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G = C2410D4order 128 = 27

5th semidirect product of C24 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2410D4, C25.51C22, C23.517C24, C24.586C23, C22.2952+ 1+4, (C22×C4)⋊33D4, C232D424C2, C23.190(C2×D4), (C22×D4)⋊9C22, C23.7Q876C2, C23.238(C4○D4), C23.23D466C2, C23.10D454C2, C23.11D455C2, C22.29(C4⋊D4), C2.22(C233D4), (C23×C4).420C22, (C22×C4).127C23, C22.342(C22×D4), C2.C4229C22, C2.33(C22.29C24), C2.36(C22.32C24), (C2×C4⋊D4)⋊22C2, (C2×C4⋊C4)⋊25C22, (C2×C4).377(C2×D4), C2.41(C2×C4⋊D4), (C2×C22≀C2)⋊10C2, (C22×C22⋊C4)⋊25C2, (C2×C22⋊C4)⋊23C22, C22.389(C2×C4○D4), SmallGroup(128,1349)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2410D4
C1C2C22C23C24C22×D4C2×C22≀C2 — C2410D4
C1C23 — C2410D4
C1C23 — C2410D4
C1C23 — C2410D4

Generators and relations for C2410D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=acd, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 996 in 412 conjugacy classes, 108 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×12], C22 [×3], C22 [×8], C22 [×62], C2×C4 [×4], C2×C4 [×36], D4 [×28], C23, C23 [×10], C23 [×62], C22⋊C4 [×28], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C22×C4 [×6], C2×D4 [×32], C24, C24 [×8], C24 [×8], C2.C42 [×6], C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×2], C22≀C2 [×8], C4⋊D4 [×4], C23×C4 [×2], C22×D4, C22×D4 [×6], C25, C23.7Q8, C23.23D4 [×2], C232D4 [×2], C23.10D4 [×4], C23.11D4 [×2], C22×C22⋊C4, C2×C22≀C2 [×2], C2×C4⋊D4, C2410D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, 2+ 1+4 [×4], C2×C4⋊D4, C233D4 [×3], C22.29C24, C22.32C24 [×2], C2410D4

Smallest permutation representation of C2410D4
On 32 points
Generators in S32
(1 25)(2 14)(3 27)(4 16)(5 7)(6 12)(8 10)(9 11)(13 23)(15 21)(17 32)(18 20)(19 30)(22 28)(24 26)(29 31)
(1 27)(2 28)(3 25)(4 26)(5 29)(6 30)(7 31)(8 32)(9 20)(10 17)(11 18)(12 19)(13 21)(14 22)(15 23)(16 24)
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)
(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 10)(2 9)(3 12)(4 11)(5 24)(6 23)(7 22)(8 21)(13 19)(14 18)(15 17)(16 20)(25 32)(26 31)(27 30)(28 29)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O2P2Q4A···4H4I···4N
order12···222222222224···44···4
size11···122224444884···48···8

32 irreducible representations

dim1111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC2410D4C23.7Q8C23.23D4C232D4C23.10D4C23.11D4C22×C22⋊C4C2×C22≀C2C2×C4⋊D4C22×C4C24C23C22
# reps1122421214444

Matrix representation of C2410D4 in GL8(𝔽5)

10000000
04000000
00400000
00040000
00001000
00000400
00000040
00000001
,
10000000
01000000
00400000
00040000
00004000
00000100
00000040
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00300000
00020000
00000010
00000001
00004000
00000400
,
01000000
10000000
00020000
00300000
00000001
00000010
00000100
00001000

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C2410D4 in GAP, Magma, Sage, TeX

C_2^4\rtimes_{10}D_4
% in TeX

G:=Group("C2^4:10D4");
// GroupNames label

G:=SmallGroup(128,1349);
// by ID

G=gap.SmallGroup(128,1349);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,185]);
// Polycyclic

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

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