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G = C24.117D4order 128 = 27

72nd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.117D4, C4.Q89C22, (C2×C8).34C23, C2.D820C22, C4⋊C4.392C23, C22⋊C811C22, C24.4C49C2, (C2×C4).292C24, (C2×D4).81C23, C23.243(C2×D4), (C22×C4).443D4, D4⋊C420C22, C22.D812C2, C23.46D42C2, M4(2)⋊C424C2, C42⋊C279C22, C4⋊D4.157C22, C23.37D410C2, C23.19D413C2, C22.46(C8⋊C22), (C23×C4).562C22, C22.552(C22×D4), C2.23(D8⋊C22), (C22×C4).1008C23, C4.99(C22.D4), (C22×D4).359C22, (C2×M4(2)).74C22, C22.64(C22.D4), C4.102(C2×C4○D4), (C2×C4).487(C2×D4), C2.29(C2×C8⋊C22), (C2×C4⋊C4)⋊118C22, (C2×C4⋊D4).58C2, (C2×C42⋊C2)⋊47C2, (C2×C4).487(C4○D4), C2.57(C2×C22.D4), SmallGroup(128,1826)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.117D4
C1C2C4C2×C4C22×C4C42⋊C2C2×C42⋊C2 — C24.117D4
C1C2C2×C4 — C24.117D4
C1C22C23×C4 — C24.117D4
C1C2C2C2×C4 — C24.117D4

Generators and relations for C24.117D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >

Subgroups: 492 in 231 conjugacy classes, 94 normal (28 characteristic)
C1, C2 [×3], C2 [×7], C4 [×4], C4 [×8], C22, C22 [×4], C22 [×21], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×22], D4 [×14], C23 [×3], C23 [×13], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C24, C24, C22⋊C8 [×4], D4⋊C4 [×8], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C42⋊C2 [×4], C42⋊C2 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C2×M4(2) [×2], C23×C4, C22×D4, C22×D4, C24.4C4, C23.37D4 [×2], M4(2)⋊C4 [×2], C22.D8 [×2], C23.46D4 [×2], C23.19D4 [×4], C2×C42⋊C2, C2×C4⋊D4, C24.117D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C2×C22.D4, C2×C8⋊C22, D8⋊C22, C24.117D4

Smallest permutation representation of C24.117D4
On 32 points
Generators in S32
(1 10)(2 31)(3 12)(4 25)(5 14)(6 27)(7 16)(8 29)(9 22)(11 24)(13 18)(15 20)(17 32)(19 26)(21 28)(23 30)
(1 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(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)
(2 22)(3 7)(4 20)(6 18)(8 24)(9 11)(10 30)(12 28)(13 15)(14 26)(16 32)(17 21)(25 27)(29 31)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4F4G···4O4P4Q8A8B8C8D
order122222222224···44···4448888
size111122224882···24···4888888

32 irreducible representations

dim11111111122244
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22D8⋊C22
kernelC24.117D4C24.4C4C23.37D4M4(2)⋊C4C22.D8C23.46D4C23.19D4C2×C42⋊C2C2×C4⋊D4C22×C4C24C2×C4C22C2
# reps11222241131822

Matrix representation of C24.117D4 in GL6(𝔽17)

13150000
1640000
0013400
009400
0000134
000094
,
100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
400000
1130000
0000160
0000151
0016100
000100
,
100000
13160000
0011600
0001600
0000160
0000151

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

C24.117D4 in GAP, Magma, Sage, TeX

C_2^4._{117}D_4
% in TeX

G:=Group("C2^4.117D4");
// GroupNames label

G:=SmallGroup(128,1826);
// by ID

G=gap.SmallGroup(128,1826);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,2019,248,4037,1027,124]);
// 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=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*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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