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G = C23.28D6order 96 = 25·3

4th non-split extension by C23 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.28D6, D6⋊C42C2, (C22×C4)⋊5S3, (C2×C4).65D6, C6.42(C2×D4), (C2×C6).37D4, Dic3⋊C43C2, (C22×C12)⋊2C2, C6.18(C4○D4), C6.D46C2, (C2×C6).47C23, C2.18(C4○D12), (C2×C12).78C22, C34(C22.D4), C22.9(C3⋊D4), (C22×S3).9C22, C22.55(C22×S3), (C22×C6).39C22, (C2×Dic3).15C22, C2.6(C2×C3⋊D4), (C2×C3⋊D4).6C2, SmallGroup(96,136)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.28D6
C1C3C6C2×C6C22×S3C2×C3⋊D4 — C23.28D6
C3C2×C6 — C23.28D6
C1C22C22×C4

Generators and relations for C23.28D6
 G = < a,b,c,d,e | a2=b2=c2=1, d6=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd5 >

Subgroups: 178 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3, C22×C6, C22.D4, Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C2×C3⋊D4, C22×C12, C23.28D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C4○D12 [×2], C2×C3⋊D4, C23.28D6

Character table of C23.28D6

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G6A6B6C6D6E6F6G12A12B12C12D12E12F12G12H
 size 1111221222222121212222222222222222
ρ1111111111111111111111111111111    trivial
ρ211111111-1-1-1-1-11-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1111-1-11-1-11-1-11-1-111-11-1-1111-1    linear of order 2
ρ41111-1-111-111-11-1-1-1-11-1-1111-111-1-1-11    linear of order 2
ρ51111-1-1-111-1-1111-1-1-11-1-111-11-1-1111-1    linear of order 2
ρ61111-1-1-11-111-1-111-1-11-1-1111-111-1-1-11    linear of order 2
ρ7111111-111111-1-1-1111111111111111    linear of order 2
ρ8111111-11-1-1-1-11-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ92222220-12222000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102222-2-20-1-222-200011-111-1-1-11-1-1111-1    orthogonal lifted from D6
ρ112222-2-20-12-2-2200011-111-1-11-111-1-1-11    orthogonal lifted from D6
ρ122-2-22-22020000000-2-2-2222-200000000    orthogonal lifted from D4
ρ132222220-1-2-2-2-2000-1-1-1-1-1-1-111111111    orthogonal lifted from D6
ρ142-2-222-202000000022-2-2-22-200000000    orthogonal lifted from D4
ρ152-2-222-20-10000000-1-1111-11--3--3-3--3-3-3--3-3    complex lifted from C3⋊D4
ρ162-2-22-220-10000000111-1-1-11--3-3-3--3--3--3-3-3    complex lifted from C3⋊D4
ρ172-2-22-220-10000000111-1-1-11-3--3--3-3-3-3--3--3    complex lifted from C3⋊D4
ρ182-2-222-20-10000000-1-1111-11-3-3--3-3--3--3-3--3    complex lifted from C3⋊D4
ρ192-22-200022i00-2i00000200-2-202i002i-2i-2i0    complex lifted from C4○D4
ρ202-22-20002-2i002i00000200-2-20-2i00-2i2i2i0    complex lifted from C4○D4
ρ2122-2-2000202i-2i000000-200-222i0-2i-2i0002i    complex lifted from C4○D4
ρ2222-2-200020-2i2i000000-200-22-2i02i2i000-2i    complex lifted from C4○D4
ρ232-22-2000-1-2i002i000-3--3-1-3--311-3i-33i-i-i3    complex lifted from C4○D12
ρ242-22-2000-1-2i002i000--3-3-1--3-3113i3-3i-i-i-3    complex lifted from C4○D12
ρ2522-2-2000-10-2i2i0000-3--31--3-31-1i-3-i-i3-33i    complex lifted from C4○D12
ρ2622-2-2000-10-2i2i0000--3-31-3--31-1i3-i-i-33-3i    complex lifted from C4○D12
ρ272-22-2000-12i00-2i000-3--3-1-3--3113-i3-3-iii-3    complex lifted from C4○D12
ρ2822-2-2000-102i-2i0000--3-31-3--31-1-i-3ii3-33-i    complex lifted from C4○D12
ρ2922-2-2000-102i-2i0000-3--31--3-31-1-i3ii-33-3-i    complex lifted from C4○D12
ρ302-22-2000-12i00-2i000--3-3-1--3-311-3-i-33-iii3    complex lifted from C4○D12

Smallest permutation representation of C23.28D6
On 48 points
Generators in S48
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 38)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 6 38 43)(2 42 39 5)(3 4 40 41)(7 12 44 37)(8 48 45 11)(9 10 46 47)(13 32 35 22)(14 21 36 31)(15 30 25 20)(16 19 26 29)(17 28 27 18)(23 34 33 24)

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

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

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

C23.28D6 is a maximal subgroup of
C22⋊C4⋊D6  C42.277D6  C24.38D6  C24.42D6  C6.2- 1+4  C6.52- 1+4  C6.62- 1+4  C4212D6  C42.96D6  C42.104D6  C4218D6  C42.113D6  C42.114D6  C4219D6  C42.115D6  C42.116D6  C42.118D6  C6.422+ 1+4  C6.442+ 1+4  C6.482+ 1+4  C6.492+ 1+4  C6.202- 1+4  C6.222- 1+4  C6.252- 1+4  C6.592+ 1+4  C6.792- 1+4  C4⋊C4.197D6  S3×C22.D4  C6.1202+ 1+4  C4⋊C428D6  C6.852- 1+4  C24.83D6  C2412D6  C6.442- 1+4  C6.1042- 1+4  C6.1452+ 1+4  C23.28D18  C62.20C23  C62.75C23  C62.56D4  C62.60D4  C62.129D4  Dic3⋊C4⋊D5  D6⋊C4⋊D5  C30.(C2×D4)  C6.D4⋊D5  C23.28D30
C23.28D6 is a maximal quotient of
(C2×C42).6S3  (C2×C42)⋊3S3  C24.20D6  C24.25D6  (C2×C6).40D8  C4⋊C4.228D6  C4⋊C4.230D6  C4⋊C4.231D6  (C2×Dic3).Q8  (C2×C12).288D4  (C2×C12).289D4  (C2×C12).290D4  C4⋊C4.233D6  C4⋊C4.236D6  C24.73D6  C24.74D6  C24.76D6  C23.28D18  C62.20C23  C62.75C23  C62.56D4  C62.60D4  C62.129D4  Dic3⋊C4⋊D5  D6⋊C4⋊D5  C30.(C2×D4)  C6.D4⋊D5  C23.28D30

Matrix representation of C23.28D6 in GL6(𝔽13)

100000
010000
000500
008000
000010
000001
,
1200000
0120000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
000010
000001
,
12110000
010000
000100
0012000
000001
0000121
,
120000
12120000
0001200
0012000
0000121
000001

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

C23.28D6 in GAP, Magma, Sage, TeX

C_2^3._{28}D_6
% in TeX

G:=Group("C2^3.28D6");
// GroupNames label

G:=SmallGroup(96,136);
// by ID

G=gap.SmallGroup(96,136);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,86,2309]);
// Polycyclic

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

Export

Character table of C23.28D6 in TeX

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