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G = C42.472C23order 128 = 27

333rd non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.472C23, C4.682+ 1+4, C8⋊D443C2, C86D411C2, C4⋊C839C22, (C4×C8)⋊48C22, C4⋊C4.162D4, D4.Q839C2, (C4×SD16)⋊55C2, D45D4.5C2, (C2×D4).176D4, C8.D427C2, (C4×Q8)⋊29C22, C4.Q829C22, C2.D840C22, D4.29(C4○D4), C22⋊SD1623C2, D4.7D448C2, C4⋊C4.415C23, C22⋊C835C22, (C2×C4).515C24, (C2×C8).103C23, Q8.D444C2, C22⋊C4.172D4, (C2×Q16)⋊33C22, C23.332(C2×D4), C22⋊Q820C22, C2.80(D4○SD16), Q8⋊C450C22, (C4×D4).165C22, (C2×D4).425C23, C4⋊D4.89C22, (C2×Q8).226C23, C2.151(D45D4), C42.C212C22, C23.20D440C2, C23.36D422C2, C23.37D417C2, C23.46D418C2, (C2×M4(2))⋊31C22, (C22×C4).328C23, C4.4D4.71C22, C22.775(C22×D4), D4⋊C4.121C22, C2.91(D8⋊C22), C22.46C246C2, (C2×SD16).159C22, (C22×D4).416C22, C42.78C2221C2, C42⋊C2.195C22, (C2×C4⋊C4)⋊62C22, C4.240(C2×C4○D4), (C2×C4).928(C2×D4), (C2×C4○D4).217C22, SmallGroup(128,2055)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.472C23
C1C2C4C2×C4C22×C4C22×D4D45D4 — C42.472C23
C1C2C2×C4 — C42.472C23
C1C22C4×D4 — C42.472C23
C1C2C2C2×C4 — C42.472C23

Generators and relations for C42.472C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2b2, ab=ba, cac-1=a-1b2, dad=ab2, eae=a-1, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >

Subgroups: 432 in 201 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×17], C8 [×4], C2×C4 [×5], C2×C4 [×15], D4 [×2], D4 [×9], Q8 [×4], C23 [×2], C23 [×7], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×5], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×2], SD16 [×3], Q16, C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×3], C24, C4×C8, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×4], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C42.C2, C42.C2, C422C2, C2×M4(2) [×2], C2×SD16 [×2], C2×Q16, C22×D4, C2×C4○D4, C23.36D4, C23.37D4, C86D4, C4×SD16, C22⋊SD16, D4.7D4, Q8.D4, C8⋊D4, C8.D4, D4.Q8, C23.46D4, C23.20D4, C42.78C22, D45D4, C22.46C24, C42.472C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2+ 1+4, D45D4, D8⋊C22, D4○SD16, C42.472C23

Character table of C42.472C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114444822224444488888444488
ρ111111111111111111111111111111    trivial
ρ21111111111111-1111-1-111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-1-111111-11-1-1-11-111-11111-1-1    linear of order 2
ρ41111-1-1-1-11111111-1-11-1-11-11-1-1-1-111    linear of order 2
ρ51111-11111-111-11-11-11-1-1-11-1-11-111-1    linear of order 2
ρ61111-11111-111-1-1-11-1-11-1-1-111-11-1-11    linear of order 2
ρ711111-1-1-11-111-1-1-1-11-1-11-111-11-11-11    linear of order 2
ρ811111-1-1-11-111-11-1-11111-1-1-11-11-11-1    linear of order 2
ρ91111-1-111-1111111-1-11-11-1-111111-1-1    linear of order 2
ρ101111-1-111-11111-11-1-1-111-11-1-1-1-1-111    linear of order 2
ρ11111111-1-1-11111-1111-1-1-1-1-1-1111111    linear of order 2
ρ12111111-1-1-11111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1311111-111-1-111-11-1-1111-11-1-1-11-11-11    linear of order 2
ρ1411111-111-1-111-1-1-1-11-1-1-11111-11-11-1    linear of order 2
ρ151111-11-1-1-1-111-1-1-11-1-1111-11-11-111-1    linear of order 2
ρ161111-11-1-1-1-111-11-11-11-1111-11-11-1-11    linear of order 2
ρ172222-220002-2-220-2-22000000000000    orthogonal lifted from D4
ρ1822222-20002-2-220-22-2000000000000    orthogonal lifted from D4
ρ192222-2-2000-2-2-2-20222000000000000    orthogonal lifted from D4
ρ20222222000-2-2-2-202-2-2000000000000    orthogonal lifted from D4
ρ212-22-200-2200-220-2i0002i000000-2i02i00    complex lifted from C4○D4
ρ222-22-2002-200-2202i000-2i000000-2i02i00    complex lifted from C4○D4
ρ232-22-2002-200-220-2i0002i0000002i0-2i00    complex lifted from C4○D4
ρ242-22-200-2200-2202i000-2i0000002i0-2i00    complex lifted from C4○D4
ρ254-44-40000004-400000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-4400000-4i004i0000000000000000    complex lifted from D8⋊C22
ρ274-4-44000004i00-4i0000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-20-2-2000    complex lifted from D4○SD16

