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G = C24.116D4order 128 = 27

71st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.116D4, C4.Q88C22, (C2×C8).33C23, C2.D819C22, C4⋊C4.391C23, (C2×C4).291C24, C24.4C48C2, (C2×D4).80C23, C23.242(C2×D4), (C22×C4).442D4, (C2×Q8).68C23, D4⋊C419C22, Q8⋊C421C22, C22⋊C8.15C22, M4(2)⋊C423C2, C23.36D410C2, C23.19D412C2, C23.20D412C2, C4⋊D4.156C22, (C23×C4).561C22, C22.551(C22×D4), C22⋊Q8.161C22, C2.22(D8⋊C22), (C22×C4).1007C23, C22.19C24.19C2, (C2×M4(2)).73C22, C42⋊C2.317C22, C4.127(C22.D4), C22.41(C22.D4), C4.101(C2×C4○D4), (C2×C4).1218(C2×D4), (C2×C42⋊C2)⋊46C2, (C2×C4).486(C4○D4), (C2×C4⋊C4).928C22, (C2×C4○D4).138C22, C2.56(C2×C22.D4), SmallGroup(128,1825)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.116D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C42⋊C2 — C24.116D4
C1C2C2×C4 — C24.116D4
C1C22C23×C4 — C24.116D4
C1C2C2C2×C4 — C24.116D4

Generators and relations for C24.116D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=ac=ca, ad=da, faf=acd, bc=cb, ebe-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >

Subgroups: 388 in 207 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×23], D4 [×8], Q8 [×2], C23, C23 [×2], C23 [×5], C42 [×5], C22⋊C4 [×9], C4⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×4], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C23×C4, C2×C4○D4, C24.4C4, C23.36D4 [×2], M4(2)⋊C4 [×2], C23.19D4 [×4], C23.20D4 [×4], C2×C42⋊C2, C22.19C24, C24.116D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4, D8⋊C22 [×2], C24.116D4

Smallest permutation representation of C24.116D4
On 32 points
Generators in S32
(2 31)(4 25)(6 27)(8 29)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)
(1 30)(2 27)(3 32)(4 29)(5 26)(6 31)(7 28)(8 25)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 22)(2 16)(3 20)(4 14)(5 18)(6 12)(7 24)(8 10)(9 30)(11 28)(13 26)(15 32)(17 27)(19 25)(21 31)(23 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I···4P4Q4R4S8A8B8C8D
order1222222224···44···44448888
size1111224482···24···48888888

32 irreducible representations

dim111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4D8⋊C22
kernelC24.116D4C24.4C4C23.36D4M4(2)⋊C4C23.19D4C23.20D4C2×C42⋊C2C22.19C24C22×C4C24C2×C4C2
# reps112244113184

Matrix representation of C24.116D4 in GL6(𝔽17)

100000
0160000
001000
0001601
00001616
000001
,
100000
010000
001001
0001016
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
040000
400000
0040010
0013017
0041304
0090013
,
0160000
1600000
000110
0001600
001100
0001501

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,1,16,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,16,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,13,4,9,0,0,0,0,13,0,0,0,0,1,0,0,0,0,10,7,4,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,1,16,1,15,0,0,1,0,0,0,0,0,0,0,0,1] >;

C24.116D4 in GAP, Magma, Sage, TeX

C_2^4._{116}D_4
% in TeX

G:=Group("C2^4.116D4");
// GroupNames label

G:=SmallGroup(128,1825);
// by ID

G=gap.SmallGroup(128,1825);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,100,2019,4037,1027,124]);
// Polycyclic

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

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