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G = C23.25D4order 64 = 26

4th non-split extension by C23 of D4 acting via D4/C4=C2

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

Aliases: C23.25D4, (C2×C8)⋊6C4, C4(C4.Q8), C4(C2.D8), C4.3(C2×Q8), C8.17(C2×C4), C4.Q813C2, C2.D813C2, C4.22(C4⋊C4), (C2×C4).20Q8, C2.2(C4○D8), (C2×C4).147D4, C4⋊C4.48C22, C22.9(C4⋊C4), (C2×C8).74C22, (C2×C4).69C23, (C22×C8).11C2, C4.26(C22×C4), C22.49(C2×D4), C42⋊C2.6C2, (C22×C4).115C22, C2.13(C2×C4⋊C4), (C2×C4)(C2.D8), (C2×C4)(C4.Q8), (C2×C4).74(C2×C4), SmallGroup(64,108)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C23.25D4
C1C2C22C2×C4C22×C4C22×C8 — C23.25D4
C1C2C4 — C23.25D4
C1C2×C4C22×C4 — C23.25D4
C1C2C2C2×C4 — C23.25D4

Generators and relations for C23.25D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 81 in 57 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×4], C22×C4, C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22×C8, C23.25D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C4○D8 [×2], C23.25D4

Character table of C23.25D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F8G8H
 size 1111221111224444444422222222
ρ11111111111111111111111111111    trivial
ρ21111111111111-11-11-11-1-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111111-11-11-11-11-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ51111-1-1-1-1-1-11111-1-1-111-1-1-11-11-111    linear of order 2
ρ61111-1-1-1-1-1-1111-1-11-1-11111-11-11-1-1    linear of order 2
ρ71111-1-1-1-1-1-111-1-1111-1-11-1-11-11-111    linear of order 2
ρ81111-1-1-1-1-1-111-111-111-1-111-11-11-1-1    linear of order 2
ρ91-1-111-11-11-11-1-ii-i-ii-iii1-1-1-1111-1    linear of order 4
ρ101-1-11-11-11-111-1-iiii-i-ii-i-11-111-11-1    linear of order 4
ρ111-1-11-11-11-111-1ii-iii-i-i-i1-11-1-11-11    linear of order 4
ρ121-1-111-11-11-11-1iii-i-i-i-ii-1111-1-1-11    linear of order 4
ρ131-1-11-11-11-111-1-i-ii-i-iiii1-11-1-11-11    linear of order 4
ρ141-1-111-11-11-11-1-i-i-iiiii-i-1111-1-1-11    linear of order 4
ρ151-1-111-11-11-11-1i-iii-ii-i-i1-1-1-1111-1    linear of order 4
ρ161-1-11-11-11-111-1i-i-i-iii-ii-11-111-11-1    linear of order 4
ρ172222-2-22222-2-20000000000000000    orthogonal lifted from D4
ρ18222222-2-2-2-2-2-20000000000000000    orthogonal lifted from D4
ρ192-2-222-2-22-22-220000000000000000    symplectic lifted from Q8, Schur index 2
ρ202-2-22-222-22-2-220000000000000000    symplectic lifted from Q8, Schur index 2
ρ2122-2-2002i2i-2i-2i0000000000-22-2-2--22-2--2    complex lifted from C4○D8
ρ222-22-2002i-2i-2i2i000000000022-2-2-2-2--2--2    complex lifted from C4○D8
ρ232-22-200-2i2i2i-2i0000000000-2-2-22-22--2--2    complex lifted from C4○D8
ρ2422-2-200-2i-2i2i2i00000000002-2-22--2-2-2--2    complex lifted from C4○D8
ρ2522-2-2002i2i-2i-2i00000000002-2--22-2-2--2-2    complex lifted from C4○D8
ρ262-22-2002i-2i-2i2i0000000000-2-2--22--22-2-2    complex lifted from C4○D8
ρ272-22-200-2i2i2i-2i000000000022--2-2--2-2-2-2    complex lifted from C4○D8
ρ2822-2-200-2i-2i2i2i0000000000-22--2-2-22--2-2    complex lifted from C4○D8

Smallest permutation representation of C23.25D4
On 32 points
Generators in S32
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 14 23 27)(2 9 24 30)(3 12 17 25)(4 15 18 28)(5 10 19 31)(6 13 20 26)(7 16 21 29)(8 11 22 32)

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

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

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

C23.25D4 is a maximal subgroup of
(C2×D4).24Q8  C8○D4⋊C4  M4(2).3Q8  M4(2).30D4  C24.100D4  C4○D4.7Q8  C4○D4.8Q8  C4×C4○D8  C42.383D4  C24.144D4  C24.110D4  (C2×C8)⋊11D4  (C2×C8)⋊12D4  C8.D4⋊C2  C42.447D4  C24.115D4  (C2×D4).301D4  (C2×D4).302D4  C42.364D4  C42.252D4
 C23.D4p: C23.9D8  C8.C42  C23.24D8  C23.40D8  C23.41D8  C23.13D8  C23.25D8  M5(2)⋊1C4 ...
 C4p.(C4⋊C4): C8.9C42  C8.4C42  C8.14C42  C8.5C42  C8.(C4⋊C4)  C42.62Q8  C42.28Q8  M5(2)⋊3C4 ...
 C4⋊C4.D2p: C42.277C23  C42.278C23  C42.279C23  C42.20C23  C42.21C23  M4(2)⋊3Q8  M4(2)⋊4Q8  C42.485C23 ...
C23.25D4 is a maximal quotient of
C42.42Q8  C42.43Q8  C42.Q8  C24.132D4  C4×C4.Q8  C4×C2.D8  C24.133D4  C42.56Q8
 C23.D4p: C23.22D8  C23.27D12  C23.22D20  C23.22D28 ...
 C4p.(C4⋊C4): C42.60Q8  C4⋊C4.234D6  C20.76(C4⋊C4)  (C2×C8)⋊6F5  C28.45(C4⋊C4) ...
 C4⋊C4.D2p: C24.71D4  C42.31Q8  (S3×C8)⋊C4  C8.27(C4×S3)  (C8×D5)⋊C4  C8.27(C4×D5)  (C8×D7)⋊C4  C8.27(C4×D7) ...

Matrix representation of C23.25D4 in GL3(𝔽17) generated by

1600
010
0416
,
1600
0160
0016
,
100
0160
0016
,
1600
020
058
,
400
0151
0142
G:=sub<GL(3,GF(17))| [16,0,0,0,1,4,0,0,16],[16,0,0,0,16,0,0,0,16],[1,0,0,0,16,0,0,0,16],[16,0,0,0,2,5,0,0,8],[4,0,0,0,15,14,0,1,2] >;

C23.25D4 in GAP, Magma, Sage, TeX

C_2^3._{25}D_4
% in TeX

G:=Group("C2^3.25D4");
// GroupNames label

G:=SmallGroup(64,108);
// by ID

G=gap.SmallGroup(64,108);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,158,963,117]);
// Polycyclic

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

Export

Character table of C23.25D4 in TeX

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