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G = C42.12C4order 64 = 26

9th non-split extension by C42 of C4 acting via C4/C2=C2

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

Aliases: C42.12C4, C4.12M4(2), C42.92C22, (C2×C4)⋊4C8, (C4×C8)⋊2C2, C42(C4⋊C8), C4⋊C817C2, C4.9(C2×C8), C42(C4⋊C8), C42(C22⋊C8), C22⋊C8.9C2, C2.3(C22×C8), C22.5(C2×C8), C4.48(C4○D4), C42(C22⋊C8), (C22×C4).16C4, (C2×C8).61C22, (C2×C42).15C2, C23.32(C2×C4), C2.5(C2×M4(2)), (C2×C4).150C23, C2.4(C42⋊C2), C22.22(C22×C4), (C22×C4).93C22, (C2×C4).57(C2×C4), SmallGroup(64,112)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C42.12C4
C1C2C4C2×C4C42C2×C42 — C42.12C4
C1C2 — C42.12C4
C1C42 — C42.12C4
C1C2C2C2×C4 — C42.12C4

Generators and relations for C42.12C4
 G = < a,b,c | a4=b4=1, c4=a2, ab=ba, cac-1=a-1b2, bc=cb >

Subgroups: 73 in 59 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×4], C22×C4 [×3], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C42.12C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], C42⋊C2, C22×C8, C2×M4(2), C42.12C4

Smallest permutation representation of C42.12C4
On 32 points
Generators in S32
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 24 13 20)(10 12 14 16)(11 18 15 22)(17 19 21 23)(25 27 29 31)
(1 21 27 12)(2 22 28 13)(3 23 29 14)(4 24 30 15)(5 17 31 16)(6 18 32 9)(7 19 25 10)(8 20 26 11)
(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)

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

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

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

C42.12C4 is a maximal subgroup of
C89M4(2)  C23.27C42  C8215C2  C822C2  C42.677C23  C42.260C23  C42.262C23  C42.678C23  C42.290C23  C42.293C23  C42.294C23  D47M4(2)  C42.693C23  C42.694C23  C42.301C23  C42.302C23  Q8.4M4(2)  C42.696C23  C42.304C23  C42.698C23  D48M4(2)  Q87M4(2)  C42.308C23  C42.309C23
 C42.D2p: C421C8  C42.20D4  C426C8  M4(2)⋊C8  C42.2Q8  C42.3Q8  C42.5Q8  C42.23D4 ...
 C2p.(C22×C8): C8×M4(2)  C82⋊C2  C8×C4○D4  C42.691C23  C42.695C23  C42.697C23  Dic3.5M4(2)  Dic5.14M4(2) ...
C42.12C4 is a maximal quotient of
C424C8  C23.32M4(2)  C425C8  C42.13C8  C42.6C8  C8.12M4(2)  C42.6F5  C42.12F5  Dic5.12M4(2)  C20.34M4(2)
 C42.D2p: C4×C22⋊C8  C4×C4⋊C8  C42.425D4  C428C8  C42.282D6  C42.200D6  C42.285D6  C42.282D10 ...
 (C2×C8).D2p: C4⋊C43C8  C22⋊C44C8  Dic3.5M4(2)  Dic5.14M4(2)  Dic7.5M4(2) ...

40 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P
order1222224···44···48···8
size1111221···12···22···2

40 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C2C4C4C8M4(2)C4○D4
kernelC42.12C4C4×C8C22⋊C8C4⋊C8C2×C42C42C22×C4C2×C4C4C4
# reps12221441644

Matrix representation of C42.12C4 in GL3(𝔽17) generated by

1300
0160
041
,
1300
0130
0013
,
1500
01315
004
G:=sub<GL(3,GF(17))| [13,0,0,0,16,4,0,0,1],[13,0,0,0,13,0,0,0,13],[15,0,0,0,13,0,0,15,4] >;

C42.12C4 in GAP, Magma, Sage, TeX

C_4^2._{12}C_4
% in TeX

G:=Group("C4^2.12C4");
// GroupNames label

G:=SmallGroup(64,112);
// by ID

G=gap.SmallGroup(64,112);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,50,88]);
// Polycyclic

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

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