23.6(C3⋊D4), C23.7(C2×Dic3), C22.D4.1S3, C23.7D6.2C2, C2.7(C23.7D6), C22.13(C6.D4), (C3×C22⋊C4)⋊2C4, (C2×C4).6(C3⋊D4), (C22×C6).14(C2×C4), (C2×C6).97(C22⋊C4), (C3×C22.D4).1C2, SmallGroup(192,98)

Series: Derived Chief Lower central Upper central

C1C22×C6 — (C22×C12)⋊C4
C1C3C6C2×C6C22×C6C6×D4C23.7D6 — (C22×C12)⋊C4
C3C6C2×C6C22×C6 — (C22×C12)⋊C4
C1C2C22C2×D4C22.D4

Generators and relations for (C22×C12)⋊C4
 G = < a,b,c,d | a2=b2=c12=d4=1, ab=ba, ac=ca, dad-1=abc6, bc=cb, dbd-1=bc6, dcd-1=abc-1 >

Subgroups: 208 in 68 conjugacy classes, 23 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, C12 [×3], C2×C6, C2×C6 [×4], C22⋊C4, C22⋊C4 [×2], C4⋊C4, M4(2), C22×C4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12 [×3], C3×D4, C22×C6 [×2], C23⋊C4, C4.D4, C22.D4, C4.Dic3, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C23.D4, C12.D4, C23.7D6, C3×C22.D4, (C22×C12)⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C23⋊C4, C6.D4, C23.D4, C23.7D6, (C22×C12)⋊C4

Character table of (C22×C12)⋊C4

 class 12A2B2C2D34A4B4C4D4E4F6A6B6C6D6E6F8A8B12A12B12C12D12E12F12G
 size 1124424448242422244824244444888
ρ1111111111111111111111111111    trivial
ρ21111111111-1-1111111-1-11111111    linear of order 2
ρ31111111-1-1-111111111-1-1-1-1-1-11-1-1    linear of order 2
ρ41111111-1-1-1-1-111111111-1-1-1-11-1-1    linear of order 2
ρ5111-111-111-1-ii11111-1-ii1111-1-1-1    linear of order 4
ρ6111-111-1-1-11-ii11111-1i-i-1-1-1-1-111    linear of order 4
ρ7111-111-111-1i-i11111-1i-i1111-1-1-1    linear of order 4
ρ8111-111-1-1-11i-i11111-1-ii-1-1-1-1-111    linear of order 4
ρ922222-1222200-1-1-1-1-1-100-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ10222-2-22200000222-2-2-2000000200    orthogonal lifted from D4
ρ112222-22-200000222-2-22000000-200    orthogonal lifted from D4
ρ1222222-12-2-2-200-1-1-1-1-1-1001111-111    orthogonal lifted from D6
ρ13222-22-1-2-2-2200-1-1-1-1-110011111-1-1    symplectic lifted from Dic3, Schur index 2
ρ14222-22-1-222-200-1-1-1-1-1100-1-1-1-1111    symplectic lifted from Dic3, Schur index 2
ρ15222-2-2-1200000-1-1-111100-3--3-3--3-1-3--3    complex lifted from C3⋊D4
ρ162222-2-1-200000-1-1-111-100-3--3-3--31--3-3    complex lifted from C3⋊D4
ρ172222-2-1-200000-1-1-111-100--3-3--3-31-3--3    complex lifted from C3⋊D4
ρ18222-2-2-1200000-1-1-111100--3-3--3-3-1--3-3    complex lifted from C3⋊D4
ρ1944-4004000000-44-4000000000000    orthogonal lifted from C23⋊C4
ρ2044-400-20000002-222-3-2-30000000000    complex lifted from C23.7D6
ρ2144-400-20000002-22-2-32-30000000000    complex lifted from C23.7D6
ρ224-4000402i-2i0000-40000002i2i-2i-2i000    complex lifted from C23.D4
ρ234-400040-2i2i0000-4000000-2i-2i2i2i000    complex lifted from C23.D4
ρ244-4000-202i-2i0002-32-2-3000004ζ34ζ3243ζ343ζ32000    complex faithful
ρ254-4000-202i-2i000-2-322-3000004ζ324ζ343ζ3243ζ3000    complex faithful
ρ264-4000-20-2i2i000-2-322-30000043ζ3243ζ34ζ324ζ3000    complex faithful
ρ274-4000-20-2i2i0002-32-2-30000043ζ343ζ324ζ34ζ32000    complex faithful

Smallest permutation representation of (C22×C12)⋊C4
On 48 points
Generators in S48
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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)
(2 36 20 42)(3 11)(4 34 22 40)(5 9)(6 32 24 38)(8 30 14 48)(10 28 16 46)(12 26 18 44)(13 19)(15 17)(21 23)(25 37 31 43)(27 47 33 41)(29 45 35 39)

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

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

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

Matrix representation of (C22×C12)⋊C4 in GL4(𝔽73) generated by

720710
072071
0010
0001
,
306000
134300
003060
001343
,
59665939
7663420
00027
004646
,
1000
727200
21434313
22524330
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,71,0,1,0,0,71,0,1],[30,13,0,0,60,43,0,0,0,0,30,13,0,0,60,43],[59,7,0,0,66,66,0,0,59,34,0,46,39,20,27,46],[1,72,21,22,0,72,43,52,0,0,43,43,0,0,13,30] >;

(C22×C12)⋊C4 in GAP, Magma, Sage, TeX

(C_2^2\times C_{12})\rtimes C_4
% in TeX

G:=Group("(C2^2xC12):C4");
// GroupNames label

G:=SmallGroup(192,98);
// by ID

G=gap.SmallGroup(192,98);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,675,297,1684,6278]);
// Polycyclic

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

Export

Character table of (C22×C12)⋊C4 in TeX

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