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G = D12.C4order 96 = 25·3

The non-split extension by D12 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.C4, C8.12D6, Dic6.C4, M4(2)⋊5S3, C24.13C22, C12.39C23, C3⋊D4.C4, (S3×C8)⋊8C2, C32(C8○D4), C4.5(C4×S3), C8⋊S36C2, D6.2(C2×C4), (C2×C4).46D6, M4(2)(C3⋊C8), C12.13(C2×C4), C4○D12.3C2, C3⋊C8.12C22, C22.1(C4×S3), (C3×M4(2))⋊6C2, C4.39(C22×S3), C6.16(C22×C4), Dic3.4(C2×C4), (C4×S3).16C22, (C2×C12).26C22, (C2×C3⋊C8)⋊3C2, C2.17(S3×C2×C4), (C2×C6).6(C2×C4), SmallGroup(96,114)

Series: Derived Chief Lower central Upper central

C1C6 — D12.C4
C1C3C6C12C4×S3C4○D12 — D12.C4
C3C6 — D12.C4
C1C4M4(2)

Generators and relations for D12.C4
 G = < a,b,c | a12=b2=1, c4=a6, bab=a-1, cac-1=a7, cbc-1=a6b >

Subgroups: 114 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C8○D4, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C3×M4(2), C4○D12, D12.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4×S3 [×2], C22×S3, C8○D4, S3×C2×C4, D12.C4

Character table of D12.C4

 class 12A2B2C2D34A4B4C4D4E6A6B8A8B8C8D8E8F8G8H8I8J12A12B12C24A24B24C24D
 size 112662112662422223333662244444
ρ1111111111111111111111111111111    trivial
ρ2111-1-11111-1-111-1-1-1-1111111111-1-1-1-1    linear of order 2
ρ31111111111111-1-1-1-1-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ4111-1-11111-1-1111111-1-1-1-1-1-11111111    linear of order 2
ρ511-11-1111-11-11-11-1-11-1-1-1-11111-1-11-11    linear of order 2
ρ611-1-11111-1-111-1-111-1-1-1-1-11111-11-11-1    linear of order 2
ρ711-11-1111-11-11-1-111-11111-1-111-11-11-1    linear of order 2
ρ811-1-11111-1-111-11-1-111111-1-111-1-11-11    linear of order 2
ρ9111-111-1-1-11-111-ii-iiii-i-ii-i-1-1-1i-i-ii    linear of order 4
ρ101111-11-1-1-1-1111i-ii-iii-i-ii-i-1-1-1-iii-i    linear of order 4
ρ1111-1111-1-11-1-11-1-i-iiiii-i-i-ii-1-11-i-iii    linear of order 4
ρ1211-1-1-11-1-11111-1ii-i-iii-i-i-ii-1-11ii-i-i    linear of order 4
ρ131111-11-1-1-1-1111-ii-ii-i-iii-ii-1-1-1i-i-ii    linear of order 4
ρ14111-111-1-1-11-111i-ii-i-i-iii-ii-1-1-1-iii-i    linear of order 4
ρ1511-1-1-11-1-11111-1-i-iii-i-iiii-i-1-11-i-iii    linear of order 4
ρ1611-1111-1-11-1-11-1ii-i-i-i-iiii-i-1-11ii-i-i    linear of order 4
ρ1722-200-122-200-11-222-2000000-1-11-11-11    orthogonal lifted from D6
ρ1822200-122200-1-1-2-2-2-2000000-1-1-11111    orthogonal lifted from D6
ρ1922200-122200-1-12222000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ2022-200-122-200-112-2-22000000-1-111-11-1    orthogonal lifted from D6
ρ2122-200-1-2-2200-112i2i-2i-2i00000011-1-i-iii    complex lifted from C4×S3
ρ2222200-1-2-2-200-1-12i-2i2i-2i000000111i-i-ii    complex lifted from C4×S3
ρ2322200-1-2-2-200-1-1-2i2i-2i2i000000111-iii-i    complex lifted from C4×S3
ρ2422-200-1-2-2200-11-2i-2i2i2i00000011-1ii-i-i    complex lifted from C4×S3
ρ252-20002-2i2i000-2000008783858002i-2i00000    complex lifted from C8○D4
ρ262-200022i-2i000-200000858878300-2i2i00000    complex lifted from C8○D4
ρ272-200022i-2i000-200000885838700-2i2i00000    complex lifted from C8○D4
ρ282-20002-2i2i000-2000008387885002i-2i00000    complex lifted from C8○D4
ρ294-4000-24i-4i0002000000000002i-2i00000    complex faithful
ρ304-4000-2-4i4i000200000000000-2i2i00000    complex faithful

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

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

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

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

D12.C4 is a maximal subgroup of
D12.2D4  D12.3D4  D12.6D4  D12.7D4  M4(2).22D6  C42.196D6  D2410C4  D247C4  M4(2)⋊26D6  S3×C8○D4  M4(2)⋊28D6  D85D6  D86D6  C24.C23  SD16.D6  D36.C4  C24.64D6  C24.D6  D12.Dic3  C3⋊C8.22D6  C24.47D6  C40.55D6  C40.35D6  D12.Dic5  D60.4C4  D60.3C4  D12.F5  Dic6.F5  D15⋊C8⋊C2
D12.C4 is a maximal quotient of
C24⋊Q8  C89D12  D6.4C42  C42.185D6  C24⋊C4⋊C2  C3⋊D4⋊C8  D6⋊C8⋊C2  C3⋊C826D4  Dic6⋊C8  C42.198D6  D12⋊C8  C122M4(2)  C42.30D6  C42.31D6  C12.88(C2×Q8)  C12.7C42  C24⋊D4  C2421D4  D6⋊C840C2  D36.C4  C24.64D6  C24.D6  D12.Dic3  C3⋊C8.22D6  C24.47D6  C40.55D6  C40.35D6  D12.Dic5  D60.4C4  D60.3C4  D12.F5  Dic6.F5  D15⋊C8⋊C2

Matrix representation of D12.C4 in GL4(𝔽5) generated by

0022
2024
1210
2004
,
1420
0310
0220
3044
,
3124
1034
4022
0330
G:=sub<GL(4,GF(5))| [0,2,1,2,0,0,2,0,2,2,1,0,2,4,0,4],[1,0,0,3,4,3,2,0,2,1,2,4,0,0,0,4],[3,1,4,0,1,0,0,3,2,3,2,3,4,4,2,0] >;

D12.C4 in GAP, Magma, Sage, TeX

D_{12}.C_4
% in TeX

G:=Group("D12.C4");
// GroupNames label

G:=SmallGroup(96,114);
// by ID

G=gap.SmallGroup(96,114);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,188,50,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^2=1,c^4=a^6,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^6*b>;
// generators/relations

Export

Character table of D12.C4 in TeX

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