Smallest permutation representation of C42.472C23
On 32 points
Generators in S32
(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)
(1 18 22 25)(2 19 23 26)(3 20 24 27)(4 17 21 28)(5 30 16 10)(6 31 13 11)(7 32 14 12)(8 29 15 9)
(1 10 24 32)(2 29 21 11)(3 12 22 30)(4 31 23 9)(5 20 14 25)(6 26 15 17)(7 18 16 27)(8 28 13 19)
(1 24)(2 4)(3 22)(5 12)(6 29)(7 10)(8 31)(9 13)(11 15)(14 30)(16 32)(17 26)(18 20)(19 28)(21 23)(25 27)
(1 4)(2 3)(5 15)(6 14)(7 13)(8 16)(9 30)(10 29)(11 32)(12 31)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)

G:=sub<Sym(32)| (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), (1,18,22,25)(2,19,23,26)(3,20,24,27)(4,17,21,28)(5,30,16,10)(6,31,13,11)(7,32,14,12)(8,29,15,9), (1,10,24,32)(2,29,21,11)(3,12,22,30)(4,31,23,9)(5,20,14,25)(6,26,15,17)(7,18,16,27)(8,28,13,19), (1,24)(2,4)(3,22)(5,12)(6,29)(7,10)(8,31)(9,13)(11,15)(14,30)(16,32)(17,26)(18,20)(19,28)(21,23)(25,27), (1,4)(2,3)(5,15)(6,14)(7,13)(8,16)(9,30)(10,29)(11,32)(12,31)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)>;

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), (1,18,22,25)(2,19,23,26)(3,20,24,27)(4,17,21,28)(5,30,16,10)(6,31,13,11)(7,32,14,12)(8,29,15,9), (1,10,24,32)(2,29,21,11)(3,12,22,30)(4,31,23,9)(5,20,14,25)(6,26,15,17)(7,18,16,27)(8,28,13,19), (1,24)(2,4)(3,22)(5,12)(6,29)(7,10)(8,31)(9,13)(11,15)(14,30)(16,32)(17,26)(18,20)(19,28)(21,23)(25,27), (1,4)(2,3)(5,15)(6,14)(7,13)(8,16)(9,30)(10,29)(11,32)(12,31)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27) );

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)], [(1,18,22,25),(2,19,23,26),(3,20,24,27),(4,17,21,28),(5,30,16,10),(6,31,13,11),(7,32,14,12),(8,29,15,9)], [(1,10,24,32),(2,29,21,11),(3,12,22,30),(4,31,23,9),(5,20,14,25),(6,26,15,17),(7,18,16,27),(8,28,13,19)], [(1,24),(2,4),(3,22),(5,12),(6,29),(7,10),(8,31),(9,13),(11,15),(14,30),(16,32),(17,26),(18,20),(19,28),(21,23),(25,27)], [(1,4),(2,3),(5,15),(6,14),(7,13),(8,16),(9,30),(10,29),(11,32),(12,31),(17,18),(19,20),(21,22),(23,24),(25,28),(26,27)])

Matrix representation of C42.472C23 in GL6(𝔽17)

010000
1600000
004900
0041300
0000138
0000134
,
100000
010000
0011500
0011600
0000162
0000161
,
400000
0130000
000010
000001
0016000
0001600
,
100000
010000
0016000
0016100
0000115
0000016
,
010000
100000
004900
0041300
000049
0000413

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,9,13,0,0,0,0,0,0,13,13,0,0,0,0,8,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,9,13,0,0,0,0,0,0,4,4,0,0,0,0,9,13] >;

C42.472C23 in GAP, Magma, Sage, TeX

C_4^2._{472}C_2^3
% in TeX

G:=Group("C4^2.472C2^3");
// GroupNames label

G:=SmallGroup(128,2055);
// by ID

G=gap.SmallGroup(128,2055);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,2019,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.472C23 in TeX

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