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G = C42.263D4order 128 = 27

245th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.263D4, C42.395C23, C4.972+ 1+4, (C2×C4)⋊4D8, C4⋊D87C2, C84D48C2, (C4×C8)⋊9C22, C4.72(C2×D8), C22⋊D86C2, C4⋊C860C22, (C2×D8)⋊5C22, C4⋊Q869C22, C4.4D811C2, (C4×D4)⋊11C22, C2.15(C22×D8), C22.24(C2×D8), D4⋊C43C22, C4⋊C4.144C23, C41D440C22, C4.25(C8⋊C22), (C2×C4).403C24, (C2×C8).158C23, (C22×C4).493D4, C23.688(C2×D4), (C2×D4).153C23, C42.12C426C2, C4⋊D4.186C22, C22⋊C8.177C22, (C2×C42).870C22, C22.663(C22×D4), (C22×C4).1074C23, C22.26C2416C2, (C22×D4).386C22, C2.74(C22.29C24), (C2×C41D4)⋊20C2, (C2×C4).864(C2×D4), C2.53(C2×C8⋊C22), SmallGroup(128,1937)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.263D4
C1C2C4C2×C4C22×C4C22×D4C2×C41D4 — C42.263D4
C1C2C2×C4 — C42.263D4
C1C22C2×C42 — C42.263D4
C1C2C2C2×C4 — C42.263D4

Generators and relations for C42.263D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=b2c3 >

Subgroups: 700 in 266 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×8], C4 [×8], C4 [×4], C22, C22 [×2], C22 [×28], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×10], D4 [×40], Q8 [×2], C23, C23 [×18], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], D8 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×6], C2×D4 [×26], C2×Q8, C4○D4 [×4], C24 [×2], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C41D4 [×4], C41D4 [×2], C4⋊Q8, C2×D8 [×8], C22×D4 [×2], C22×D4 [×2], C2×C4○D4, C42.12C4, C22⋊D8 [×4], C4⋊D8 [×4], C4.4D8 [×2], C84D4 [×2], C2×C41D4, C22.26C24, C42.263D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C8⋊C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C22×D8, C2×C8⋊C22, C42.263D4

Smallest permutation representation of C42.263D4
On 32 points
Generators in S32
(1 14 26 23)(2 15 27 24)(3 16 28 17)(4 9 29 18)(5 10 30 19)(6 11 31 20)(7 12 32 21)(8 13 25 22)
(1 7 5 3)(2 29 6 25)(4 31 8 27)(9 20 13 24)(10 16 14 12)(11 22 15 18)(17 23 21 19)(26 32 30 28)
(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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A···4H4I4J4K4L8A···8H
order1222222···24···444448···8
size1111228···82···244884···4

32 irreducible representations

dim1111111122244
type+++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D8C8⋊C222+ 1+4
kernelC42.263D4C42.12C4C22⋊D8C4⋊D8C4.4D8C84D4C2×C41D4C22.26C24C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.263D4 in GL6(𝔽17)

120000
16160000
00161500
001100
0024162
0002161
,
120000
16160000
001000
000100
00150160
001515016
,
660000
1400000
001615115
0001161
0000162
000001
,
060000
300000
00160160
00016116
000010
000001

G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,16,1,2,0,0,0,15,1,4,2,0,0,0,0,16,16,0,0,0,0,2,1],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,1,0,15,15,0,0,0,1,0,15,0,0,0,0,16,0,0,0,0,0,0,16],[6,14,0,0,0,0,6,0,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,1,16,16,0,0,0,15,1,2,1],[0,3,0,0,0,0,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,16,1,1,0,0,0,0,16,0,1] >;

C42.263D4 in GAP, Magma, Sage, TeX

C_4^2._{263}D_4
% in TeX

G:=Group("C4^2.263D4");
// GroupNames label

G:=SmallGroup(128,1937);
// by ID

G=gap.SmallGroup(128,1937);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,675,4037,1027,124]);
// Polycyclic

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

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