Copied to
clipboard

G = C42.225D4order 128 = 27

207th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.225D4, C42.701C23, (C4×D8)⋊20C2, C4⋊C866C22, (C4×C8)⋊14C22, (C4×SD16)⋊1C2, (C4×D4)⋊6C22, C43(D4.Q8), D4.Q855C2, C43(C22⋊D8), (C4×Q8)⋊6C22, D4.3(C4○D4), C22⋊D8.5C2, C4⋊C4.58C23, C2.D857C22, C4.Q866C22, C43(D4.2D4), C43(C22⋊SD16), C22⋊SD1637C2, D4.2D454C2, (C2×C4).303C24, (C2×C8).148C23, (C2×D4).87C23, C23.670(C2×D4), (C22×C4).720D4, (C2×Q8).74C23, D4⋊C479C22, C4.148(C8⋊C22), Q8⋊C470C22, (C2×D8).123C22, C22.29(C4○D8), C4.4D454C22, C42.C231C22, C42.12C429C2, C43(C23.46D4), C43(C23.48D4), C23.46D437C2, C4⋊D4.161C22, C23.48D438C2, C22⋊C8.175C22, (C2×C42).830C22, C22.563(C22×D4), C22⋊Q8.166C22, C23.36C232C2, (C22×C4).1019C23, (C2×SD16).139C22, (C22×D4).574C22, C2.104(C22.19C24), (C2×C4×D4)⋊63C2, C2.24(C2×C4○D8), (C2×C4)(D4.Q8), C4.188(C2×C4○D4), (C2×C4).494(C2×D4), C2.30(C2×C8⋊C22), (C2×C4⋊C4).934C22, (C2×C4)(C23.48D4), SmallGroup(128,1837)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.225D4
C1C2C4C2×C4C42C4×D4C2×C4×D4 — C42.225D4
C1C2C2×C4 — C42.225D4
C1C2×C4C2×C42 — C42.225D4
C1C2C2C2×C4 — C42.225D4

Generators and relations for C42.225D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, bc=cb, dbd=a2b, dcd=a2c3 >

Subgroups: 460 in 227 conjugacy classes, 92 normal (42 characteristic)
C1, C2 [×3], C2 [×7], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×21], C8 [×4], C2×C4 [×6], C2×C4 [×23], D4 [×4], D4 [×10], Q8 [×2], C23, C23 [×11], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×10], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C24, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×4], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×D8 [×2], C2×SD16 [×2], C23×C4, C22×D4, C42.12C4, C4×D8 [×2], C4×SD16 [×2], C22⋊D8, C22⋊SD16, D4.2D4 [×2], D4.Q8 [×2], C23.46D4, C23.48D4, C2×C4×D4, C23.36C23, C42.225D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×2], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C4○D8, C2×C8⋊C22, C42.225D4

Smallest permutation representation of C42.225D4
On 32 points
Generators in S32
(1 29 5 25)(2 8 6 4)(3 31 7 27)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(26 32 30 28)
(1 15 31 17)(2 16 32 18)(3 9 25 19)(4 10 26 20)(5 11 27 21)(6 12 28 22)(7 13 29 23)(8 14 30 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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 32)(8 31)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E···4J4K···4P4Q4R4S8A···8H
order1222222222244444···44···44448···8
size1111224444811112···24···48884···4

38 irreducible representations

dim11111111111122224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D8C8⋊C22
kernelC42.225D4C42.12C4C4×D8C4×SD16C22⋊D8C22⋊SD16D4.2D4D4.Q8C23.46D4C23.48D4C2×C4×D4C23.36C23C42C22×C4D4C22C4
# reps11221122111122882

Matrix representation of C42.225D4 in GL4(𝔽17) generated by

16000
16100
00115
00116
,
4000
0400
0049
00413
,
16200
0100
00011
00311
,
11500
01600
00011
00140
G:=sub<GL(4,GF(17))| [16,16,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,4,0,0,0,0,4,4,0,0,9,13],[16,0,0,0,2,1,0,0,0,0,0,3,0,0,11,11],[1,0,0,0,15,16,0,0,0,0,0,14,0,0,11,0] >;

C42.225D4 in GAP, Magma, Sage, TeX

C_4^2._{225}D_4
% in TeX

G:=Group("C4^2.225D4");
// GroupNames label

G:=SmallGroup(128,1837);
// by ID

G=gap.SmallGroup(128,1837);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,304,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